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Construction of block designs admitting an Abelian automorphism group

2005-10-10
Dean Crnković , Sanja Rukavina

Self-orthogonal codes from the strongly regular graphs on up to 40 vertices

2016-08-01
Dean Crnković , Marija Maksimović , Bernardo Rodrigues +1 more
This paper outlines a method for constructing self-orthogonal codes from orbit matrices of strongly regular graphs admitting an automorphism group $G$ which acts with orbits of length $w$, where $w$ … This paper outlines a method for constructing self-orthogonal codes from orbit matrices of strongly regular graphs admitting an automorphism group $G$ which acts with orbits of length $w$, where $w$ divides $|G|$.We apply this method to construct self-orthogonal codes from orbit matrices of the strongly regular graphs with at most 40 vertices. In particular, we construct codes from adjacency or orbit matrices of graphs with parameters $(36, 15, 6, 6)$, $(36, 14, 4, 6)$,$(35, 16, 6, 8)$ and their complements, and from the graphs with parameters $(40, 12, 2, 4)$ and their complements.That completes the classification of self-orthogonal codes spanned by the adjacency matrices or orbit matrices of thestrongly regular graphs with at most 40 vertices.Furthermore, we construct ternary codes of $2$-$(27,9,4)$ designs obtained as residual designs of the symmetric $(40, 13, 4)$ designs (complementary designs of the symmetric $(40, 27, 18)$ designs), and their ternary hulls. Some of the obtained codes are optimal, and some are best known for the given length and dimension.
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Self-orthogonal codes from orbit matrices of 2-designs

2013-01-01
Dean Crnković , Bernardo Rodrigues , Sanja Rukavina +1 more
In this paper we present a method for constructing self-orthogonal codes from orbit matrices of $2$-designs that admit an automorphism group $G$ which acts with orbit lengths $1$ and $w$, … In this paper we present a method for constructing self-orthogonal codes from orbit matrices of $2$-designs that admit an automorphism group $G$ which acts with orbit lengths $1$ and $w$, where $w$ divides $|G|$. This is a generalization of an earlier method proposed by Tonchev for constructing self-orthogonal codes from orbit matrices of $2$-designs with a fixed-point-free automorphism of prime order. As an illustration of our method we provide a classification of self-orthogonal codes obtained from the non-fixed parts of the orbit matrices of the symmetric $2$-$(56,11,2)$ designs, some symmetric designs $2$-$(71,15,3)$ (and their residual designs), and some non-symmetric $2$-designs, namely those with parameters $2$-$(15,3,1)$, $2$-$(25,4,1)$, $2$-$(37,4,1)$, and $2$-$(45,5,1)$, respectively with automorphisms of order $p$, where $p$ is an odd prime. We establish that the codes with parameters $[10,4,6]_3$ and $[11,4,6]_3$are optimal two-weight codes. Further, we construct an optimal binary self-orthogonal $[16,5,8]$ code from the non-fixed part of the orbit matrix of the $2$-$(64,8,1)$ design with respect to an automorphism group of order four.

A classification of all symmetric block designs of order nine with an automorphism of order six

2005-10-21
Dean Crnković , Sanja Rukavina , Marcel Schmidt
Abstract We complete the classification of all symmetric designs of order nine admitting an automorphism of order six. As a matter of fact, the classification for the parameters (35,17,8), (56,11,2), … Abstract We complete the classification of all symmetric designs of order nine admitting an automorphism of order six. As a matter of fact, the classification for the parameters (35,17,8), (56,11,2), and (91,10,1) had already been done, and in this paper we present the results for the parameters (36,15,6), (40,13,4), and (45,12,3). We also provide information about the order and the structure of the full automorphism groups of the constructed designs. © 2005 Wiley Periodicals, Inc. J Combin Designs 14: 301–312, 2006

New strongly regular graphs from orthogonal groups <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml142" display="inline" overflow="scroll" altimg="si142.gif"><mml:msup><mml:mrow><mml:mi>O</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mn>6</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml143" display="inline" overflow="…

2018-07-14
Dean Crnković , Sanja Rukavina , Andrea Švob

Ternary codes from the strongly regular (45, 12, 3, 3) graphs and orbit matrices of 2-(45, 12, 3) designs

2012-07-20
Dean Crnković , Bernardo Rodrigues , Sanja Rukavina +1 more

Self-orthogonal codes from equitable partitions of association schemes

2022-01-15
Dean Crnković , Sanja Rukavina , Andrea Švob
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Self-dual codes from extended orbit matrices of symmetric designs

2015-01-27
Dean Crnković , Sanja Rukavina

On triplanes of order twelve admitting an automorphism of order six and their binary and ternary codes

2017-01-01
Dean Crnković , Sanja Rukavina , Loredana Simčić
In this paper we present the complete classification of triplanes (71, 15, 3) admitting an action of the cyclic automorphism group of order six. Up to isomorphism there are 146 … In this paper we present the complete classification of triplanes (71, 15, 3) admitting an action of the cyclic automorphism group of order six. Up to isomorphism there are 146 such triplanes and these are all triplanes of order 12 known up to now. Further, we analyze binary and ternary codes spanned by the incidence matrices of triplanes (71, 15, 3) and their residual designs. The constructed binary codes are self- complementary, and the ternary codes are self- orthogonal. In addition, we study ternary self-orthogonal codes constructed from the orbit matrices for Z_3 acting on the 146 symmetric 2-(71, 15, 3) designs. Some of the obtained codes have minimum distance one or two less than the best known codes with the same length and dimension. Finally, we discuss k-geodetic graphs from the symmetric (71, 15, 3) designs and their residual and derived designs.

On some distance-regular graphs with many vertices

2019-05-18
Dean Crnković , Sanja Rukavina , Andrea Švob
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Extremal ternary self-dual codes of length 36 and symmetric 2-(36, 15, 6) designs with an automorphism of order 2

2022-12-29
Sanja Rukavina , Vladimir D. Tonchev
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Self-dual codes from quotient matrices of symmetric divisible designs with the dual property

2015-10-01
Dean Crnković , Nina Mostarac , Sanja Rukavina

Quasi-symmetric 2-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml22" display="inline" overflow="scroll" altimg="si2.gif"><mml:mrow><mml:mo>(</mml:mo><mml:mn>64</mml:mn><mml:mo>,</mml:mo><mml:mn>24</mml:mn><mml:mo>,</mml:mo><mml:mn>46</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math> designs derived from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml23" display="inline" overflow="scroll" altimg="si23.gif"><mml:mi>A</mml:mi><mml:mi>G</mml:mi><mml:mrow><mml:mo>(</…

2017-07-05
Dean Crnković , Bernardo Rodrigues , Sanja Rukavina +1 more

On the 2-Y-Homogeneous Condition of the Incidence Graphs of 2-Designs

2022-07-12
Blas Fernández , Sanja Rukavina

Construction of extremal $ \mathbb{Z}_{4} $-codes using a neighborhood search algorithm

2023-10-17
Dean Crnković , Matteo Mravić , Sanja Rukavina
In this paper, we present a method for constructing extremal $ \mathbb{Z}_{4} $-codes based on random neighborhood search. This method is used to find new extremal Type Ⅰ and Type … In this paper, we present a method for constructing extremal $ \mathbb{Z}_{4} $-codes based on random neighborhood search. This method is used to find new extremal Type Ⅰ and Type Ⅱ $ \mathbb{Z}_{4} $-codes of lengths 32 and 40. For the length 32, at least 182 new Type Ⅱ extremal $ \mathbb{Z}_{4} $-codes of types $ 4^{k}2^{32-2k} $, $ k\in\left\{9,10,12,13,14,15,16\right\} $ are constructed. In addition, we obtained at least 762 new extremal Type Ⅰ $ \mathbb{Z}_{4} $-codes of types $ 4^{k}2^{32-2k} $, $ k\in\left\{7,9,10,12,13,14,15,16\right\} $. For the length 40, constructed extremal $ \mathbb{Z}_{4} $-codes are of types $ 4^{k}2^{40-2k} $, $ k\in\left\{7,10,11,15,16\right\} $. There are at least 40 new Type Ⅱ extremal $ \mathbb{Z}_{4} $-codes, and at least 4144 new Type Ⅰ extremal $ \mathbb{Z}_{4} $-codes.
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Binary doubly-even self-dual codes of length 72 with large automorphism groups ∗

2013-04-28
Dean Crnković , Sanja Rukavina , Loredana Simčić
We study binary linear codes constructed from fifty-four Hadamard 2-(71; 35; 17) designs. The constructed codes are self-dual, doubly-even and self-complementary. Since most of these codes have large automorphism groups, … We study binary linear codes constructed from fifty-four Hadamard 2-(71; 35; 17) designs. The constructed codes are self-dual, doubly-even and self-complementary. Since most of these codes have large automorphism groups, they are suitable for permutation decoding. Therefore we study PD-sets of the obtained codes. We also discuss the error- correcting capability of the obtained codes by majority logic decoding. Further, we de- scribe a construction of a 3-(72; 12; 11) design and a 3-(72; 16; 2010) design from a bi- nary (72; 16; 12) code, and a construction of a strongly regular graph with parameters (126; 25; 8; 4) from a binary (35; 8; 4) code related to a derived 2-(35; 17; 16) design. AMS subject classications : 94B05, 05B20, 05B05, 05E30

Symmetric (66, 26, 10) designs having Frob_55 as an automorphism group

2000-01-01
Dean Crnković , Sanja Rukavina
Up to isomorphism there are three symmetric (66; 26; 10) designs with automorphism group isomorphic to F rob55 . Among them there is one self-dual and one pair of dual … Up to isomorphism there are three symmetric (66; 26; 10) designs with automorphism group isomorphic to F rob55 . Among them there is one self-dual and one pair of dual designs. Full automorphism groups of dual designs are isomorphic to F rob55, and full automorphism group of the self-dual design is isomorphic to F rob55 D10. For those three designs corresponding derived and residual designs with respect to a block are constructed.

On some new extremal Type II Z4-codes of length 40

2020-09-23
Sara Ban , Sanja Rukavina
Using the building-up method and a modification of the doubling method we construct new extremal Type II Z4-codes of length 40. The constructed codes of type $4^{k_1}2^{k_2}$, for $ k_1\in … Using the building-up method and a modification of the doubling method we construct new extremal Type II  Z4-codes of length 40. The constructed codes of type $4^{k_1}2^{k_2}$, for $ k_1\in \{8,9,10,11,12,14,15\}$, are the first examples of extremal Type II  Z4-codes of given type and length 40 whose residue codes have minimum weight greater than or equal to 8. Further, we use minimum weight codewords for a construction of 1-designs, some of which are self-orthogonal.

LDPC codes constructed from cubic symmetric graphs

2020-10-13
Dean Crnković , Sanja Rukavina , Marina Šimac
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The Influence of Curriculum on the Concept of Function: An Empirical Study of Pre-Service Teachers

2022-04-22
Gabrijela Kehler-Poljak , Ljerka Jukić Matić , Sanja Rukavina
This article reports on the understanding of the function concept by pre-service mathematics teachers from two countries (Germany and Croatia). We focused on investigating students’ concept definition and concept image … This article reports on the understanding of the function concept by pre-service mathematics teachers from two countries (Germany and Croatia). We focused on investigating students’ concept definition and concept image of the function in relation to their curriculum experiences. Data were collected using a questionnaire in the form of open-ended questions followed by interviews. The results indicate that the curriculum has a great influence on the development of the concept definition and concept image. The curriculum strongly influenced the theoretical background of the function concept and thus the gap between the formal and the personal definition of function. Later and more intensive work with the formal definition of function led to a better development of the function concept in general. The curriculum also had an influence on the range of the concept image developed by the pre-service mathematics teachers, with no proportional dependence in relation to the better developed understanding of the concept of function.
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New examples of self-dual near-extremal ternary codes of length 48 derived from 2-(47,23,11) designs

2023-01-01
Sanja Rukavina , Vladimir D. Tonchev
In a recent paper [M. Araya, M. Harada, Some restrictions on the weight enumerators of near-extremal ternary self-dual codes and quaternary Hermitian self-dual codes, Des. Codes Cryptogr., 91 (2023), 1813--1843], … In a recent paper [M. Araya, M. Harada, Some restrictions on the weight enumerators of near-extremal ternary self-dual codes and quaternary Hermitian self-dual codes, Des. Codes Cryptogr., 91 (2023), 1813--1843], Araya and Harada gave examples of self-dual near-extremal ternary codes of length 48 for $145$ distinct values of the number $A_{12}$ of codewords of minimum weight 12, and raised the question about the existence of codes for other values of $A_{12}$. In this note, we use symmetric 2-$(47,23,11)$ designs with an automorphism group of order 6 to construct self-dual near-extremal ternary codes of length 48 for $150$ new values of $A_{12}$.
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New examples of self-dual near-extremal ternary codes of length 48 derived from 2-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg" display="inline" id="d1e1118"><mml:mrow><mml:mo>(</mml:mo><mml:mn>47</mml:mn><mml:mo>,</mml:mo><mml:mn>23</mml:mn><mml:mo>,</mml:mo><mml:mn>11</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math> designs

2023-12-15
Sanja Rukavina , Vladimir D. Tonchev
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ENUMERATION OF SYMMETRIC (69,17,4) DESIGNS ADMITTING Z6 AS AN AUTOMORPHISM GROUP

1999-12-01
Sanja Rukavina
All symmetric (69,17,4) designs admitting the cyclic group of order 6 as an automorphism group are classified and their full automorphism groups are determined. All symmetric (69,17,4) designs admitting the cyclic group of order 6 as an automorphism group are classified and their full automorphism groups are determined.

Classification of quasi-symmetric 2-(64,24,46) designs of Blokhuis-Haemers type

2016-02-17
Dean Crnković , Bernardo Rodrigues , Sanja Rukavina +1 more
This paper completes the classification of quasi-symmetric 2-$(64,24,46)$ designs of Blokhuis-Haemers type supported by the dual code $C^{\perp}$ of the binary linear code $C$ spanned by the lines of $AG(3,2^2)$ … This paper completes the classification of quasi-symmetric 2-$(64,24,46)$ designs of Blokhuis-Haemers type supported by the dual code $C^{\perp}$ of the binary linear code $C$ spanned by the lines of $AG(3,2^2)$ initiated in \cite{bgr-vdt}. It is shown that $C^{\perp}$ contains exactly 30,264 nonisomorphic quasi-symmetric 2-$(64,24,46)$ designs obtainable from maximal arcs in $AG(2,2^2)$ via the Blokhuis-Haemers construction. The related strongly regular graphs are also discussed.

Strongly regular graphs from orthogonal groups $O^+(6,2)$ and $O^-(6,2)$

2016-01-01
Dean Crnković , Sanja Rukavina , Andrea Švob
In this paper we construct all strongly regular graphs, with at most 600 vertices, admitting a transitive action of the orthogonal group $O^+(6,2)$ or $O^-(6,2)$. Consequently, we prove the existence … In this paper we construct all strongly regular graphs, with at most 600 vertices, admitting a transitive action of the orthogonal group $O^+(6,2)$ or $O^-(6,2)$. Consequently, we prove the existence of strongly regular graphs with parameters (216,40,4,8) and (540,187,58,68). We also construct a strongly regular graph with parameters (540,224,88,96) that was to the best of our knowledge previously unknown. Further, we show that under certain conditions an orbit matrix $M$ of a strongly regular graph $Γ$ can be used to define a new strongly regular graph $\widetildeΓ$, where the vertices of the graph $\widetildeΓ$ correspond to the orbits of $Γ$ (the rows of $M$). We show that some of the obtained graphs are related to each other in a way that one can be constructed from an orbit matrix of the other.

On some distance-regular graphs with many vertices

2018-09-26
Dean Crnković , Sanja Rukavina , Andrea Švob
We construct distance-regular graphs, including strongly regular graphs, admitting a transitive action of the Chevalley groups $G_2(4)$ and $G_2(5)$, the orthogonal group $O(7,3)$ and the Tits group $T=$$^2F_4(2)'$. Most of … We construct distance-regular graphs, including strongly regular graphs, admitting a transitive action of the Chevalley groups $G_2(4)$ and $G_2(5)$, the orthogonal group $O(7,3)$ and the Tits group $T=$$^2F_4(2)'$. Most of the constructed graphs have more than 1000 vertices, and the number of vertices goes up to 28431. Some of the obtained graphs are new.

LDPC codes constructed from cubic symmetric graphs.

2020-02-16
Dean Crnković , Sanja Rukavina , Marina Šimac
Low-density parity-check (LDPC) codes have been the subject of much interest due to the fact that they can perform near the Shannon limit. In this paper we present a construction … Low-density parity-check (LDPC) codes have been the subject of much interest due to the fact that they can perform near the Shannon limit. In this paper we present a construction of LDPC codes from cubic symmetric graphs. The constructed codes are $(3,3)$-regular and the vast majority of the corresponding Tanner graphs have girth greater than four. We analyse properties of the obtained codes and present bounds for the code parameters, the dimension and the minimum distance. Furthermore, we give an expression for the variance of the syndrome weight of the constructed codes. Information on the LDPC codes constructed from bipartite cubic symmetric graphs with less than 200 vertices is presented as well. Some of the constructed codes are optimal, and some have an additional property of being self-orthogonal or linear codes with complementary dual (LCD codes).

On automorphism groups of a biplane (121,16,2)

2020-10-24
Dean Crnković , Doris Dumičić Danilović , Sanja Rukavina
The existence of a biplane with parameters $(121,16,2)$ is an open problem. Recently, it has been proved by Alavi, Daneshkhah and Praeger that the order of an automorphism group of … The existence of a biplane with parameters $(121,16,2)$ is an open problem. Recently, it has been proved by Alavi, Daneshkhah and Praeger that the order of an automorphism group of a of possible biplane ${\mathcal D}$ of order $14$ divides $2^7\cdot3^2\cdot5\cdot7\cdot11\cdot13$. In this paper we show that such a biplane do not have an automorphism of order $11$ or $13$, and thereby establish that $|Aut({\mathcal D})|$ divides $2^7\cdot3^2\cdot5\cdot7.$ Further, we study a possible action of an automorphism of order five or seven, and some small groups of order divisible by five or seven, on a biplane with parameters $(121,16,2)$.

Type IV-II codes over Z4 constructed from generalized bent functions

2021-01-01
Sara Ban , Sanja Rukavina
A Type IV-II Z4-code is a self-dual code over Z4 with the property that all Euclidean weights are divisible by eight and all codewords have even Hamming weight. In this … A Type IV-II Z4-code is a self-dual code over Z4 with the property that all Euclidean weights are divisible by eight and all codewords have even Hamming weight. In this paper we use generalized bent functions for a construction of self-orthogonal codes over Z4 of length $2^m$, for $m$ odd, $m \geq 3$, and prove that for $m \geq 5$ those codes can be extended to Type IV-II Z4-codes. From that family of Type IV-II Z4-codes, we obtain a family of self-dual Type II binary codes by using Gray map. We also consider the weight distributions of the obtained codes and the structure of the supports of the minimum weight codewords.
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On automorphism groups of a biplane (121,16,2)

2021-06-17
Dean Crnković , Doris Dumičić Danilović , Sanja Rukavina
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Self-orthogonal codes from equitable partitions of distance-regular graphs

2022-03-03
Dean Crnković , Sanja Rukavina , Andrea Švob
<p style='text-indent:20px;'>We give two methods for a construction of self-orthogonal linear codes from equitable partitions of distance-regular graphs. By applying these methods, we construct self-orthogonal codes from equitable partitions of … <p style='text-indent:20px;'>We give two methods for a construction of self-orthogonal linear codes from equitable partitions of distance-regular graphs. By applying these methods, we construct self-orthogonal codes from equitable partitions of the graph of unitals in <inline-formula><tex-math id="M1">\begin{document}$ PG(2,4) $\end{document}</tex-math></inline-formula> and the only known strongly regular graph with parameters <inline-formula><tex-math id="M2">\begin{document}$ (216,40,4,8) $\end{document}</tex-math></inline-formula>. Some of the codes obtained are optimal.
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On automorphism groups of a biplane (121,16,2)

2020-01-01
Dean Crnković , Doris Dumičić Danilović , Sanja Rukavina
The existence of a biplane with parameters $(121,16,2)$ is an open problem. Recently, it has been proved by Alavi, Daneshkhah and Praeger that the order of an automorphism group of … The existence of a biplane with parameters $(121,16,2)$ is an open problem. Recently, it has been proved by Alavi, Daneshkhah and Praeger that the order of an automorphism group of a of possible biplane ${\mathcal D}$ of order $14$ divides $2^7\cdot3^2\cdot5\cdot7\cdot11\cdot13$. In this paper we show that such a biplane do not have an automorphism of order $11$ or $13$, and thereby establish that $|Aut({\mathcal D})|$ divides $2^7\cdot3^2\cdot5\cdot7.$ Further, we study a possible action of an automorphism of order five or seven, and some small groups of order divisible by five or seven, on a biplane with parameters $(121,16,2)$.

LDPC codes constructed from cubic symmetric graphs

2020-01-01
Dean Crnković , Sanja Rukavina , Marina Šimac
Low-density parity-check (LDPC) codes have been the subject of much interest due to the fact that they can perform near the Shannon limit. In this paper we present a construction … Low-density parity-check (LDPC) codes have been the subject of much interest due to the fact that they can perform near the Shannon limit. In this paper we present a construction of LDPC codes from cubic symmetric graphs. The constructed codes are $(3,3)$-regular and the vast majority of the corresponding Tanner graphs have girth greater than four. We analyse properties of the obtained codes and present bounds for the code parameters, the dimension and the minimum distance. Furthermore, we give an expression for the variance of the syndrome weight of the constructed codes. Information on the LDPC codes constructed from bipartite cubic symmetric graphs with less than 200 vertices is presented as well. Some of the constructed codes are optimal, and some have an additional property of being self-orthogonal or linear codes with complementary dual (LCD codes).

Self-orthogonal codes from equitable partitions of association schemes

2019-01-01
Dean Crnković , Sanja Rukavina , Andrea Švob
We give a method of constructing self-orthogonal codes from equitable partitions of association schemes. By applying this method we construct self-orthogonal codes from some distance-regular graphs. Some of the obtained … We give a method of constructing self-orthogonal codes from equitable partitions of association schemes. By applying this method we construct self-orthogonal codes from some distance-regular graphs. Some of the obtained codes are optimal. Further, we introduce a notion of self-orthogonal subspace codes. We show that under some conditions equitable partitions of association schemes yield such self-orthogonal subspace codes and we give some examples from distance-regular graphs.

On some distance-regular graphs with many vertices

2018-01-01
Dean Crnković , Sanja Rukavina , Andrea Švob
We construct distance-regular graphs, including strongly regular graphs, admitting a transitive action of the Chevalley groups $G_2(4)$ and $G_2(5)$, the orthogonal group $O(7,3)$ and the Tits group $T=$$^2F_4(2)'$. Most of … We construct distance-regular graphs, including strongly regular graphs, admitting a transitive action of the Chevalley groups $G_2(4)$ and $G_2(5)$, the orthogonal group $O(7,3)$ and the Tits group $T=$$^2F_4(2)'$. Most of the constructed graphs have more than 1000 vertices, and the number of vertices goes up to 28431. Some of the obtained graphs are new.
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Classification of quasi-symmetric 2-(64,24,46) designs of Blokhuis-Haemers type

2016-01-01
Dean Crnković , Bernardo Rodrigues , Sanja Rukavina +1 more
This paper completes the classification of quasi-symmetric 2-$(64,24,46)$ designs of Blokhuis-Haemers type supported by the dual code $C^{\perp}$ of the binary linear code $C$ spanned by the lines of $AG(3,2^2)$ … This paper completes the classification of quasi-symmetric 2-$(64,24,46)$ designs of Blokhuis-Haemers type supported by the dual code $C^{\perp}$ of the binary linear code $C$ spanned by the lines of $AG(3,2^2)$ initiated in \cite{bgr-vdt}. It is shown that $C^{\perp}$ contains exactly 30,264 nonisomorphic quasi-symmetric 2-$(64,24,46)$ designs obtainable from maximal arcs in $AG(2,2^2)$ via the Blokhuis-Haemers construction. The related strongly regular graphs are also discussed.
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Extremal ternary self-dual codes of length 36 and symmetric 2-(36,15,6) designs with an automorphism of order 2

2022-01-01
Sanja Rukavina , Vladimir D. Tonchev
In this note we report the classification of all symmetric 2-(36,15,6) designs that admit an automorphism of order 2 and their incidence matrices generate an extremal ternary self-dual code. It … In this note we report the classification of all symmetric 2-(36,15,6) designs that admit an automorphism of order 2 and their incidence matrices generate an extremal ternary self-dual code. It is shown that up to isomorphism, there exists only one such design, having a full automorphism group of order 24, and the ternary code spanned by its incidence matrix is equivalent to the Pless symmetry code.
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On symmetric 2-(70,24,8) designs with an automorphism of order 6

2022-01-01
Sanja Rukavina , Vladimir D. Tonchev
In this paper we analyze possible actions of an automorphism of order six on a $2$-$(70, 24, 8)$ design, and give a complete classification for the action of the cyclic … In this paper we analyze possible actions of an automorphism of order six on a $2$-$(70, 24, 8)$ design, and give a complete classification for the action of the cyclic automorphism group of order six $G= \langle \rho \rangle \cong Z_6 \cong Z_2 \times Z_3$ where $\rho^3$ fixes exactly $14$ points (blocks) and $\rho^2$ fixes $4$ points (blocks). Up to isomorphism, there are $3718$ such designs. This result significantly increases the number of known $2$-$(70,24,8)$ designs.
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Characterizing bipartite distance-regularized graphs with vertices of eccentricity 4

2023-01-01
Blas Fernández , Marija Maksimović , Sanja Rukavina
The characterization of bipartite distance-regularized graphs, where some vertices have eccentricity less than four, in terms of the incidence structures of which they are incidence graphs, is known. In this … The characterization of bipartite distance-regularized graphs, where some vertices have eccentricity less than four, in terms of the incidence structures of which they are incidence graphs, is known. In this paper we prove that there is a one-to-one correspondence between the incidence graphs of quasi-symmetric SPBIBDs with parameters $(v,b,r,k, \lambda_1,0)$ of type $(k-1,t)$ with intersection numbers $x=0$ and $y>0$, where $0< y\leq t<k$ , and bipartite distance-regularized graphs with $D=D'=4$.
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New extremal Type II $\mathbb{Z}_4$-codes of length 64 by the doubling method

2023-01-01
Sara Ban , Sanja Rukavina
Extremal Type II $\mathbb{Z}_4$-codes are a class of self-dual $\mathbb{Z}_4$-codes with Euclidean weights divisible by eight and the largest possible minimum Euclidean weight for a given length. A small number … Extremal Type II $\mathbb{Z}_4$-codes are a class of self-dual $\mathbb{Z}_4$-codes with Euclidean weights divisible by eight and the largest possible minimum Euclidean weight for a given length. A small number of such codes is known for lengths greater than or equal to $48.$ The doubling method is a method for constructing Type II $\mathbb{Z}_4$-codes from a given Type II $\mathbb{Z}_4$-code. Based on the doubling method, in this paper we develop a method to construct new extremal Type II $\mathbb{Z}_4$-codes starting from an extremal Type II $\mathbb{Z}_4$-code of type $4^k$ with an extremal residue code and length $48, 56$ or $64$. Using this method, we construct three new extremal Type II $\mathbb{Z}_4$-codes of length $64$ and type $4^{31}2^2$. Extremal Type II $\mathbb{Z}_4$-codes of length $64$ of this type were not known before. Moreover, the residue codes of the constructed extremal $\mathbb{Z}_4$-codes are new best known $[64,31]$ binary codes and the supports of the minimum weight codewords of the residue code and the torsion code of one of these codes form self-orthogonal $1$-designs.
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Construction of self-orthogonal $$\mathbb {Z}_{2^k}$$-codes

2023-12-19
Sara Ban , Sanja Rukavina

On symmetric \(2\)-\((70,24,8)\) designs with an automorphism of order \(6\)

2023-12-20
Sanja Rukavina , Vladimir D. Tonchev
In this paper we analyze possible actions of an automorphism of order six on a \(2\)-\((70, 24, 8)\) design, and give a complete classification for the action of the cyclic … In this paper we analyze possible actions of an automorphism of order six on a \(2\)-\((70, 24, 8)\) design, and give a complete classification for the action of the cyclic group of order six \(G= \langle \rho \rangle \cong Z_6 \cong Z_2 \times Z_3\), where \(\rho^3\) fixes exactly \(14\) points (blocks) and \(\rho^2\) fixes \(4\) points (blocks). Up to isomorphism there are \(3718\) such designs. This result significantly increases the number of previously known \(2\)-\((70,24,8)\) designs.

Symmetric 2$-$(36,15,6) designs with an automorphism of order two

2024-03-05
Sanja Rukavina , Vladimir D. Tonchev
The parameters 2-(36,15,6) are the smallest parameters of symmetric designs for which a complete classification up to isomorphism is yet unknown. Bouyukliev, Fack and Winne classified all 2-$(36,15,6)$ designs that … The parameters 2-(36,15,6) are the smallest parameters of symmetric designs for which a complete classification up to isomorphism is yet unknown. Bouyukliev, Fack and Winne classified all 2-$(36,15,6)$ designs that admit an automorphism of odd prime order, and gave a partial classification of such designs that admit an automorphism of order 2. In this paper, we give the classification of all symmetric 2-$(36,15,6)$ designs that admit an automorphism of order two. It is shown that there are exactly $1 547 701$ nonisomorphic such designs, $135 779$ of which are self-dual designs. The ternary linear codes spanned by the incidence matrices of these designs are computed. Among these codes, there are near-extremal self-dual codes with previously unknown weight distributions.
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Characterizing Bipartite Distance-Regularized Graphs with Vertices of Eccentricity 4

2024-04-29
Blas Fernández , Marija Maksimović , Sanja Rukavina
Abstract Consider a bipartite distance-regularized graph $$\Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> with color partitions Y and $$Y'$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>Y</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:math> . Notably, all vertices in … Abstract Consider a bipartite distance-regularized graph $$\Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> with color partitions Y and $$Y'$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>Y</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:math> . Notably, all vertices in partition Y (and similarly in $$Y'$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>Y</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:math> ) exhibit a shared eccentricity denoted as D (and $$D'$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>D</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:math> , respectively). The characterization of bipartite distance-regularized graphs, specifically those with $$D \le 3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> , in relation to the incidence structures they represent is well established. However, when $$D=4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> , there are only two possible scenarios: either $$D'=3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>D</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> or $$D'=4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>D</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> . The instance where $$D=4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> and $$D'=3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>D</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> has been previously investigated. In this paper, we establish a one-to-one correspondence between the incidence graphs of quasi-symmetric SPBIBDs with parameters $$(v, b, r, k, \lambda _1, 0)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>v</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo>,</mml:mo> <mml:mi>r</mml:mi> <mml:mo>,</mml:mo> <mml:mi>k</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>λ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> of type $$(k-1, t)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , featuring intersection numbers $$x=0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> and $$y&gt;0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>y</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> (where $$y \le t &lt; k$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>y</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>t</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mi>k</mml:mi> </mml:mrow> </mml:math> ), and bipartite distance-regularized graphs with $$D=D'=4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>D</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> . Moreover, our investigations result in the systematic classification of 2- Y -homogeneous bipartite distance-regularized graphs, which are incidence graphs of quasi-symmetric SPBIBDs with parameters $$(v,b,r,k, \lambda _1,0)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>v</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo>,</mml:mo> <mml:mi>r</mml:mi> <mml:mo>,</mml:mo> <mml:mi>k</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>λ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> of type $$(k-1,t)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> with intersection numbers $$x=0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> and $$y=1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>y</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> .
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Construction of extremal Type II $\mathbb{Z}_{8}$-codes via doubling method

2024-05-01
Sara Ban , Sanja Rukavina
Extremal Type II $\mathbb{Z}_{8}$-codes are a class of self-dual $\mathbb{Z}_{8}$-codes with Euclidean weights divisible by $16$ and the largest possible minimum Euclidean weight for a given length. We introduce a … Extremal Type II $\mathbb{Z}_{8}$-codes are a class of self-dual $\mathbb{Z}_{8}$-codes with Euclidean weights divisible by $16$ and the largest possible minimum Euclidean weight for a given length. We introduce a doubling method for constructing a Type II $\mathbb{Z}_{2k}$-code of length $n$ from a known Type II $\mathbb{Z}_{2k}$-code of length $n$. Based on this method, we develop an algorithm to construct new extremal Type II $\mathbb{Z}_8$-codes starting from an extremal Type II $\mathbb{Z}_8$-code of type $(\frac{n}{2},0,0)$ with an extremal $\mathbb{Z}_4$-residue code and length $24, 32$ or $40$. We construct at least ten new extremal Type II $\mathbb{Z}_8$-codes of length $32$ and type $(15,1,1)$. Extremal Type II $\mathbb{Z}_8$-codes of length $32$ of this type were not known before. Moreover, the binary residue codes of the constructed extremal $\mathbb{Z}_8$-codes are optimal $[32,15]$ binary codes.
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Symmetric 2‐(36,15,6) $(36,15,6)$ designs with an automorphism of order two

2024-06-17
Sanja Rukavina , Vladimir D. Tonchev
Abstract Bouyukliev, Fack and Winne classified all 2‐ designs that admit an automorphism of odd prime order, and gave a partial classification of such designs that admit an automorphism of … Abstract Bouyukliev, Fack and Winne classified all 2‐ designs that admit an automorphism of odd prime order, and gave a partial classification of such designs that admit an automorphism of order 2. In this paper, we give the classification of all symmetric 2‐ designs that admit an automorphism of order two. It is shown that there are exactly nonisomorphic such designs, of which are self‐dual designs. The ternary linear codes spanned by the incidence matrices of these designs are computed. Among these codes, there are near‐extremal self‐dual codes with previously unknown weight distributions.
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New extremal Type II $${\mathbb {Z}}_4$$-codes of length 64

2024-12-11
Sara Ban , Sanja Rukavina

Construction of directed strongly regular graphs with nontrivial automorphisms

2024-12-14
Marija Maksimović , Sanja Rukavina
In this paper we present a method for constructing directed strongly regular graphs with assumed action of an automorphism group. The application of this method leads to first examples of … In this paper we present a method for constructing directed strongly regular graphs with assumed action of an automorphism group. The application of this method leads to first examples of directed strongly regular graphs with parameters $(22,9,6,3,4)$. We have shown that an automorphism of prime order acting on such a graph can only be of order two or three. Furthermore, we have constructed $472$ directed strongly regular graphs with parameters $(22,9,6,3,4)$ and classified all these graphs with an automorphism of order three.
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Construction of directed strongly regular graphs with nontrivial automorphisms

2024-12-14
Marija Maksimović , Sanja Rukavina
In this paper we present a method for constructing directed strongly regular graphs with assumed action of an automorphism group. The application of this method leads to first examples of … In this paper we present a method for constructing directed strongly regular graphs with assumed action of an automorphism group. The application of this method leads to first examples of directed strongly regular graphs with parameters $(22,9,6,3,4)$. We have shown that an automorphism of prime order acting on such a graph can only be of order two or three. Furthermore, we have constructed $472$ directed strongly regular graphs with parameters $(22,9,6,3,4)$ and classified all these graphs with an automorphism of order three.
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New extremal Type II $${\mathbb {Z}}_4$$-codes of length 64

2024-12-11
Sara Ban , Sanja Rukavina

Symmetric 2‐(36,15,6) $(36,15,6)$ designs with an automorphism of order two

2024-06-17
Sanja Rukavina , Vladimir D. Tonchev
Abstract Bouyukliev, Fack and Winne classified all 2‐ designs that admit an automorphism of odd prime order, and gave a partial classification of such designs that admit an automorphism of … Abstract Bouyukliev, Fack and Winne classified all 2‐ designs that admit an automorphism of odd prime order, and gave a partial classification of such designs that admit an automorphism of order 2. In this paper, we give the classification of all symmetric 2‐ designs that admit an automorphism of order two. It is shown that there are exactly nonisomorphic such designs, of which are self‐dual designs. The ternary linear codes spanned by the incidence matrices of these designs are computed. Among these codes, there are near‐extremal self‐dual codes with previously unknown weight distributions.
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Construction of extremal Type II $\mathbb{Z}_{8}$-codes via doubling method

2024-05-01
Sara Ban , Sanja Rukavina
Extremal Type II $\mathbb{Z}_{8}$-codes are a class of self-dual $\mathbb{Z}_{8}$-codes with Euclidean weights divisible by $16$ and the largest possible minimum Euclidean weight for a given length. We introduce a … Extremal Type II $\mathbb{Z}_{8}$-codes are a class of self-dual $\mathbb{Z}_{8}$-codes with Euclidean weights divisible by $16$ and the largest possible minimum Euclidean weight for a given length. We introduce a doubling method for constructing a Type II $\mathbb{Z}_{2k}$-code of length $n$ from a known Type II $\mathbb{Z}_{2k}$-code of length $n$. Based on this method, we develop an algorithm to construct new extremal Type II $\mathbb{Z}_8$-codes starting from an extremal Type II $\mathbb{Z}_8$-code of type $(\frac{n}{2},0,0)$ with an extremal $\mathbb{Z}_4$-residue code and length $24, 32$ or $40$. We construct at least ten new extremal Type II $\mathbb{Z}_8$-codes of length $32$ and type $(15,1,1)$. Extremal Type II $\mathbb{Z}_8$-codes of length $32$ of this type were not known before. Moreover, the binary residue codes of the constructed extremal $\mathbb{Z}_8$-codes are optimal $[32,15]$ binary codes.
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Characterizing Bipartite Distance-Regularized Graphs with Vertices of Eccentricity 4

2024-04-29
Blas Fernández , Marija Maksimović , Sanja Rukavina
Abstract Consider a bipartite distance-regularized graph $$\Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> with color partitions Y and $$Y'$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>Y</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:math> . Notably, all vertices in … Abstract Consider a bipartite distance-regularized graph $$\Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> with color partitions Y and $$Y'$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>Y</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:math> . Notably, all vertices in partition Y (and similarly in $$Y'$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>Y</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:math> ) exhibit a shared eccentricity denoted as D (and $$D'$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>D</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:math> , respectively). The characterization of bipartite distance-regularized graphs, specifically those with $$D \le 3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> , in relation to the incidence structures they represent is well established. However, when $$D=4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> , there are only two possible scenarios: either $$D'=3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>D</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> or $$D'=4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>D</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> . The instance where $$D=4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> and $$D'=3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>D</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> has been previously investigated. In this paper, we establish a one-to-one correspondence between the incidence graphs of quasi-symmetric SPBIBDs with parameters $$(v, b, r, k, \lambda _1, 0)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>v</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo>,</mml:mo> <mml:mi>r</mml:mi> <mml:mo>,</mml:mo> <mml:mi>k</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>λ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> of type $$(k-1, t)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , featuring intersection numbers $$x=0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> and $$y&gt;0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>y</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> (where $$y \le t &lt; k$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>y</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>t</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mi>k</mml:mi> </mml:mrow> </mml:math> ), and bipartite distance-regularized graphs with $$D=D'=4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>D</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> . Moreover, our investigations result in the systematic classification of 2- Y -homogeneous bipartite distance-regularized graphs, which are incidence graphs of quasi-symmetric SPBIBDs with parameters $$(v,b,r,k, \lambda _1,0)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>v</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo>,</mml:mo> <mml:mi>r</mml:mi> <mml:mo>,</mml:mo> <mml:mi>k</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>λ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> of type $$(k-1,t)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> with intersection numbers $$x=0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> and $$y=1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>y</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> .
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Symmetric 2$-$(36,15,6) designs with an automorphism of order two

2024-03-05
Sanja Rukavina , Vladimir D. Tonchev
The parameters 2-(36,15,6) are the smallest parameters of symmetric designs for which a complete classification up to isomorphism is yet unknown. Bouyukliev, Fack and Winne classified all 2-$(36,15,6)$ designs that … The parameters 2-(36,15,6) are the smallest parameters of symmetric designs for which a complete classification up to isomorphism is yet unknown. Bouyukliev, Fack and Winne classified all 2-$(36,15,6)$ designs that admit an automorphism of odd prime order, and gave a partial classification of such designs that admit an automorphism of order 2. In this paper, we give the classification of all symmetric 2-$(36,15,6)$ designs that admit an automorphism of order two. It is shown that there are exactly $1 547 701$ nonisomorphic such designs, $135 779$ of which are self-dual designs. The ternary linear codes spanned by the incidence matrices of these designs are computed. Among these codes, there are near-extremal self-dual codes with previously unknown weight distributions.
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On symmetric \(2\)-\((70,24,8)\) designs with an automorphism of order \(6\)

2023-12-20
Sanja Rukavina , Vladimir D. Tonchev
In this paper we analyze possible actions of an automorphism of order six on a \(2\)-\((70, 24, 8)\) design, and give a complete classification for the action of the cyclic … In this paper we analyze possible actions of an automorphism of order six on a \(2\)-\((70, 24, 8)\) design, and give a complete classification for the action of the cyclic group of order six \(G= \langle \rho \rangle \cong Z_6 \cong Z_2 \times Z_3\), where \(\rho^3\) fixes exactly \(14\) points (blocks) and \(\rho^2\) fixes \(4\) points (blocks). Up to isomorphism there are \(3718\) such designs. This result significantly increases the number of previously known \(2\)-\((70,24,8)\) designs.

Construction of self-orthogonal $$\mathbb {Z}_{2^k}$$-codes

2023-12-19
Sara Ban , Sanja Rukavina

New examples of self-dual near-extremal ternary codes of length 48 derived from 2-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si84.svg" display="inline" id="d1e1118"><mml:mrow><mml:mo>(</mml:mo><mml:mn>47</mml:mn><mml:mo>,</mml:mo><mml:mn>23</mml:mn><mml:mo>,</mml:mo><mml:mn>11</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math> designs

2023-12-15
Sanja Rukavina , Vladimir D. Tonchev
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Construction of extremal $ \mathbb{Z}_{4} $-codes using a neighborhood search algorithm

2023-10-17
Dean Crnković , Matteo Mravić , Sanja Rukavina
In this paper, we present a method for constructing extremal $ \mathbb{Z}_{4} $-codes based on random neighborhood search. This method is used to find new extremal Type Ⅰ and Type … In this paper, we present a method for constructing extremal $ \mathbb{Z}_{4} $-codes based on random neighborhood search. This method is used to find new extremal Type Ⅰ and Type Ⅱ $ \mathbb{Z}_{4} $-codes of lengths 32 and 40. For the length 32, at least 182 new Type Ⅱ extremal $ \mathbb{Z}_{4} $-codes of types $ 4^{k}2^{32-2k} $, $ k\in\left\{9,10,12,13,14,15,16\right\} $ are constructed. In addition, we obtained at least 762 new extremal Type Ⅰ $ \mathbb{Z}_{4} $-codes of types $ 4^{k}2^{32-2k} $, $ k\in\left\{7,9,10,12,13,14,15,16\right\} $. For the length 40, constructed extremal $ \mathbb{Z}_{4} $-codes are of types $ 4^{k}2^{40-2k} $, $ k\in\left\{7,10,11,15,16\right\} $. There are at least 40 new Type Ⅱ extremal $ \mathbb{Z}_{4} $-codes, and at least 4144 new Type Ⅰ extremal $ \mathbb{Z}_{4} $-codes.
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Characterizing bipartite distance-regularized graphs with vertices of eccentricity 4

2023-01-01
Blas Fernández , Marija Maksimović , Sanja Rukavina
The characterization of bipartite distance-regularized graphs, where some vertices have eccentricity less than four, in terms of the incidence structures of which they are incidence graphs, is known. In this … The characterization of bipartite distance-regularized graphs, where some vertices have eccentricity less than four, in terms of the incidence structures of which they are incidence graphs, is known. In this paper we prove that there is a one-to-one correspondence between the incidence graphs of quasi-symmetric SPBIBDs with parameters $(v,b,r,k, \lambda_1,0)$ of type $(k-1,t)$ with intersection numbers $x=0$ and $y>0$, where $0< y\leq t<k$ , and bipartite distance-regularized graphs with $D=D'=4$.
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New extremal Type II $\mathbb{Z}_4$-codes of length 64 by the doubling method

2023-01-01
Sara Ban , Sanja Rukavina
Extremal Type II $\mathbb{Z}_4$-codes are a class of self-dual $\mathbb{Z}_4$-codes with Euclidean weights divisible by eight and the largest possible minimum Euclidean weight for a given length. A small number … Extremal Type II $\mathbb{Z}_4$-codes are a class of self-dual $\mathbb{Z}_4$-codes with Euclidean weights divisible by eight and the largest possible minimum Euclidean weight for a given length. A small number of such codes is known for lengths greater than or equal to $48.$ The doubling method is a method for constructing Type II $\mathbb{Z}_4$-codes from a given Type II $\mathbb{Z}_4$-code. Based on the doubling method, in this paper we develop a method to construct new extremal Type II $\mathbb{Z}_4$-codes starting from an extremal Type II $\mathbb{Z}_4$-code of type $4^k$ with an extremal residue code and length $48, 56$ or $64$. Using this method, we construct three new extremal Type II $\mathbb{Z}_4$-codes of length $64$ and type $4^{31}2^2$. Extremal Type II $\mathbb{Z}_4$-codes of length $64$ of this type were not known before. Moreover, the residue codes of the constructed extremal $\mathbb{Z}_4$-codes are new best known $[64,31]$ binary codes and the supports of the minimum weight codewords of the residue code and the torsion code of one of these codes form self-orthogonal $1$-designs.
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New examples of self-dual near-extremal ternary codes of length 48 derived from 2-(47,23,11) designs

2023-01-01
Sanja Rukavina , Vladimir D. Tonchev
In a recent paper [M. Araya, M. Harada, Some restrictions on the weight enumerators of near-extremal ternary self-dual codes and quaternary Hermitian self-dual codes, Des. Codes Cryptogr., 91 (2023), 1813--1843], … In a recent paper [M. Araya, M. Harada, Some restrictions on the weight enumerators of near-extremal ternary self-dual codes and quaternary Hermitian self-dual codes, Des. Codes Cryptogr., 91 (2023), 1813--1843], Araya and Harada gave examples of self-dual near-extremal ternary codes of length 48 for $145$ distinct values of the number $A_{12}$ of codewords of minimum weight 12, and raised the question about the existence of codes for other values of $A_{12}$. In this note, we use symmetric 2-$(47,23,11)$ designs with an automorphism group of order 6 to construct self-dual near-extremal ternary codes of length 48 for $150$ new values of $A_{12}$.
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Extremal ternary self-dual codes of length 36 and symmetric 2-(36, 15, 6) designs with an automorphism of order 2

2022-12-29
Sanja Rukavina , Vladimir D. Tonchev
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On the 2-Y-Homogeneous Condition of the Incidence Graphs of 2-Designs

2022-07-12
Blas Fernández , Sanja Rukavina

The Influence of Curriculum on the Concept of Function: An Empirical Study of Pre-Service Teachers

2022-04-22
Gabrijela Kehler-Poljak , Ljerka Jukić Matić , Sanja Rukavina
This article reports on the understanding of the function concept by pre-service mathematics teachers from two countries (Germany and Croatia). We focused on investigating students’ concept definition and concept image … This article reports on the understanding of the function concept by pre-service mathematics teachers from two countries (Germany and Croatia). We focused on investigating students’ concept definition and concept image of the function in relation to their curriculum experiences. Data were collected using a questionnaire in the form of open-ended questions followed by interviews. The results indicate that the curriculum has a great influence on the development of the concept definition and concept image. The curriculum strongly influenced the theoretical background of the function concept and thus the gap between the formal and the personal definition of function. Later and more intensive work with the formal definition of function led to a better development of the function concept in general. The curriculum also had an influence on the range of the concept image developed by the pre-service mathematics teachers, with no proportional dependence in relation to the better developed understanding of the concept of function.
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Self-orthogonal codes from equitable partitions of distance-regular graphs

2022-03-03
Dean Crnković , Sanja Rukavina , Andrea Švob
<p style='text-indent:20px;'>We give two methods for a construction of self-orthogonal linear codes from equitable partitions of distance-regular graphs. By applying these methods, we construct self-orthogonal codes from equitable partitions of … <p style='text-indent:20px;'>We give two methods for a construction of self-orthogonal linear codes from equitable partitions of distance-regular graphs. By applying these methods, we construct self-orthogonal codes from equitable partitions of the graph of unitals in <inline-formula><tex-math id="M1">\begin{document}$ PG(2,4) $\end{document}</tex-math></inline-formula> and the only known strongly regular graph with parameters <inline-formula><tex-math id="M2">\begin{document}$ (216,40,4,8) $\end{document}</tex-math></inline-formula>. Some of the codes obtained are optimal.
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Self-orthogonal codes from equitable partitions of association schemes

2022-01-15
Dean Crnković , Sanja Rukavina , Andrea Švob
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Extremal ternary self-dual codes of length 36 and symmetric 2-(36,15,6) designs with an automorphism of order 2

2022-01-01
Sanja Rukavina , Vladimir D. Tonchev
In this note we report the classification of all symmetric 2-(36,15,6) designs that admit an automorphism of order 2 and their incidence matrices generate an extremal ternary self-dual code. It … In this note we report the classification of all symmetric 2-(36,15,6) designs that admit an automorphism of order 2 and their incidence matrices generate an extremal ternary self-dual code. It is shown that up to isomorphism, there exists only one such design, having a full automorphism group of order 24, and the ternary code spanned by its incidence matrix is equivalent to the Pless symmetry code.
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On symmetric 2-(70,24,8) designs with an automorphism of order 6

2022-01-01
Sanja Rukavina , Vladimir D. Tonchev
In this paper we analyze possible actions of an automorphism of order six on a $2$-$(70, 24, 8)$ design, and give a complete classification for the action of the cyclic … In this paper we analyze possible actions of an automorphism of order six on a $2$-$(70, 24, 8)$ design, and give a complete classification for the action of the cyclic automorphism group of order six $G= \langle \rho \rangle \cong Z_6 \cong Z_2 \times Z_3$ where $\rho^3$ fixes exactly $14$ points (blocks) and $\rho^2$ fixes $4$ points (blocks). Up to isomorphism, there are $3718$ such designs. This result significantly increases the number of known $2$-$(70,24,8)$ designs.
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On automorphism groups of a biplane (121,16,2)

2021-06-17
Dean Crnković , Doris Dumičić Danilović , Sanja Rukavina
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Type IV-II codes over Z4 constructed from generalized bent functions

2021-01-01
Sara Ban , Sanja Rukavina
A Type IV-II Z4-code is a self-dual code over Z4 with the property that all Euclidean weights are divisible by eight and all codewords have even Hamming weight. In this … A Type IV-II Z4-code is a self-dual code over Z4 with the property that all Euclidean weights are divisible by eight and all codewords have even Hamming weight. In this paper we use generalized bent functions for a construction of self-orthogonal codes over Z4 of length $2^m$, for $m$ odd, $m \geq 3$, and prove that for $m \geq 5$ those codes can be extended to Type IV-II Z4-codes. From that family of Type IV-II Z4-codes, we obtain a family of self-dual Type II binary codes by using Gray map. We also consider the weight distributions of the obtained codes and the structure of the supports of the minimum weight codewords.
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On automorphism groups of a biplane (121,16,2)

2020-10-24
Dean Crnković , Doris Dumičić Danilović , Sanja Rukavina
The existence of a biplane with parameters $(121,16,2)$ is an open problem. Recently, it has been proved by Alavi, Daneshkhah and Praeger that the order of an automorphism group of … The existence of a biplane with parameters $(121,16,2)$ is an open problem. Recently, it has been proved by Alavi, Daneshkhah and Praeger that the order of an automorphism group of a of possible biplane ${\mathcal D}$ of order $14$ divides $2^7\cdot3^2\cdot5\cdot7\cdot11\cdot13$. In this paper we show that such a biplane do not have an automorphism of order $11$ or $13$, and thereby establish that $|Aut({\mathcal D})|$ divides $2^7\cdot3^2\cdot5\cdot7.$ Further, we study a possible action of an automorphism of order five or seven, and some small groups of order divisible by five or seven, on a biplane with parameters $(121,16,2)$.

LDPC codes constructed from cubic symmetric graphs

2020-10-13
Dean Crnković , Sanja Rukavina , Marina Šimac
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On some new extremal Type II Z4-codes of length 40

2020-09-23
Sara Ban , Sanja Rukavina
Using the building-up method and a modification of the doubling method we construct new extremal Type II Z4-codes of length 40. The constructed codes of type $4^{k_1}2^{k_2}$, for $ k_1\in … Using the building-up method and a modification of the doubling method we construct new extremal Type II  Z4-codes of length 40. The constructed codes of type $4^{k_1}2^{k_2}$, for $ k_1\in \{8,9,10,11,12,14,15\}$, are the first examples of extremal Type II  Z4-codes of given type and length 40 whose residue codes have minimum weight greater than or equal to 8. Further, we use minimum weight codewords for a construction of 1-designs, some of which are self-orthogonal.

LDPC codes constructed from cubic symmetric graphs.

2020-02-16
Dean Crnković , Sanja Rukavina , Marina Šimac
Low-density parity-check (LDPC) codes have been the subject of much interest due to the fact that they can perform near the Shannon limit. In this paper we present a construction … Low-density parity-check (LDPC) codes have been the subject of much interest due to the fact that they can perform near the Shannon limit. In this paper we present a construction of LDPC codes from cubic symmetric graphs. The constructed codes are $(3,3)$-regular and the vast majority of the corresponding Tanner graphs have girth greater than four. We analyse properties of the obtained codes and present bounds for the code parameters, the dimension and the minimum distance. Furthermore, we give an expression for the variance of the syndrome weight of the constructed codes. Information on the LDPC codes constructed from bipartite cubic symmetric graphs with less than 200 vertices is presented as well. Some of the constructed codes are optimal, and some have an additional property of being self-orthogonal or linear codes with complementary dual (LCD codes).

On automorphism groups of a biplane (121,16,2)

2020-01-01
Dean Crnković , Doris Dumičić Danilović , Sanja Rukavina
The existence of a biplane with parameters $(121,16,2)$ is an open problem. Recently, it has been proved by Alavi, Daneshkhah and Praeger that the order of an automorphism group of … The existence of a biplane with parameters $(121,16,2)$ is an open problem. Recently, it has been proved by Alavi, Daneshkhah and Praeger that the order of an automorphism group of a of possible biplane ${\mathcal D}$ of order $14$ divides $2^7\cdot3^2\cdot5\cdot7\cdot11\cdot13$. In this paper we show that such a biplane do not have an automorphism of order $11$ or $13$, and thereby establish that $|Aut({\mathcal D})|$ divides $2^7\cdot3^2\cdot5\cdot7.$ Further, we study a possible action of an automorphism of order five or seven, and some small groups of order divisible by five or seven, on a biplane with parameters $(121,16,2)$.

LDPC codes constructed from cubic symmetric graphs

2020-01-01
Dean Crnković , Sanja Rukavina , Marina Šimac
Low-density parity-check (LDPC) codes have been the subject of much interest due to the fact that they can perform near the Shannon limit. In this paper we present a construction … Low-density parity-check (LDPC) codes have been the subject of much interest due to the fact that they can perform near the Shannon limit. In this paper we present a construction of LDPC codes from cubic symmetric graphs. The constructed codes are $(3,3)$-regular and the vast majority of the corresponding Tanner graphs have girth greater than four. We analyse properties of the obtained codes and present bounds for the code parameters, the dimension and the minimum distance. Furthermore, we give an expression for the variance of the syndrome weight of the constructed codes. Information on the LDPC codes constructed from bipartite cubic symmetric graphs with less than 200 vertices is presented as well. Some of the constructed codes are optimal, and some have an additional property of being self-orthogonal or linear codes with complementary dual (LCD codes).

On some distance-regular graphs with many vertices

2019-05-18
Dean Crnković , Sanja Rukavina , Andrea Švob
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Self-orthogonal codes from equitable partitions of association schemes

2019-01-01
Dean Crnković , Sanja Rukavina , Andrea Švob
We give a method of constructing self-orthogonal codes from equitable partitions of association schemes. By applying this method we construct self-orthogonal codes from some distance-regular graphs. Some of the obtained … We give a method of constructing self-orthogonal codes from equitable partitions of association schemes. By applying this method we construct self-orthogonal codes from some distance-regular graphs. Some of the obtained codes are optimal. Further, we introduce a notion of self-orthogonal subspace codes. We show that under some conditions equitable partitions of association schemes yield such self-orthogonal subspace codes and we give some examples from distance-regular graphs.

On some distance-regular graphs with many vertices

2018-09-26
Dean Crnković , Sanja Rukavina , Andrea Švob
We construct distance-regular graphs, including strongly regular graphs, admitting a transitive action of the Chevalley groups $G_2(4)$ and $G_2(5)$, the orthogonal group $O(7,3)$ and the Tits group $T=$$^2F_4(2)'$. Most of … We construct distance-regular graphs, including strongly regular graphs, admitting a transitive action of the Chevalley groups $G_2(4)$ and $G_2(5)$, the orthogonal group $O(7,3)$ and the Tits group $T=$$^2F_4(2)'$. Most of the constructed graphs have more than 1000 vertices, and the number of vertices goes up to 28431. Some of the obtained graphs are new.

New strongly regular graphs from orthogonal groups <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml142" display="inline" overflow="scroll" altimg="si142.gif"><mml:msup><mml:mrow><mml:mi>O</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mn>6</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml143" display="inline" overflow="…

2018-07-14
Dean Crnković , Sanja Rukavina , Andrea Švob

On some distance-regular graphs with many vertices

2018-01-01
Dean Crnković , Sanja Rukavina , Andrea Švob
We construct distance-regular graphs, including strongly regular graphs, admitting a transitive action of the Chevalley groups $G_2(4)$ and $G_2(5)$, the orthogonal group $O(7,3)$ and the Tits group $T=$$^2F_4(2)'$. Most of … We construct distance-regular graphs, including strongly regular graphs, admitting a transitive action of the Chevalley groups $G_2(4)$ and $G_2(5)$, the orthogonal group $O(7,3)$ and the Tits group $T=$$^2F_4(2)'$. Most of the constructed graphs have more than 1000 vertices, and the number of vertices goes up to 28431. Some of the obtained graphs are new.
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Quasi-symmetric 2-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml22" display="inline" overflow="scroll" altimg="si2.gif"><mml:mrow><mml:mo>(</mml:mo><mml:mn>64</mml:mn><mml:mo>,</mml:mo><mml:mn>24</mml:mn><mml:mo>,</mml:mo><mml:mn>46</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math> designs derived from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml23" display="inline" overflow="scroll" altimg="si23.gif"><mml:mi>A</mml:mi><mml:mi>G</mml:mi><mml:mrow><mml:mo>(</…

2017-07-05
Dean Crnković , Bernardo Rodrigues , Sanja Rukavina +1 more

On triplanes of order twelve admitting an automorphism of order six and their binary and ternary codes

2017-01-01
Dean Crnković , Sanja Rukavina , Loredana Simčić
In this paper we present the complete classification of triplanes (71, 15, 3) admitting an action of the cyclic automorphism group of order six. Up to isomorphism there are 146 … In this paper we present the complete classification of triplanes (71, 15, 3) admitting an action of the cyclic automorphism group of order six. Up to isomorphism there are 146 such triplanes and these are all triplanes of order 12 known up to now. Further, we analyze binary and ternary codes spanned by the incidence matrices of triplanes (71, 15, 3) and their residual designs. The constructed binary codes are self- complementary, and the ternary codes are self- orthogonal. In addition, we study ternary self-orthogonal codes constructed from the orbit matrices for Z_3 acting on the 146 symmetric 2-(71, 15, 3) designs. Some of the obtained codes have minimum distance one or two less than the best known codes with the same length and dimension. Finally, we discuss k-geodetic graphs from the symmetric (71, 15, 3) designs and their residual and derived designs.

Self-orthogonal codes from the strongly regular graphs on up to 40 vertices

2016-08-01
Dean Crnković , Marija Maksimović , Bernardo Rodrigues +1 more
This paper outlines a method for constructing self-orthogonal codes from orbit matrices of strongly regular graphs admitting an automorphism group $G$ which acts with orbits of length $w$, where $w$ … This paper outlines a method for constructing self-orthogonal codes from orbit matrices of strongly regular graphs admitting an automorphism group $G$ which acts with orbits of length $w$, where $w$ divides $|G|$.We apply this method to construct self-orthogonal codes from orbit matrices of the strongly regular graphs with at most 40 vertices. In particular, we construct codes from adjacency or orbit matrices of graphs with parameters $(36, 15, 6, 6)$, $(36, 14, 4, 6)$,$(35, 16, 6, 8)$ and their complements, and from the graphs with parameters $(40, 12, 2, 4)$ and their complements.That completes the classification of self-orthogonal codes spanned by the adjacency matrices or orbit matrices of thestrongly regular graphs with at most 40 vertices.Furthermore, we construct ternary codes of $2$-$(27,9,4)$ designs obtained as residual designs of the symmetric $(40, 13, 4)$ designs (complementary designs of the symmetric $(40, 27, 18)$ designs), and their ternary hulls. Some of the obtained codes are optimal, and some are best known for the given length and dimension.
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Classification of quasi-symmetric 2-(64,24,46) designs of Blokhuis-Haemers type

2016-02-17
Dean Crnković , Bernardo Rodrigues , Sanja Rukavina +1 more
This paper completes the classification of quasi-symmetric 2-$(64,24,46)$ designs of Blokhuis-Haemers type supported by the dual code $C^{\perp}$ of the binary linear code $C$ spanned by the lines of $AG(3,2^2)$ … This paper completes the classification of quasi-symmetric 2-$(64,24,46)$ designs of Blokhuis-Haemers type supported by the dual code $C^{\perp}$ of the binary linear code $C$ spanned by the lines of $AG(3,2^2)$ initiated in \cite{bgr-vdt}. It is shown that $C^{\perp}$ contains exactly 30,264 nonisomorphic quasi-symmetric 2-$(64,24,46)$ designs obtainable from maximal arcs in $AG(2,2^2)$ via the Blokhuis-Haemers construction. The related strongly regular graphs are also discussed.

Strongly regular graphs from orthogonal groups $O^+(6,2)$ and $O^-(6,2)$

2016-01-01
Dean Crnković , Sanja Rukavina , Andrea Švob
In this paper we construct all strongly regular graphs, with at most 600 vertices, admitting a transitive action of the orthogonal group $O^+(6,2)$ or $O^-(6,2)$. Consequently, we prove the existence … In this paper we construct all strongly regular graphs, with at most 600 vertices, admitting a transitive action of the orthogonal group $O^+(6,2)$ or $O^-(6,2)$. Consequently, we prove the existence of strongly regular graphs with parameters (216,40,4,8) and (540,187,58,68). We also construct a strongly regular graph with parameters (540,224,88,96) that was to the best of our knowledge previously unknown. Further, we show that under certain conditions an orbit matrix $M$ of a strongly regular graph $Γ$ can be used to define a new strongly regular graph $\widetildeΓ$, where the vertices of the graph $\widetildeΓ$ correspond to the orbits of $Γ$ (the rows of $M$). We show that some of the obtained graphs are related to each other in a way that one can be constructed from an orbit matrix of the other.

Classification of quasi-symmetric 2-(64,24,46) designs of Blokhuis-Haemers type

2016-01-01
Dean Crnković , Bernardo Rodrigues , Sanja Rukavina +1 more
This paper completes the classification of quasi-symmetric 2-$(64,24,46)$ designs of Blokhuis-Haemers type supported by the dual code $C^{\perp}$ of the binary linear code $C$ spanned by the lines of $AG(3,2^2)$ … This paper completes the classification of quasi-symmetric 2-$(64,24,46)$ designs of Blokhuis-Haemers type supported by the dual code $C^{\perp}$ of the binary linear code $C$ spanned by the lines of $AG(3,2^2)$ initiated in \cite{bgr-vdt}. It is shown that $C^{\perp}$ contains exactly 30,264 nonisomorphic quasi-symmetric 2-$(64,24,46)$ designs obtainable from maximal arcs in $AG(2,2^2)$ via the Blokhuis-Haemers construction. The related strongly regular graphs are also discussed.
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Self-dual codes from quotient matrices of symmetric divisible designs with the dual property

2015-10-01
Dean Crnković , Nina Mostarac , Sanja Rukavina

Self-dual codes from extended orbit matrices of symmetric designs

2015-01-27
Dean Crnković , Sanja Rukavina

Binary doubly-even self-dual codes of length 72 with large automorphism groups ∗

2013-04-28
Dean Crnković , Sanja Rukavina , Loredana Simčić
We study binary linear codes constructed from fifty-four Hadamard 2-(71; 35; 17) designs. The constructed codes are self-dual, doubly-even and self-complementary. Since most of these codes have large automorphism groups, … We study binary linear codes constructed from fifty-four Hadamard 2-(71; 35; 17) designs. The constructed codes are self-dual, doubly-even and self-complementary. Since most of these codes have large automorphism groups, they are suitable for permutation decoding. Therefore we study PD-sets of the obtained codes. We also discuss the error- correcting capability of the obtained codes by majority logic decoding. Further, we de- scribe a construction of a 3-(72; 12; 11) design and a 3-(72; 16; 2010) design from a bi- nary (72; 16; 12) code, and a construction of a strongly regular graph with parameters (126; 25; 8; 4) from a binary (35; 8; 4) code related to a derived 2-(35; 17; 16) design. AMS subject classications : 94B05, 05B20, 05B05, 05E30

Self-orthogonal codes from orbit matrices of 2-designs

2013-01-01
Dean Crnković , Bernardo Rodrigues , Sanja Rukavina +1 more
In this paper we present a method for constructing self-orthogonal codes from orbit matrices of $2$-designs that admit an automorphism group $G$ which acts with orbit lengths $1$ and $w$, … In this paper we present a method for constructing self-orthogonal codes from orbit matrices of $2$-designs that admit an automorphism group $G$ which acts with orbit lengths $1$ and $w$, where $w$ divides $|G|$. This is a generalization of an earlier method proposed by Tonchev for constructing self-orthogonal codes from orbit matrices of $2$-designs with a fixed-point-free automorphism of prime order. As an illustration of our method we provide a classification of self-orthogonal codes obtained from the non-fixed parts of the orbit matrices of the symmetric $2$-$(56,11,2)$ designs, some symmetric designs $2$-$(71,15,3)$ (and their residual designs), and some non-symmetric $2$-designs, namely those with parameters $2$-$(15,3,1)$, $2$-$(25,4,1)$, $2$-$(37,4,1)$, and $2$-$(45,5,1)$, respectively with automorphisms of order $p$, where $p$ is an odd prime. We establish that the codes with parameters $[10,4,6]_3$ and $[11,4,6]_3$are optimal two-weight codes. Further, we construct an optimal binary self-orthogonal $[16,5,8]$ code from the non-fixed part of the orbit matrix of the $2$-$(64,8,1)$ design with respect to an automorphism group of order four.

Ternary codes from the strongly regular (45, 12, 3, 3) graphs and orbit matrices of 2-(45, 12, 3) designs

2012-07-20
Dean Crnković , Bernardo Rodrigues , Sanja Rukavina +1 more

A classification of all symmetric block designs of order nine with an automorphism of order six

2005-10-21
Dean Crnković , Sanja Rukavina , Marcel Schmidt
Abstract We complete the classification of all symmetric designs of order nine admitting an automorphism of order six. As a matter of fact, the classification for the parameters (35,17,8), (56,11,2), … Abstract We complete the classification of all symmetric designs of order nine admitting an automorphism of order six. As a matter of fact, the classification for the parameters (35,17,8), (56,11,2), and (91,10,1) had already been done, and in this paper we present the results for the parameters (36,15,6), (40,13,4), and (45,12,3). We also provide information about the order and the structure of the full automorphism groups of the constructed designs. © 2005 Wiley Periodicals, Inc. J Combin Designs 14: 301–312, 2006

Construction of block designs admitting an Abelian automorphism group

2005-10-10
Dean Crnković , Sanja Rukavina

Symmetric (66, 26, 10) designs having Frob_55 as an automorphism group

2000-01-01
Dean Crnković , Sanja Rukavina
Up to isomorphism there are three symmetric (66; 26; 10) designs with automorphism group isomorphic to F rob55 . Among them there is one self-dual and one pair of dual … Up to isomorphism there are three symmetric (66; 26; 10) designs with automorphism group isomorphic to F rob55 . Among them there is one self-dual and one pair of dual designs. Full automorphism groups of dual designs are isomorphic to F rob55, and full automorphism group of the self-dual design is isomorphic to F rob55 D10. For those three designs corresponding derived and residual designs with respect to a block are constructed.

ENUMERATION OF SYMMETRIC (69,17,4) DESIGNS ADMITTING Z6 AS AN AUTOMORPHISM GROUP

1999-12-01
Sanja Rukavina
All symmetric (69,17,4) designs admitting the cyclic group of order 6 as an automorphism group are classified and their full automorphism groups are determined. All symmetric (69,17,4) designs admitting the cyclic group of order 6 as an automorphism group are classified and their full automorphism groups are determined.
Coauthor Papers Together
Dean Crnković 28
Vladimir D. Tonchev 11
Andrea Švob 8
Sara Ban 6
Bernardo Rodrigues 6
Marija Maksimović 4
Loredana Simčić 4
Blas Fernández 3
Marina Šimac 3
Doris Dumičić Danilović 3
Gabrijela Kehler-Poljak 1
Ljerka Jukić Matić 1
Marcel Schmidt 1
Nina Mostarac 1
Matteo Mravić 1