Vladimir D. Tonchev

Search for another author

Author Description

Ask a Question About This Mathematician

Entanglement-assisted quantum low-density parity-check codes

2010-10-29

Yuichiro Fujiwara , David Clark , Peter Vandendriessche +2 more

This article develops a general method for constructing entanglement-assisted quantum low-density parity-check (LDPC) codes, which is based on combinatorial design theory. Explicit constructions are given for entanglement-assisted quantum error-correcting codes … This article develops a general method for constructing entanglement-assisted quantum low-density parity-check (LDPC) codes, which is based on combinatorial design theory. Explicit constructions are given for entanglement-assisted quantum error-correcting codes with many desirable properties. These properties include the requirement of only one initial entanglement bit, high error-correction performance, high rates, and low decoding complexity. The proposed method produces several infinite families of codes with a wide variety of parameters and entanglement requirements. Our framework encompasses the previously known entanglement-assisted quantum LDPC codes having the best error-correction performance and many other codes with better block error rates in simulations over the depolarizing channel. We also determine important parameters of several well-known classes of quantum and classical LDPC codes for previously unsettled cases.

View PDF Chat

On symmetric and quasi-symmetric designs with the symmetric difference property and their codes

1992-01-01

Dieter Jungnickel , Vladimir D. Tonchev

Spreads in strongly regular graphs

1996-05-01

Willem H. Haemers , Vladimir D. Tonchev

View PDF Chat

Quasi-symmetric designs, codes, quadrics, and hyperplane sections

1993-12-01

Vladimir D. Tonchev

Classification of affine resolvable 2-(27, 9, 4) designs

1996-12-01

Clement Lam , Vladimir D. Tonchev

Polarities, quasi-symmetric designs, and Hamada’s conjecture

2008-11-26

Dieter Jungnickel , Vladimir D. Tonchev

Algebraic techniques in designing quantum synchronizable codes

2013-07-16

Yuichiro Fujiwara , Vladimir D. Tonchev , Tony W. H. Wong

Quantum synchronizable codes are quantum error-correcting codes that can correct the effects of quantum noise as well as block synchronization errors. We improve the previously known general framework for designing … Quantum synchronizable codes are quantum error-correcting codes that can correct the effects of quantum noise as well as block synchronization errors. We improve the previously known general framework for designing quantum synchronizable codes through more extensive use of the theory of finite fields. This makes it possible to widen the range of tolerable magnitude of block synchronization errors while giving mathematical insight into the algebraic mechanism of synchronization recovery. Also given are families of quantum synchronizable codes based on punctured Reed-Muller codes and their ambient spaces.

View PDF Chat

Unitals and codes

2002-12-27

Anton Betten , Dieter Betten , Vladimir D. Tonchev

Designs with the symmetric difference property on 64 points and their groups

1994-07-01

Christopher Parker , Edward Spence , Vladimir D. Tonchev

The number of designs with geometric parameters grows exponentially

2009-05-22

Dieter Jungnickel , Vladimir D. Tonchev

Maximal arcs and disjoint maximal arcs in projective planes of order 16

2000-03-01

Nicholas Hamilton , Stoicho D. Stoichev , Vladimir D. Tonchev

A characterization of designs related to the Witt systemS(5,8,24)

1986-06-01

Vladimir D. Tonchev

A Characterization of Entanglement-Assisted Quantum Low-Density Parity-Check Codes

2013-02-14

Yuichiro Fujiwara , Vladimir D. Tonchev

As in classical coding theory, quantum analogues of low-density parity-check (LDPC) codes have offered good error correction performance and low decoding complexity by employing the Calderbank-Shor-Steane (CSS) construction. However, special … As in classical coding theory, quantum analogues of low-density parity-check (LDPC) codes have offered good error correction performance and low decoding complexity by employing the Calderbank-Shor-Steane (CSS) construction. However, special requirements in the quantum setting severely limit the structures such quantum codes can have. While the entanglement-assisted stabilizer formalism overcomes this limitation by exploiting maximally entangled states (ebits), excessive reliance on ebits is a substantial obstacle to implementation. This paper gives necessary and sufficient conditions for the existence of quantum LDPC codes which are obtainable from pairs of identical LDPC codes and consume only one ebit, and studies the spectrum of attainable code parameters.

View PDF Chat

Bent Vectorial Functions, Codes and Designs

2019-06-13

Cunsheng Ding , Akihiro Munemasa , Vladimir D. Tonchev

Bent functions, or equivalently, Hadamard difference sets in the elementary Abelian group (GF(2 2m ), +), have been employed to construct symmetric and quasi-symmetric designs having the symmetric … Bent functions, or equivalently, Hadamard difference sets in the elementary Abelian group (GF(2 2m ), +), have been employed to construct symmetric and quasi-symmetric designs having the symmetric difference property. The main objective of this paper is to use bent vectorial functions for a construction of a two-parameter family of binary linear codes that do not satisfy the conditions of the Assmus-Mattson theorem, but nevertheless hold 2-designs. A new coding-theoretic characterization of bent vectorial functions is presented.

View PDF Chat

A formula for the number of Steiner quadruple systems on 2n points of 2‐rank 2n−n

2003-01-01

Vladimir D. Tonchev

Abstract Assmus [ 1 ] gave a description of the binary code spanned by the blocks of a Steiner triple or quadruple system according to the 2‐rank of the incidence … Abstract Assmus [ 1 ] gave a description of the binary code spanned by the blocks of a Steiner triple or quadruple system according to the 2‐rank of the incidence matrix. Using this description, the author [ 13 ] found a formula for the total number of distinct Steiner triple systems on 2 n −1 points of 2‐rank 2 n ‐ n . In this paper, a similar formula is found for the number of Steiner quadruple systems on 2 n points of 2‐rank 2 n ‐ n . The formula can be used for deriving bounds on the number of pairwise non‐isomorphic systems for large n , and for the classification of all non‐isomorphic systems of small orders. The formula implies that the number of non‐isomorphic Steiner quadruple systems on 2 n points of 2‐rank 2 n ‐ n grows exponentially. As an application, the Steiner quadruple systems on 16 points of 2‐rank 12 are classified up to isomorphism. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 260–274, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10036

View PDF Chat

A Mass Formula for Steiner Triple Systems STS(2n−1) of 2-Rank 2n−n

2001-08-01

Vladimir D. Tonchev

High-Rate Self-Synchronizing Codes

2013-03-13

Yuichiro Fujiwara , Vladimir D. Tonchev

Self-synchronization under the presence of additive noise can be achieved by allocating a certain number of bits of each codeword as markers for synchronization. Difference systems of sets are combinatorial … Self-synchronization under the presence of additive noise can be achieved by allocating a certain number of bits of each codeword as markers for synchronization. Difference systems of sets are combinatorial designs which specify the positions of synchronization markers in codewords in such a way that the resulting error-tolerant self-synchronizing codes may be realized as cosets of linear codes. Ideally, difference systems of sets should sacrifice as few bits as possible for a given code length, alphabet size, and error-tolerance capability. However, it seems difficult to attain optimality with respect to known bounds when the noise level is relatively low. In fact, the majority of known optimal difference systems of sets are for exceptionally noisy channels, requiring a substantial amount of bits for synchronization. To address this problem, we present constructions for difference systems of sets that allow for higher information rates while sacrificing optimality to only a small extent. Our constructions utilize optimal difference systems of sets as ingredients and, when applied carefully, generate asymptotically optimal ones with higher information rates. We also give direct constructions for optimal difference systems of sets with high information rates and error-tolerance that generate binary and ternary self-synchronizing codes.

View PDF Chat

Hadamard matrices of order 36 with automorphisms of order 17

1986-12-01

Vladimir D. Tonchev

A Hadamard matrix of order n is an n by n matrix of 1’s and − 1’s such that HH t − nI . In such a matrix n is … A Hadamard matrix of order n is an n by n matrix of 1’s and − 1’s such that HH t − nI . In such a matrix n is necessarily 1, 2 or a multiple of 4. Two Hadamard matrices H 1 and H 2 are called equivalent if there exist monomial matrices P, Q with PH 1 Q = H 2 . An automorphism of a Hadamard matrix H is an equivalence of the matrix to itself, i.e. a pair ( P, Q ) of monomial matrices such that PHQ = H . In other words, an automorphism of H is a permutation of its rows followed by multiplication of some rows by − 1, which leads to reordering of its columns and multiplication of some columns by − 1. The set of all automorphisms form a group under composition called the automorphism group (Aut H ) of H . For a detailed study of the basic properties and applications of Hadamard matrices see, e.g. [1], [7, Chap. 14], [8].

View PDF Chat

The [52, 26, 10] Binary Self-Dual Codes with an Automorphism of Order 7

2001-04-01

W. Cary Huffman , Vladimir D. Tonchev

Decompositions of Difference Sets

1999-07-01

Dieter Jungnickel , Vladimir D. Tonchev

On the binary codes of Steiner triple systems

1996-05-01

Alphonse Baartmans , Ivan Landjev , Vladimir D. Tonchev

Mutually disjoint designs and new 5‐designs derived from groups and codes

2010-02-19

Makoto Araya , Masaaki Harada , Vladimir D. Tonchev +1 more

Abstract The article gives constructions of disjoint 5‐designs obtained from permutation groups and extremal self‐dual codes. Several new simple 5‐designs are found with parameters that were left open in the … Abstract The article gives constructions of disjoint 5‐designs obtained from permutation groups and extremal self‐dual codes. Several new simple 5‐designs are found with parameters that were left open in the table of 5‐designs given in (G. B. Khosrovshahi and R. Laue, t ‐Designs with t ⩾3, in “Handbook of Combinatorial Designs”, 2nd edn, C. J. Colbourn and J. H. Dinitz (Editors), Chapman & Hall/CRC, Boca Raton, FL, 2007, pp. 79–101), namely, 5−( v, k , λ) designs with ( v, k , λ)=(18, 8, 2 m ) ( m =6, 9), (19, 9, 7 m ) ( m =6, 9), (24, 9, 6 m ) ( m =3, 4, 5), (25, 9, 30), (25, 10, 24 m ) ( m =4, 5), (26, 10, 126), (30, 12, 440), (32, 6, 3 m ) ( m =2, 3, 4), (33, 7, 84), and (36, 12, 45 n ) for 2⩽ n ⩽17. These results imply that a simple 5−( v, k , λ) design with ( v, k )=(24, 9), (25, 9), (26, 10), (32, 6), or (33, 7) exists for all admissible values of λ. © 2010 Wiley Periodicals, Inc. J Combin Designs 18: 305–317, 2010

The twisted Grassmann graph is the block graph of a design

2011-01-01

Akihiro Munemasa , Vladimir D. Tonchev

In this note, we show that the twisted Grassmann graph constructed by Van Dam and Koolen is the block graph of the design constructed by Jungnickel and Tonchev.We also show … In this note, we show that the twisted Grassmann graph constructed by Van Dam and Koolen is the block graph of the design constructed by Jungnickel and Tonchev.We also show that the full automorphism group of the design is isomorphic to that of the twisted Grassmann graph.

View PDF Chat

On Infinite Families of Narrow-Sense Antiprimitive BCH Codes Admitting 3-Transitive Automorphism Groups and Their Consequences

2021-12-30

Qi Liu , Cunsheng Ding , Sihem Mesnager +2 more

The Bose-Chaudhuri-Hocquenghem (BCH) codes are a well-studied subclass of cyclic codes that have found numerous applications in error correction and notably in quantum information processing. They are widely used in … The Bose-Chaudhuri-Hocquenghem (BCH) codes are a well-studied subclass of cyclic codes that have found numerous applications in error correction and notably in quantum information processing. They are widely used in data storage and communication systems. A subclass of attractive BCH codes is the narrow-sense BCH codes over the Galois field <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathrm {GF}}(q)$ </tex-math></inline-formula> with length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q+1$ </tex-math></inline-formula> , which are closely related to the action of the projective general linear group of degree two on the projective line. Despite its interest, not much is known about this class of BCH codes. This paper aims to study some of the codes within this class and specifically narrow-sense antiprimitive BCH codes (these codes are also linear complementary duals (LCD) codes that have interesting practical recent applications in cryptography, among other benefits). We shall use tools and combine arguments from algebraic coding theory, combinatorial designs, and group theory (group actions, representation theory of finite groups, etc.) to investigate narrow-sense antiprimitive BCH Codes and extend results from the recent literature. Notably, the dimension, the minimum distance of some <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> -ary BCH codes with length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q+1$ </tex-math></inline-formula> , and their duals are determined in this paper. The dual codes of the narrow-sense antiprimitive BCH codes derived in this paper include almost MDS codes. Furthermore, the classification of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathrm {PGL}}(2, p^{m})$ </tex-math></inline-formula> -invariant codes over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathrm {GF}}(p^{h})$ </tex-math></inline-formula> is completed. As an application of this result, the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> -ranks of all incidence structures invariant under the projective general linear group <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathrm {PGL}}(2, p^{m})$ </tex-math></inline-formula> are determined. Furthermore, infinite families of narrow-sense BCH codes admitting a 3-transitive automorphism group are obtained. Via these BCH codes, a coding-theory approach to constructing the Witt spherical geometry designs is presented. The BCH codes proposed in this paper are good candidates for permutation decoding, as they have a relatively large group of automorphisms.

View PDF Chat

Affine geometry designs, polarities, and Hamada's conjecture

2010-07-17

David Clark , Dieter Jungnickel , Vladimir D. Tonchev

On affine designs and Hadamard designs with line spreads

2007-06-07

V. C. Mavron , T. P. McDonough , Vladimir D. Tonchev

View PDF Chat

The uniformly packed binary [27,21,3] and [35,29,3] codes

1996-02-01

Vladimir D. Tonchev

Automorphisms of 2-(22, 8, 4) designs

1989-09-01

I. Landgev , Vladimir D. Tonchev

Counting Steiner triple systems with classical parameters and prescribed rank

2018-10-02

Dieter Jungnickel , Vladimir D. Tonchev

View PDF Chat

Linear codes of 2-designs associated with subcodes of the ternary generalized Reed–Muller codes

2019-12-11

Cunsheng Ding , Chunming Tang , Vladimir D. Tonchev

View PDF Chat

Cyclic 2-(91, 6, 1) designs with multiplier automorphisms

1991-12-01

Zvonimir Janko , Vladimir D. Tonchev

Classification of Generalized Hadamard Matrices $H(6,3)$ and Quaternary Hermitian Self-Dual Codes of Length 18

2010-12-10

Masaaki Harada , Clement Lam , Akihiro Munemasa +1 more

All generalized Hadamard matrices of order 18 over a group of order 3, $H(6,3)$, are enumerated in two different ways: once, as class regular symmetric $(6,3)$-nets, or symmetric transversal designs … All generalized Hadamard matrices of order 18 over a group of order 3, $H(6,3)$, are enumerated in two different ways: once, as class regular symmetric $(6,3)$-nets, or symmetric transversal designs on 54 points and 54 blocks with a group of order 3 acting semi-regularly on points and blocks, and secondly, as collections of full weight vectors in quaternary Hermitian self-dual codes of length 18. The second enumeration is based on the classification of Hermitian self-dual $[18,9]$ codes over $GF(4)$, completed in this paper. It is shown that up to monomial equivalence, there are 85 generalized Hadamard matrices $H(6,3)$, and 245 inequivalent Hermitian self-dual codes of length 18 over $GF(4)$.

View PDF Chat

The classification of antipodal two-weight linear codes

2018-01-04

Dieter Jungnickel , Vladimir D. Tonchev

The classification of Steiner triple systems on 27 points with 3-rank 24

2018-06-15

Dieter Jungnickel , Spyros S. Magliveras , Vladimir D. Tonchev +1 more

On Pless symmetry codes, ternary QR codes, and related Hadamard matrices and designs

2021-11-11

Vladimir D. Tonchev

View PDF Chat

On Kirkman triple systems of order 33

1992-09-01

Vladimir D. Tonchev , Scott A. Vanstone

Unitals in the H�lz design on 28 points

1991-06-01

Vladimir D. Tonchev

Maximal arcs and extended cyclic codes

2018-07-06

Stefaan De Winter , Cunsheng Ding , Vladimir D. Tonchev

View PDF Chat

Maximal arcs in projective planes of order 16 and related designs

2018-03-22

Mustafa Gezek , Vladimir D. Tonchev , Tim Wagner

Abstract The resolutions and maximal sets of compatible resolutions of all 2-(120,8,1) designs arising from maximal (120,8)-arcs, and the 2-(52,4,1) designs arising from previously known maximal (52,4)-arcs, as well as … Abstract The resolutions and maximal sets of compatible resolutions of all 2-(120,8,1) designs arising from maximal (120,8)-arcs, and the 2-(52,4,1) designs arising from previously known maximal (52,4)-arcs, as well as some newly discovered maximal (52,4)-arcs in the known projective planes of order 16, are computed. It is shown that each 2-(120,8,1) design associated with a maximal (120,8)-arc is embeddable in a unique way in a projective plane of order 16. This result suggests a possible strengthening of the Bose–Shrikhande theorem about the embeddability of the complement of a hyperoval in a projective plane of even order. The computations of the maximal sets of compatible resolutions of the 2-(52,4,1) designs associated with maximal (52,4)-arcs show that five of the known projective planes of order 16 contain maximal arcs whose associated designs are embeddable in two nonisomorphic planes of order 16.

View PDF Chat

Embedding of the Witt-Mathieu system S(3, 6, 22) in a symmetric 2-(78, 22, 6) design

1987-01-01

Vladimir D. Tonchev

On generalized Hadamard matrices of minimum rank

2004-01-16

Vladimir D. Tonchev

The projective general linear group $${\mathrm {PGL}}(2,2^m)$$ and linear codes of length $$2^m+1$$

2021-05-21

Cunsheng Ding , Chunming Tang , Vladimir D. Tonchev

Computational results for the known biplanes of order 9

1997-08-14

J. D. Key , Vladimir D. Tonchev

The ternary codes associated with the five known biplanes of order 9 were examined using the computer language Magma. The computations showed that each biplane is the only one to … The ternary codes associated with the five known biplanes of order 9 were examined using the computer language Magma. The computations showed that each biplane is the only one to be found among the weight-11 vectors of its ternary code, and that none of the biplanes can be extended to a 3-(57,12,2) design. The residual designs of the biplanes, and designs associated with {12; 3}-arcs were also examined.

Linear codes and doubly transitive symmetric designs

1995-09-01

Christopher Parker , Vladimir D. Tonchev

Transitive Steiner triple systems of order 25

1987-11-01

Vladimir D. Tonchev

New Designs with Block Size 7

1998-07-01

Zvonimir Janko , Vladimir D. Tonchev

The isomorphism of the Cohen, Haemers-van Lint and De Clerck-Dye-Thas partial geometries

1984-04-01

Vladimir D. Tonchev

On resolvable Steiner 2-designs and maximal arcs in projective planes

2016-07-02

Vladimir D. Tonchev

View PDF Chat

Maximal Arcs, Codes, and New Links Between Projective Planes of Order 16

2020-03-19

Mustafa Gezek , Rudi Mathon , Vladimir D. Tonchev

In this paper we consider binary linear codes spanned by incidence matrices of Steiner 2-designs associated with maximal arcs in projective planes of even order, and their dual codes. Upper … In this paper we consider binary linear codes spanned by incidence matrices of Steiner 2-designs associated with maximal arcs in projective planes of even order, and their dual codes. Upper and lower bounds on the 2-rank of the incidence matrices are derived. A lower bound on the minimum distance of the dual codes is proved, and it is shown that the bound is achieved if and only if the related maximal arc contains a hyperoval of the plane. The binary linear codes of length 52 spanned by the incidence matrices of 2-$(52,4,1)$ designs associated with previously known and some newly found maximal arcs of degree 4 in projective planes of order 16 are analyzed and classified up to equivalence. The classification shows that some designs associated with maximal arcs in nonisomorphic planes generate equivalent codes. This phenomenon establishes new links between several of the known planes. A conjecture concerning the codes of maximal arcs in $PG(2,2^m)$ is formulated.

View PDF Chat

Strongly regular graphs with parameters (81, 30, 9, 12) and a new partial geometry

2021-01-30

Dean Crnković , Andrea Švob , Vladimir D. Tonchev

View PDF Chat

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.

View PDF Chat

On Projective Planes of Order 16 Associated with 1-rotational 2-(52, 4, 1) Designs

2024-03-05

Zazil Santizo Huerta , Melissa Keranen , Vladimir D. Tonchev

A maximal arc of degree k in a finite projective plane P of order q = ks is a set of (q-s+1)k points that meets every line of P in … A maximal arc of degree k in a finite projective plane P of order q = ks is a set of (q-s+1)k points that meets every line of P in either k or 0 points. The collection of the nonempty intersections of a maximal arc with the lines of P is a resolvable Steiner 2-((q-s+1)k, k, 1) design. Necessary and sufficient conditions for a resolvable Steiner 2- design to be embeddable as a maximal arc in a projective plane were proved recently in [8]. Steiner designs associated with maximal arcs in the known projective planes of order 16 were analyzed in [6], where it was shown that some of the associated designs are embeddable in two non-isomorphic planes. Using MAGMA, we conducted an analysis to ascertain whether any of the 22 non-isomorphic 1-rotational 2-(52,4,1) designs, previously classified in [3], could be embedded in maximal arcs of degree 4 within projective planes of order 16. This paper presents a summary of our findings, revealing that precisely only one out of the the twenty-two 1-rotational designs from [3] is embeddable in a plane of order 16, being the Desarguesian plane P G(2, 16).

View PDF Chat

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.

View PDF Chat

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.

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

View PDF Chat

Hadamard matrices of orders 60 and 64 with automorphisms of orders 29 and 31

2023-01-01

Makoto Araya , Masaaki Harada , Vladimir D. Tonchev

A classification of Hadamard matrices of order $2p+2$ with an automorphism of order $p$ is given for $p=29$ and $31$. The ternary self-dual codes spanned by the newly found Hadamard … A classification of Hadamard matrices of order $2p+2$ with an automorphism of order $p$ is given for $p=29$ and $31$. The ternary self-dual codes spanned by the newly found Hadamard matrices of order $60$ with an automorphism of order $29$ are computed, as well as the binary doubly even self-dual codes of length $120$ with generator matrices defined by related Hadamard designs. Several new ternary near-extremal self-dual codes, as well as binary near-extremal doubly even self-dual codes with previously unknown weight enumerators are found.

On optimal constant weight codes derived from $ω$-circulant balanced generalized weighing matrices

2023-01-01

Hadi Kharaghani , Thomas Pender , Vladimir D. Tonchev

A family of $\omega$-circulant balanced weighing matrices with classical parameters is used for the construction of optimal constant weight codes over an alphabet of size $g+1$ and length $n=(q^m -1)/(q-1)$, … A family of $\omega$-circulant balanced weighing matrices with classical parameters is used for the construction of optimal constant weight codes over an alphabet of size $g+1$ and length $n=(q^m -1)/(q-1)$, where $q$ is an odd prime power, $m>1$, and $g$ is a divisor of $q-1$.

View PDF Chat

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 $A_{12}$.

View PDF Chat

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

View PDF Chat

On the classification of unitals on 28 points of low rank

2022-02-08

Vladimir D. Tonchev , Alfred Wassermann

Abstract The classification of unitals with parameters 2-(28, 4, 1) according to the 2-rank of their incidence matrices was initiated by McGuire, Tonchev and Ward, who proved that the 2-rank … Abstract The classification of unitals with parameters 2-(28, 4, 1) according to the 2-rank of their incidence matrices was initiated by McGuire, Tonchev and Ward, who proved that the 2-rank of any unital on 28 points is greater than or equal to 19, and up to isomorphism, there is a unique unital with 2-rank equal to 19. Jaffe and Tonchev investigated the next two 2-ranks, 20 and 21, and showed that there are no unitals on 28 points with 2-rank equal to 20, and there are exactly 4 isomorphism classes of unitals of rank 21. The subject of this paper is the classification of unitals having 2-rank 22, 23 and 24.

View PDF Chat

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.

View PDF Chat

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.

View PDF Chat

On Infinite Families of Narrow-Sense Antiprimitive BCH Codes Admitting 3-Transitive Automorphism Groups and Their Consequences

2021-12-30

Qi Liu , Cunsheng Ding , Sihem Mesnager +2 more

View PDF Chat

On Pless symmetry codes, ternary QR codes, and related Hadamard matrices and designs

2021-11-11

Vladimir D. Tonchev

View PDF Chat

On Pless symmetry codes, ternary QR codes, and related Hadamard matrices and designs.

2021-09-12

Vladimir D. Tonchev

It is proved that a code $L(q)$ which is monomially equivalent to the Pless symmetry code $C(q)$ of length $2q+2$ contains the (0,1)-incidence matrix of a Hadamard 3-$(2q+2,q+1,(q-1)/2)$ design $D(q)$ … It is proved that a code $L(q)$ which is monomially equivalent to the Pless symmetry code $C(q)$ of length $2q+2$ contains the (0,1)-incidence matrix of a Hadamard 3-$(2q+2,q+1,(q-1)/2)$ design $D(q)$ associated with a Paley-Hadamard matrix of type II. Similarly, any ternary extended quadratic residue code contains the incidence matrix of a Hadamard 3-design associated with a Paley-Hadamard matrix of type I. If $q=5, 11, 17, 23$, then the full permutation automorphism group of $L(q)$ coincides with the full automorphism group of $D(q)$, and a similar result holds for the ternary extended quadratic residue codes of lengths 24 and 48. All Hadamard matrices of order 36 formed by codewords of the Pless symmetry code $C(17)$ are enumerated and classified up to equivalence. There are two equivalence classes of such matrices: the Paley-Hadamard matrix $H$ of type I with a full automorphism group of order 19584, and a second regular Hadamard matrix $H'$ such that the symmetric 2-$(36,15,6)$ design $D$ associated with $H'$ has trivial full automorphism group, and the incidence matrix of $D$ spans a ternary code equivalent to $C(17)$.

The projective general linear group $${\mathrm {PGL}}(2,2^m)$$ and linear codes of length $$2^m+1$$

2021-05-21

Cunsheng Ding , Chunming Tang , Vladimir D. Tonchev

Strongly regular graphs with parameters (81, 30, 9, 12) and a new partial geometry

2021-01-30

Dean Crnković , Andrea Švob , Vladimir D. Tonchev

View PDF Chat

On partial geometries arising from maximal arcs

2021-01-07

Mustafa Gezek , Vladimir D. Tonchev

View PDF Chat

On Pless symmetry codes, ternary QR codes, and related Hadamard matrices and designs

2021-01-01

Vladimir D. Tonchev

View PDF Chat

On Infinite Families of Narrow-Sense Antiprimitive BCH Codes Admitting 3-Transitive Automorphism Groups and their Consequences

2021-01-01

Qi Liu , Cunsheng Ding , Sihem Mesnager +2 more

On partial geometries arising from maximal arcs

2020-08-30

Mustafa Gezek , Vladimir D. Tonchev

The subject of this paper are partial geometries $pg(s,t,\alpha)$ with parameters $s=d(d'-1), t=d'(d-1), \alpha=(d-1)(d'-1)$, $d, d' \ge 2$. In all known examples, $q=dd'$ is a power of 2 and the … The subject of this paper are partial geometries $pg(s,t,\alpha)$ with parameters $s=d(d'-1), t=d'(d-1), \alpha=(d-1)(d'-1)$, $d, d' \ge 2$. In all known examples, $q=dd'$ is a power of 2 and the partial geometry arises from a maximal arc of degree $d$ or $d'$ in a projective plane of order $q$ via a known construction due to Thas \cite{Thas73} and Wallis \cite{W}, with a single known exception of a partial geometry $pg(4,6,3)$ found by Mathon \cite{Math} that is not associated with a maximal arc in the projective plane of order 8. A parallel class of lines is a set of pairwise disjoint lines that covers the point set. Two parallel classes are called orthogonal if they share exactly one line. An upper bound on the maximum number of pairwise orthogonal parallel classes in a partial geometry $G$ with parameters $pg(d(d'-1),d'(d-1),(d-1)(d'-1))$ is proved, and it is shown that a necessary and sufficient condition for $G$ to arise from a maximal arc of degree $d$ or $d'$ in a projective plane of order $q=dd'$ is that both $G$ and its dual geometry contain sets of pairwise orthogonal parallel classes that meet the upper bound. An alternative construction of Mathon's partial geometry is presented, and the new necessary condition is used to demonstrate why this partial geometry is not associated with any maximal arc in the projective plane of order 8. The partial geometries associated with all known maximal arcs in projective planes of order 16 are classified up to isomorphism, and their parallel classes of lines and the 2-rank of their incidence matrices are computed. Based on these results, some open problems and conjectures are formulated.

Ternary codes, biplanes, and the nonexistence of some quasisymmetric and quasi‐3 designs

2020-06-22

Akihiro Munemasa , Vladimir D. Tonchev

Abstract The dual codes of the ternary linear codes of the residual designs of biplanes on 56 points are used to prove the nonexistence of quasisymmetric 2‐ and 2‐ designs … Abstract The dual codes of the ternary linear codes of the residual designs of biplanes on 56 points are used to prove the nonexistence of quasisymmetric 2‐ and 2‐ designs with intersection numbers 0 and 3, and the nonexistence of a 2‐ quasi‐3 design. The nonexistence of a 2‐ quasi‐3 design is also proved.

View PDF Chat

Maximal Arcs, Codes, and New Links Between Projective Planes of Order 16

2020-03-19

Mustafa Gezek , Rudi Mathon , Vladimir D. Tonchev

View PDF Chat

Ternary codes, biplanes, and the nonexistence of some quasi-symmetric and quasi-3 designs

2020-03-09

Akihiro Munemasa , Vladimir D. Tonchev

The dual codes of the ternary linear codes of the residual designs of biplanes on 56 points are used to prove the nonexistence of quasi-symmetric 2-$(56,12,9)$ and 2-$(57,12,11)$ designs with … The dual codes of the ternary linear codes of the residual designs of biplanes on 56 points are used to prove the nonexistence of quasi-symmetric 2-$(56,12,9)$ and 2-$(57,12,11)$ designs with intersection numbers 0 and 3, and the nonexistence of a 2-$(267,57,12)$ quasi-3 design. The nonexistence of a 2-$(149,37,9)$ quasi-3 design is also proved.

Maximal arcs, codes, and new links between projective planes

2020-01-01

Mustafa Gezek , Rudi Mathon , Vladimir D. Tonchev

Strongly regular graphs with parameters (81,30,9,12) and a new partial geometry pg(5,5,2)

2020-01-01

Dean Crnković , Andrea Švob , Vladimir D. Tonchev

Twelve new strongly regular graphs with parameters (81,30,9,12) are found as graphs invariant under certain subgroups of the automorphism groups of the two previously known graphs that arise from 2-weight … Twelve new strongly regular graphs with parameters (81,30,9,12) are found as graphs invariant under certain subgroups of the automorphism groups of the two previously known graphs that arise from 2-weight codes. One of these new graphs is geometric and yields a partial geometry with parameters pg(5,5,2) that is not isomorphic to the partial geometry discovered by J. H. van Lint and A. Schrijver in 1981.

The Projective General Linear Group $\mathrm{PGL}_2(\mathrm{GF}(2^m))$ and Linear Codes of Length $2^m+1$

2020-01-01

Cunsheng Ding , Chunming Tang , Vladimir D. Tonchev

The projective general linear group $\mathrm{PGL}_2(\mathrm{GF}(2^m))$ acts as a $3$-transitive permutation group on the set of points of the projective line. The first objective of this paper is to prove … The projective general linear group $\mathrm{PGL}_2(\mathrm{GF}(2^m))$ acts as a $3$-transitive permutation group on the set of points of the projective line. The first objective of this paper is to prove that all linear codes over $\mathrm{GF}(2^h)$ that are invariant under $\mathrm{PGL}_2(\mathrm{GF}(2^m))$ are trivial codes: the repetition code, the whole space $\mathrm{GF}(2^h)^{2^m+1}$, and their dual codes. As an application of this result, the $2$-ranks of the (0,1)-incidence matrices of all $3$-$(q+1,k,\lambda)$ designs that are invariant under $\mathrm{PGL}_2(\mathrm{GF}(2^m))$ are determined. The second objective is to present two infinite families of cyclic codes over $\mathrm{GF}(2^m)$ such that the set of the supports of all codewords of any fixed nonzero weight is invariant under $\mathrm{PGL}_2(\mathrm{GF}(2^m))$, therefore, the codewords of any nonzero weight support a 3-design. A code from the first family has parameters $[q+1,q-3,4]_q$, where $q=2^m$, and $m\ge 4$ is even. The exact number of the codewords of minimum weight is determined, and the codewords of minimum weight support a 3-$(q+1,4,2)$ design. A code from the second family has parameters $[q+1,4,q-4]_q$, $q=2^m$, $m\ge 4$ even, and the minimum weight codewords support a 3-$(q +1,q-4,(q-4)(q-5)(q-6)/60)$ design, whose complementary 3-$(q +1, 5, 1)$ design is isomorphic to the Witt spherical geometry with these parameters. A lower bound on the dimension of a linear code over $\mathrm{GF}(q)$ that can support a 3-$(q +1,q-4,(q-4)(q-5)(q-6)/60)$ design is proved, and it is shown that the designs supported by the codewords of minimum weight in the codes from the second family of codes meet this bound.

On partial geometries arising from maximal arcs

2020-01-01

Mustafa Gezek , Vladimir D. Tonchev

The subject of this paper are partial geometries $pg(s,t,\alpha)$ with parameters $s=d(d'-1), \ t=d'(d-1), \ \alpha=(d-1)(d'-1)$, $d, d' \ge 2$. In all known examples, $q=dd'$ is a power of 2 … The subject of this paper are partial geometries $pg(s,t,\alpha)$ with parameters $s=d(d'-1), \ t=d'(d-1), \ \alpha=(d-1)(d'-1)$, $d, d' \ge 2$. In all known examples, $q=dd'$ is a power of 2 and the partial geometry arises from a maximal arc of degree $d$ or $d'$ in a projective plane of order $q$ via a known construction due to Thas \cite{Thas73} and Wallis \cite{W}, with a single known exception of a partial geometry $pg(4,6,3)$ found by Mathon \cite{Math} that is not associated with a maximal arc in the projective plane of order 8. A parallel class of lines is a set of pairwise disjoint lines that covers the point set. Two parallel classes are called orthogonal if they share exactly one line. An upper bound on the maximum number of pairwise orthogonal parallel classes in a partial geometry $G$ with parameters $pg(d(d'-1),d'(d-1),(d-1)(d'-1))$ is proved, and it is shown that a necessary and sufficient condition for $G$ to arise from a maximal arc of degree $d$ or $d'$ in a projective plane of order $q=dd'$ is that both $G$ and its dual geometry contain sets of pairwise orthogonal parallel classes that meet the upper bound. An alternative construction of Mathon's partial geometry is presented, and the new necessary condition is used to demonstrate why this partial geometry is not associated with any maximal arc in the projective plane of order 8. The partial geometries associated with all known maximal arcs in projective planes of order 16 are classified up to isomorphism, and their parallel classes of lines and the 2-rank of their incidence matrices are computed. Based on these results, some open problems and conjectures are formulated.

Ternary codes, biplanes, and the nonexistence of some quasi-symmetric and quasi-3 designs

2020-01-01

Akihiro Munemasa , Vladimir D. Tonchev

Linear codes of 2-designs associated with subcodes of the ternary generalized Reed–Muller codes

2019-12-11

Cunsheng Ding , Chunming Tang , Vladimir D. Tonchev

View PDF Chat

Bent Vectorial Functions, Codes and Designs

2019-06-13

Cunsheng Ding , Akihiro Munemasa , Vladimir D. Tonchev

View PDF Chat

Linear codes of 2-designs associated with subcodes of the ternary generalized Reed-Muller codes

2019-01-01

Cunsheng Ding , Chunming Tang , Vladimir D. Tonchev

In this paper, the 3-rank of the incidence matrices of 2-designs supported by the minimum weight codewords in a family of ternary linear codes considered in [C. Ding, C. Li, … In this paper, the 3-rank of the incidence matrices of 2-designs supported by the minimum weight codewords in a family of ternary linear codes considered in [C. Ding, C. Li, Infinite families of 2-designs and 3-designs from linear codes, Discrete Mathematics 340(10) (2017) 2415--2431] are computed. A lower bound on the minimum distance of the ternary codes spanned by the incidence matrices of these designs is derived, and it is proved that the codes are subcodes of the 4th order generalized Reed-Muller codes.

View PDF Chat

Counting Steiner triple systems with classical parameters and prescribed rank

2018-10-02

Dieter Jungnickel , Vladimir D. Tonchev

View PDF Chat

Maximal arcs and extended cyclic codes

2018-07-06

Stefaan De Winter , Cunsheng Ding , Vladimir D. Tonchev

View PDF Chat

The classification of Steiner triple systems on 27 points with 3-rank 24

2018-06-15

Dieter Jungnickel , Spyros S. Magliveras , Vladimir D. Tonchev +1 more

Maximal arcs in projective planes of order 16 and related designs

2018-03-22

Mustafa Gezek , Vladimir D. Tonchev , Tim Wagner

View PDF Chat

The classification of antipodal two-weight linear codes

2018-01-04

Dieter Jungnickel , Vladimir D. Tonchev

Bent Vectorial Functions, Codes and Designs

2018-01-01

Cunsheng Ding , Akihiro Munemasa , Vladimir D. Tonchev

Bent functions, or equivalently, Hadamard difference sets in the elementary Abelian group $(\gf(2^{2m}), +)$, have been employed to construct symmetric and quasi-symmetric designs having the symmetric difference property. The main … Bent functions, or equivalently, Hadamard difference sets in the elementary Abelian group $(\gf(2^{2m}), +)$, have been employed to construct symmetric and quasi-symmetric designs having the symmetric difference property. The main objective of this paper is to use bent vectorial functions for a construction of a two-parameter family of binary linear codes that do not satisfy the conditions of the Assmus-Mattson theorem, but nevertheless hold $2$-designs. A new coding-theoretic characterization of bent vectorial functions is presented.

Counting Steiner triple systems with classical parameters and prescribed rank

2017-09-18

Dieter Jungnickel , Vladimir D. Tonchev

By a famous result of Doyen, Hubaut and Vandensavel \cite{DHV}, the 2-rank of a Steiner triple system on $2^n-1$ points is at least $2^n -1 -n$, and equality holds only … By a famous result of Doyen, Hubaut and Vandensavel \cite{DHV}, the 2-rank of a Steiner triple system on $2^n-1$ points is at least $2^n -1 -n$, and equality holds only for the classical point-line design in the projective geometry $PG(n-1,2)$. It follows from results of Assmus \cite{A} that, given any integer $t$ with $1 \leq t \leq n-1$, there is a code $C_{n,t}$ containing representatives of all isomorphism classes of STS$(2^n-1)$ with 2-rank at most $2^n -1 -n + t$. Using a mixture of coding theoretic, geometric, design theoretic and combinatorial arguments, we prove a general formula for the number of distinct STS$(2^n-1)$ with 2-rank at most $2^n -1 -n + t$ contained in this code. This generalizes the only previously known cases, $t=1$, proved by Tonchev \cite{T01} in 2001, $t=2$, proved by V. Zinoviev and D. Zinoviev \cite{ZZ12} in 2012, and $t=3$ (V. Zinoviev and D. Zinoviev \cite{ZZ13}, \cite{ZZ13a} (2013), D. Zinoviev \cite{Z16} (2016)), while also unifying and simplifying the proofs. This enumeration result allows us to prove lower and upper bounds for the number of isomorphism classes of STS$(2^n-1)$ with 2-rank exactly (or at most) $2^n -1 -n + t$. Finally, using our recent systematic study of the ternary block codes of Steiner triple systems \cite{JT}, we obtain analogous results for the ternary case, that is, for STS$(3^n)$ with 3-rank at most (or exactly) $3^n -1 -n + t$. We note that this work provides the first two infinite families of 2-designs for which one has non-trivial lower and upper bounds for the number of non-isomorphic examples with a prescribed $p$-rank in almost the entire range of possible ranks.

All Binary Linear Codes That Are Invariant Under <inline-formula> <tex-math notation="LaTeX">${\mathrm{PSL}}_2(n)$ </tex-math> </inline-formula>

2017-08-30

Cunsheng Ding , Hao Liu , Vladimir D. Tonchev

The projective special linear group PSL 2 (n) is 2-transitive for all primes n and 3-homogeneous for n = 3 (mod 4) on the set (0, 1, ... … The projective special linear group PSL 2 (n) is 2-transitive for all primes n and 3-homogeneous for n = 3 (mod 4) on the set (0, 1, ... , n - 1, ∞). It is known that the extended odd-like quadratic residue codes are invariant under PSL 2 (n). Hence, the extended quadratic residue codes hold an infinite family of 2-designs for primes n = 1 (mod 4), an infinite family of 3-designs for primes n = 3 (mod 4). To construct more t-designs with t ∈ (2, 31, one would search for other extended cyclic codes over finite fields that are invariant under the action of PSL 2 (n). The objective of this paper is to prove that the extended quadratic residue binary codes are the only nontrivial extended binary cyclic codes that are invariant under PSL 2 (n).

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

Self‐Dual Codes and the Nonexistence of a Quasi‐Symmetric 2‐(37,9,8) Design with Intersection Numbers 1 and 3

2017-03-21

Masaaki Harada , Akihiro Munemasa , Vladimir D. Tonchev

Abstract We prove that a certain binary linear code associated with the incidence matrix of a quasi‐symmetric 2‐(37, 9, 8) design with intersection numbers 1 and 3 must be contained … Abstract We prove that a certain binary linear code associated with the incidence matrix of a quasi‐symmetric 2‐(37, 9, 8) design with intersection numbers 1 and 3 must be contained in an extremal doubly even self‐dual code of length 40. Using the classification of extremal doubly even self‐dual codes of length 40, we show that a quasi‐symmetric 2‐(37, 9, 8) design with intersection numbers 1 and 3 does not exist.

View PDF Chat

Extension sets, affine designs, and Hamada’s conjecture

2017-03-01

Dieter Jungnickel , Yue Zhou , Vladimir D. Tonchev

All binary linear codes that are invariant under $\PSL_2(n)$

2017-01-01

Cunsheng Ding , Hao Liu , Vladimir D. Tonchev

The projective special linear group $\PSL_2(n)$ is $2$-transitive for all primes $n$ and $3$-homogeneous for $n \equiv 3 \pmod{4}$ on the set $\{0,1, \cdots, n-1, \infty\}$. It is known that … The projective special linear group $\PSL_2(n)$ is $2$-transitive for all primes $n$ and $3$-homogeneous for $n \equiv 3 \pmod{4}$ on the set $\{0,1, \cdots, n-1, \infty\}$. It is known that the extended odd-like quadratic residue codes are invariant under $\PSL_2(n)$. Hence, the extended quadratic residue codes hold an infinite family of $2$-designs for primes $n \equiv 1 \pmod{4}$, an infinite family of $3$-designs for primes $n \equiv 3 \pmod{4}$. To construct more $t$-designs with $t \in \{2, 3\}$, one would search for other extended cyclic codes over finite fields that are invariant under the action of $\PSL_2(n)$. The objective of this paper is to prove that the extended quadratic residue binary codes are the only nontrivial extended binary cyclic codes that are invariant under $\PSL_2(n)$.

View PDF Chat

On Classifying Steiner Triple Systems by Their 3-Rank

2017-01-01

Dieter Jungnickel , Spyros S. Magliveras , Vladimir D. Tonchev +1 more

Maximal arcs and extended cyclic codes

2017-01-01

Stefaan De Winter , Cunsheng Ding , Vladimir D. Tonchev

It is proved that for every $d\ge 2$ such that $d-1$ divides $q-1$, where $q$ is a power of 2, there exists a Denniston maximal arc $A$ of degree $d$ … It is proved that for every $d\ge 2$ such that $d-1$ divides $q-1$, where $q$ is a power of 2, there exists a Denniston maximal arc $A$ of degree $d$ in $\PG(2,q)$, being invariant under a cyclic linear group that fixes one point of $A$ and acts regularly on the set of the remaining points of ${A}$. Two alternative proofs are given, one geometric proof based on Abatangelo-Larato's characterization of Denniston arcs, and a second coding-theoretical proof based on cyclotomy and the link between maximal arcs and two-weight codes.

View PDF Chat

Counting Steiner triple systems with classical parameters and prescribed rank

2017-01-01

Dieter Jungnickel , Vladimir D. Tonchev

View PDF Chat

Maximal (120,8)-arcs in projective planes of order 16 and related designs

2017-01-01

Vladimir D. Tonchev , Tim Wagner

The resolutions and maximal sets of compatible resolutions of all 2-(120,8,1) designs arising frommaximal (120,8)-arcs in the known projective planes of order 16 are computed. It is shown that each … The resolutions and maximal sets of compatible resolutions of all 2-(120,8,1) designs arising frommaximal (120,8)-arcs in the known projective planes of order 16 are computed. It is shown that each of these designs is embeddable in a unique way in a projective plane of order 16.

View PDF Chat

Linearly embeddable designs

2016-11-18

Vladimir D. Tonchev

View PDF Chat

The weight distribution of the self-dual $[128,64]$ polarity design code

2016-08-01

Vladimir D. Tonchev , Ethan Novak , Masaaki Harada

The weight distribution of the binary self-dual $[128,64]$ code being the extended code $C^{*}$ of the code $C$ spanned by the incidence vectors of the blocks of the polarity design … The weight distribution of the binary self-dual $[128,64]$ code being the extended code $C^{*}$ of the code $C$ spanned by the incidence vectors of the blocks of the polarity design in $PG(6,2)$ [11] is computed. It is shown also that $R(3,7)$ and $C^{*}$ have no self-dual $[128,64,d]$ neighbor with $d \in \{ 20, 24 \}$.

View PDF Chat

Coauthor	Papers Together
Dieter Jungnickel	17
Cunsheng Ding	12
Sanja Rukavina	11
Akihiro Munemasa	11
David Clark	7
Mustafa Gezek	6
Chunming Tang	6
Masaaki Harada	5
Dean Crnković	5
Yuichiro Fujiwara	4
Spyros S. Magliveras	4
Bernardo Rodrigues	4
Clement Lam	4
Alfred Wassermann	4
Edward Spence	3
Ivan Landjev	2
Rudi Mathon	2
Alphonse Baartmans	2
Andrea Švob	2
Zvonimir Janko	2
Stefaan De Winter	2
I. Landgev	2
Christopher Parker	2
Sihem Mesnager	2
Makoto Araya	2
Willem H. Haemers	2
Chekad Sarami	1
Tran van Trung	1
Leo G. Chouinard	1
Gordon Royle	1
Nicholas Hamilton	1
Peter J. Dukes	1
Earl S. Kramer	1
Peter Vandendriessche	1
Anton Betten	1
Zazil Santizo Huerta	1
Qi Liu	1
Hao Liu	1
Hao Liu	1
Qi Liu	1
Andries E. Brouwer	1
Hugh Williams	1
Masaaki Harada	1
L. Thiel	1
Scott A. Vanstone	1
Jeffrey H. Dinitz	1
V. C. Mavron	1
Richard Wilson	1
T. P. McDonough	1
Michel Deza	1