This paper delves into the classification of regular semi groups and their various sub classifications based on their interior ideals. By introducing and examining concepts such as strongly prime, prime, …
This paper delves into the classification of regular semi groups and their various sub classifications based on their interior ideals. By introducing and examining concepts such as strongly prime, prime, semi prime, strongly irreducible, and irreducible interior ideals, we present a new framework for understanding regular semi groups. In addition, we investigate the relationships among these different types of ideals, offering a thorough exploration of their interconnections.
An element e of an ordered semigroup (S; ;6) is called an ordered idempotent if e 6 e2. We call an ordered semigroup S idempotent ordered semigroup if every element …
An element e of an ordered semigroup (S; ;6) is called an ordered idempotent if e 6 e2. We call an ordered semigroup S idempotent ordered semigroup if every element of S is an ordered idempotent. Every idempotent semigroup is a complete semilattice of rectangular idempotent semigroups and in this way we arrive to many other important classes of idempotent ordered semigroups.
A code is called $(n, k, r, t)$ information symbol locally repairable code \big($(n, k, r, t)_i$ LRC\big) if each information coordinate can be achieved by at least $t$ disjoint …
A code is called $(n, k, r, t)$ information symbol locally repairable code \big($(n, k, r, t)_i$ LRC\big) if each information coordinate can be achieved by at least $t$ disjoint repair sets, containing at most $r$ other coordinates. This paper considers a class of $(n, k, r, t)_i$ LRCs, where each repair set contains exactly one parity coordinate. We explore the systematic code in terms of the standard parity check matrix. First, some structural features of the parity check matrix are proposed by showing some connections with the membership matrix and the minimum distance optimality of the code. Next to that, parity check matrix based proofs of various bounds associated with the code are placed. In addition to this, we provide several constructions of optimal $(n, k, r, t)_i$ LRCs, with the help of two Cayley tables of a finite field. Finally, we generalize a result of $q$-ary $(n, k, r)$ LRCs to $q$-ary $(n, k, r, t)$ LRCs.
The purpose of this paper is to study the generalization of inverse semigroups (without order). An ordered semigroup S is called an inverse ordered semigroup if for every a 2 …
The purpose of this paper is to study the generalization of inverse semigroups (without order). An ordered semigroup S is called an inverse ordered semigroup if for every a 2 S, any two inverses of a are H-related. We prove that an ordered semigroup is complete semilattice of t-simple ordered semigroups if and only if it is completely regular and inverse. Furthermore characterizations of inverse ordered semigroups have been characterized by their ordered idempotents.
Lee and Kwon [12] defined an ordered semigroup S to be completely regular if a 2 (a2Sa2] for every a 2 S. We characterize every completely regular ordered semigroup as …
Lee and Kwon [12] defined an ordered semigroup S to be completely regular if a 2 (a2Sa2] for every a 2 S. We characterize every completely regular ordered semigroup as a union of t-simple subsemigroups, and every Clifford ordered semigroup as a complete semilattice of t-simple subsemigroups. Green's Theorem for the completely regular ordered semigroups has been established. In an ordered semigroup S, we call an element e an ordered idempotent if it satisfies e ? e2. Different characterizations of the regular, completely regular and Clifford ordered semigroups are done by their ordered idempotents. Thus a foundation for the completely regular ordered semigroups and Clifford ordered semigroups has been developed
In this paper, nil extensions of some special type of ordered semigroups, such as, simple regular ordered semigroups, left simple and right regular ordered semigroup. Moreover, we have characterized complete …
In this paper, nil extensions of some special type of ordered semigroups, such as, simple regular ordered semigroups, left simple and right regular ordered semigroup. Moreover, we have characterized complete semilattice decomposition of all ordered semigroups which are nil extension of ordered semigroup.
Here we characterize regular and completely regular ordered semigroups by their minimal bi-ideals. A minimal bi-ideal is expressed as a product of a minimal right ideal and a minimal left …
Here we characterize regular and completely regular ordered semigroups by their minimal bi-ideals. A minimal bi-ideal is expressed as a product of a minimal right ideal and a minimal left ideal. Furthermore, we show that every bi-ideal in a completely regular ordered semigroup is minimal and hence a regular ordered semigroup S is completely regular if and only if S is union its of minimal bi-ideals.
In this paper we describe all those ordered semigroups which are the nil extension of Clifford, left Clifford, group like, left group like ordered semigroups.
In this paper we describe all those ordered semigroups which are the nil extension of Clifford, left Clifford, group like, left group like ordered semigroups.
A regular ordered semigroup $S$ is called right inverse if every principal left ideal of $S$ is generated by an $\mathcal{R}$-unique ordered idempotent. Here we explore the theory of right …
A regular ordered semigroup $S$ is called right inverse if every principal left ideal of $S$ is generated by an $\mathcal{R}$-unique ordered idempotent. Here we explore the theory of right inverse ordered semigroups. We show that a regular ordered semigroup is right inverse if and only if any two right inverses of an element $a\in S$ are $\mathcal{R}$-related. Furthermore, different characterizations of right Clifford, right group-like, group like ordered semigroups are done by right inverse ordered semigroups. Thus a foundation of right inverse semigroups has been developed.
An element e of an ordered semigroup $(S,\cdot,\leq)$ is called an ordered idempotent if $e\leq e^2$. We call an ordered semigroup $S$ idempotent ordered semigroup if every element of $S$ …
An element e of an ordered semigroup $(S,\cdot,\leq)$ is called an ordered idempotent if $e\leq e^2$. We call an ordered semigroup $S$ idempotent ordered semigroup if every element of $S$ is an ordered idempotent. Every idempotent semigroup is a complete semilattice of rectangular idempotent semigroups and in this way we arrive to many other important classes of idempotent ordered semigroups.
Lee and Kwon [12] defined an ordered semigroup S to be completely regular if a 2 (a2Sa2] for every a 2 S. We characterize every completely regular ordered semigroup as …
Lee and Kwon [12] defined an ordered semigroup S to be completely regular if a 2 (a2Sa2] for every a 2 S. We characterize every completely regular ordered semigroup as a union of t-simple subsemigroups, and every Clifford ordered semigroup as a complete semilattice of t-simple subsemigroups. Green's Theorem for the completely regular ordered semigroups has been established. In an ordered semigroup S, we call an element e an ordered idempotent if it satisfies e ? e2. Different characterizations of the regular, completely regular and Clifford ordered semigroups are done by their ordered idempotents. Thus a foundation for the completely regular ordered semigroups and Clifford ordered semigroups has been developed
Exactly as in semigroups, Green′s relations play an important role in the theory of ordered semigroups—especially for decompositions of such semigroups. In this paper we deal with the ℐ ‐trivial …
Exactly as in semigroups, Green′s relations play an important role in the theory of ordered semigroups—especially for decompositions of such semigroups. In this paper we deal with the ℐ ‐trivial ordered semigroups which are defined via the Green′s relation ℐ , and with the nil and Δ ‐ordered semigroups. We prove that every nil ordered semigroup is ℐ ‐trivial which means that there is no ordered semigroup which is 0‐simple and nil at the same time. We show that in nil ordered semigroups which are chains with respect to the divisibility ordering, every complete congruence is a Rees congruence, and that this type of ordered semigroups are △‐ordered semigroups, that is, ordered semigroups for which the complete congruences form a chain. Moreover, the homomorphic images of △‐ordered semigroups are △‐ordered semigroups as well. Finally, we prove that the ideals of a nil ordered semigroup S form a chain under inclusion if and only if S is a chain with respect to the divisibility ordering.
In this paper, we consider two questions: one is to characterize the structure of ordered inverse semigroups and the other is to give a condition in order that an inverse …
In this paper, we consider two questions: one is to characterize the structure of ordered inverse semigroups and the other is to give a condition in order that an inverse semigroup is orderable. The solution of the first question is carried out in terms of three types of mappings. Two of these consist of mappings of an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper R"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-class onto an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper R"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-class, while one of these consists of mappings of a principal ideal of the semilattice <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding="application/x-tex">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> constituted by idempotents onto a principal ideal of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding="application/x-tex">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. As for the second question, we give a theorem which extends a well-known result about groups that a group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with the identity <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="e"> <mml:semantics> <mml:mi>e</mml:mi> <mml:annotation encoding="application/x-tex">e</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is orderable if and only if there exists a subsemigroup <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P"> <mml:semantics> <mml:mi>P</mml:mi> <mml:annotation encoding="application/x-tex">P</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P union upper P Superscript negative 1 Baseline equals upper G comma upper P intersection upper P Superscript negative 1 Baseline equals StartSet e EndSet"> <mml:semantics> <mml:mrow> <mml:mi>P</mml:mi> <mml:mo>∪<!-- ∪ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>P</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mi>P</mml:mi> <mml:mo>∩<!-- ∩ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>P</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:mo>=</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi>e</mml:mi> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">P \cup {P^{ - 1}} = G,P \cap {P^{ - 1}} = \{ e\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x upper P x Superscript negative 1 subset-of-above-equals upper P"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mi>P</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>x</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:mo>⫅<!-- ⫅ --></mml:mo> <mml:mi>P</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">xP{x^{ - 1}} \subseteqq P</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for every <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x element-of upper G"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mi>G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">x \in G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.
Abstract The ideal extensions of semigroups -without order- have been first considered by Clifford (Clifford, A. H. (1950). Extension of semigroups. Trans. Amer. Math. Soc. 68: 165–173). The aim of …
Abstract The ideal extensions of semigroups -without order- have been first considered by Clifford (Clifford, A. H. (1950). Extension of semigroups. Trans. Amer. Math. Soc. 68: 165–173). The aim of this paper is to give the main theorem of the ideal extensions for ordered semigroups. Key Words: Left (right) translationBitranslationPermutable bitranslationsInner left (right) translationInner bitranslationPartial homomorphismRamification mapping(Ideal) extension of an ordered semigroup S by an ordered semigroup Q (Q having a zero) Acknowledgments This research was supported by the Ministry of Development, General Secretariat of Research and Technology-International Cooperation Division (Greece-Russia), Grant No. 70/3/4967. We express our warmest thanks to the editor of the journal Professor Peter M. Higgins and to the referee of the paper for their very useful comments, to Professor Mikhail V. Volkov for reading the manuscript carefully before submitting it and his very useful commends. The question if there are two ordered semigroups such that there is no ordered semigroup extension of the first one by the second one is due to the referee. Notes †To the memory of Professor Bernhard H. Neumann.
In this paper we deal with the decomposition of some classes of ordered semigroups into archimedean components. In particular, we prove that in ordered semigroups the concepts of semilattices of …
In this paper we deal with the decomposition of some classes of ordered semigroups into archimedean components. In particular, we prove that in ordered semigroups the concepts of semilattices of archimedean semigroups and of complete semilattices of archimedean semigroups are the same. In addition, we discuss similarities and differences between semigroups (without order) and ordered semigroups.
Our aim is to show the way we pass from the results of ordered semigroups (or semigroups) to ordered $\Gamma$-semigroups (or $\Gamma$-semigroups). The results of this note have been transferred …
Our aim is to show the way we pass from the results of ordered semigroups (or semigroups) to ordered $\Gamma$-semigroups (or $\Gamma$-semigroups). The results of this note have been transferred from ordered semigroups. The concept of strongly regular $po$-$\Gamma$-semigroups has been first introduced here and a characterization of strongly regular $po$-$\Gamma$-semigroups is given.
A semiring $S$ is said to be a $t$-$k$-simple semiring if it has no non-trivial proper left $k$-ideal and no non-trivial proper right $k$-ideal. We introduce the notion of $t$-$k$-simple …
A semiring $S$ is said to be a $t$-$k$-simple semiring if it has no non-trivial proper left $k$-ideal and no non-trivial proper right $k$-ideal. We introduce the notion of $t$-$k$-simple semirings and characterize the semirings in $\mathbb{SL^{+}}$, the variety of all semirings with a semilattice additive reduct, which are distributive lattices of $t$-$k$-simple subsemirings. A semiring $S$ is a distributive lattice of $t$-$k$-simple subsemirings if and only if every $k$-bi-ideal in $S$ is completely semiprime $k$-ideal. Also the semirings for which every $k$-bi-ideal is completely prime has been characterized.
Generalizing the notion of regular ring in the sense of Von Neumann, Bourne, Adhikari, Sen and Wienert introduced the notion of k-regular semiring. In this paper, we investigate Q-ideals of …
Generalizing the notion of regular ring in the sense of Von Neumann, Bourne, Adhikari, Sen and Wienert introduced the notion of k-regular semiring. In this paper, we investigate Q-ideals of the semiring of non-negative integers for which the quotient semiring is a semifield and a k-regular semiring. Also we prove that a semiring R is k-regular if and only if the quotient semiring R/I is k-regular for every Q-ideal I of R. Finally we prove that if R is an additively idempotent semiring with identity, then R is k-regular if and only if the matrix semiring R n×n is k-regular.
A right chain ordered semigroup is an ordered semigroup whose right ideals form a chain. In this paper we study the ideal theory of right chain ordered semigroups in terms …
A right chain ordered semigroup is an ordered semigroup whose right ideals form a chain. In this paper we study the ideal theory of right chain ordered semigroups in terms of prime ideals, completely prime ideals and prime segments, extending to these semigroups results on right chain semigroups proved in Ferrero et al. (J Algebra 292:574–584, 2005).
The aim of writing this paper is given in the title. We want to show that not only the ideals but also the ideal elements play an essential role in …
The aim of writing this paper is given in the title. We want to show that not only the ideals but also the ideal elements play an essential role in studying the structure of some ordered semigroups.We first prove that a $\vee e$-semigroup $S$ is a semilattice of left simple $\vee e$-semigroups if and only if it is decomposable into some pairwise disjoint left simple $\vee e$-subsemigroups of $S$ indexed by a semilattice $Y$. Then we give an example of a semilattice of left simple $\vee e$-semigroups that leads to a characterization of the semilattices of left simple and the chains of left simple $\vee e$-semigroups in terms of left ideal elements.
Lee and Kwon [12] defined an ordered semigroup S to be completely regular if a 2 (a2Sa2] for every a 2 S. We characterize every completely regular ordered semigroup as …
Lee and Kwon [12] defined an ordered semigroup S to be completely regular if a 2 (a2Sa2] for every a 2 S. We characterize every completely regular ordered semigroup as a union of t-simple subsemigroups, and every Clifford ordered semigroup as a complete semilattice of t-simple subsemigroups. Green's Theorem for the completely regular ordered semigroups has been established. In an ordered semigroup S, we call an element e an ordered idempotent if it satisfies e ? e2. Different characterizations of the regular, completely regular and Clifford ordered semigroups are done by their ordered idempotents. Thus a foundation for the completely regular ordered semigroups and Clifford ordered semigroups has been developed
In this paper, a class of two-weight and three-weight linear codes over GF(p) is constructed, and their application in secret sharing is investigated. Some of the linear codes obtained are …
In this paper, a class of two-weight and three-weight linear codes over GF(p) is constructed, and their application in secret sharing is investigated. Some of the linear codes obtained are optimal in the sense that they meet certain bounds on linear codes. These codes have applications also in authentication codes, association schemes, and strongly regular graphs, in addition to their applications in consumer electronics, communication and data storage systems.
Linear codes with a few weights have applications in consumer electronics, communication, data storage system, secret sharing, authentication codes, association schemes, and strongly regular graphs. This paper first generalizes the …
Linear codes with a few weights have applications in consumer electronics, communication, data storage system, secret sharing, authentication codes, association schemes, and strongly regular graphs. This paper first generalizes the method of constructing two-weight and three-weight linear codes of Ding et al. and Zhou et al. to general weakly regular bent functions and determines the weight distributions of these linear codes. It solves an open problem proposed by Ding et al. Furthermore, this paper constructs new linear codes with two or three weights and presents their weight distributions. They contain some optimal codes meeting certain bound on linear codes.
Minimal linear codes have interesting applications in secret sharing schemes and secure two-party computation. This paper uses characteristic functions of some subsets of $\mathbb{F}_q$ to construct minimal linear codes. By …
Minimal linear codes have interesting applications in secret sharing schemes and secure two-party computation. This paper uses characteristic functions of some subsets of $\mathbb{F}_q$ to construct minimal linear codes. By properties of characteristic functions, we can obtain more minimal binary linear codes from known minimal binary linear codes, which generalizes results of Ding et al. [IEEE Trans. Inf. Theory, vol. 64, no. 10, pp. 6536-6545, 2018]. By characteristic functions corresponding to some subspaces of $\mathbb{F}_q$, we obtain many minimal linear codes, which generalizes results of [IEEE Trans. Inf. Theory, vol. 64, no. 10, pp. 6536-6545, 2018] and [IEEE Trans. Inf. Theory, vol. 65, no. 11, pp. 7067-7078, 2019]. Finally, we use characteristic functions to present a characterization of minimal linear codes from the defining set method and present a class of minimal linear codes.
<p style='text-indent:20px;'>In this paper, we give a geometric characterization of minimal linear codes. In particular, we relate minimal linear codes to cutting blocking sets, introduced in a recent paper by …
<p style='text-indent:20px;'>In this paper, we give a geometric characterization of minimal linear codes. In particular, we relate minimal linear codes to cutting blocking sets, introduced in a recent paper by Bonini and Borello. Using this characterization, we derive some bounds on the length and the distance of minimal codes, according to their dimension and the underlying field size. Furthermore, we show that the family of minimal codes is asymptotically good. Finally, we provide some geometrical constructions of minimal codes as cutting blocking sets.</p>
In this paper, we first study in detail the relationship between minimal linear codes and cutting blocking sets, recently introduced by Bonini and Borello, and then completely characterize minimal linear …
In this paper, we first study in detail the relationship between minimal linear codes and cutting blocking sets, recently introduced by Bonini and Borello, and then completely characterize minimal linear codes as cutting blocking sets. As a direct result, minimal projective codes of dimension 3 and t-fold blocking sets with t ≥ 2 in projective planes are identical objects. Some bounds on the parameters of minimal codes are derived from this characterization. Using this new link between minimal codes and blocking sets, we also present new general primary and secondary constructions of minimal linear codes. As a result, infinite families of minimal linear codes not satisfying the Aschikhmin-Barg's condition are obtained. In addition to this, open problems on the parameters and the weight distributions of some generated linear codes are presented.
Minimal linear codes are in one-to-one correspondence with special types of blocking sets of projective spaces over a finite field, which are called strong or cutting blocking sets. Minimal linear …
Minimal linear codes are in one-to-one correspondence with special types of blocking sets of projective spaces over a finite field, which are called strong or cutting blocking sets. Minimal linear codes have been studied since decades but their tight connection with cutting blocking sets of finite projective spaces was unfolded only in the past few years, and it has not been fully exploited yet. In this paper we apply finite geometric and probabilistic arguments to contribute to the field of minimal codes. We prove an upper bound on the minimal length of minimal codes of dimension <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> over the <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula>-element Galois field which is linear in both <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>, hence improve the previous superlinear bounds. This result determines the minimal length up to a small constant factor. We also improve the lower and upper bounds on the size of so called higgledy-piggledy line sets in projective spaces and apply these results to present improved bounds on the size of covering codes and saturating sets in projective spaces as well.
Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>PG</m:mi><m:mo></m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mi>r</m:mi><m:mo>,</m:mo><m:mi>q</m:mi><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> {\operatorname{PG}(r,q)} be the r -dimensional projective space over the finite field <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>GF</m:mi><m:mo></m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mi>q</m:mi><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> {\operatorname{GF}(q)} . A set <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi mathvariant="script">𝒳</m:mi></m:math> {\mathcal{X}} of …
Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>PG</m:mi><m:mo></m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mi>r</m:mi><m:mo>,</m:mo><m:mi>q</m:mi><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> {\operatorname{PG}(r,q)} be the r -dimensional projective space over the finite field <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>GF</m:mi><m:mo></m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mi>q</m:mi><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> {\operatorname{GF}(q)} . A set <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi mathvariant="script">𝒳</m:mi></m:math> {\mathcal{X}} of points of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>PG</m:mi><m:mo></m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mi>r</m:mi><m:mo>,</m:mo><m:mi>q</m:mi><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> {\operatorname{PG}(r,q)} is a cutting blocking set if for each hyperplane Π of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>PG</m:mi><m:mo></m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mi>r</m:mi><m:mo>,</m:mo><m:mi>q</m:mi><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> {\operatorname{PG}(r,q)} the set <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi mathvariant="normal">Π</m:mi><m:mo>∩</m:mo><m:mi mathvariant="script">𝒳</m:mi></m:mrow></m:math> {\Pi\cap\mathcal{X}} spans Π. Cutting blocking sets give rise to saturating sets and minimal linear codes, and those having size as small as possible are of particular interest. We observe that from a cutting blocking set obtained in [20], by using a set of pairwise disjoint lines, there arises a minimal linear code whose length grows linearly with respect to its dimension. We also provide two distinct constructions: a cutting blocking set of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>PG</m:mi><m:mo></m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mn>3</m:mn><m:mo>,</m:mo><m:msup><m:mi>q</m:mi><m:mn>3</m:mn></m:msup><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> {\operatorname{PG}(3,q^{3})} of size <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mn>3</m:mn><m:mo></m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mrow><m:mi>q</m:mi><m:mo>+</m:mo><m:mn>1</m:mn></m:mrow><m:mo stretchy="false">)</m:mo></m:mrow><m:mo></m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mrow><m:msup><m:mi>q</m:mi><m:mn>2</m:mn></m:msup><m:mo>+</m:mo><m:mn>1</m:mn></m:mrow><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> {3(q+1)(q^{2}+1)} as a union of three pairwise disjoint q -order subgeometries, and a cutting blocking set of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>PG</m:mi><m:mo></m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mn>5</m:mn><m:mo>,</m:mo><m:mi>q</m:mi><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> {\operatorname{PG}(5,q)} of size <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mn>7</m:mn><m:mo></m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mrow><m:mi>q</m:mi><m:mo>+</m:mo><m:mn>1</m:mn></m:mrow><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> {7(q+1)} from seven lines of a Desarguesian line spread of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>PG</m:mi><m:mo></m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mn>5</m:mn><m:mo>,</m:mo><m:mi>q</m:mi><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> {\operatorname{PG}(5,q)} . In both cases, the cutting blocking sets obtained are smaller than the known ones. As a byproduct, we further improve on the upper bound of the smallest size of certain saturating sets and on the minimum length of a minimal q -ary linear code having dimension 4 and 6.
We develop three approaches of combinatorial flavor to study the structure of minimal codes and cutting blocking sets in finite geometry, each of which has a particular application. The first …
We develop three approaches of combinatorial flavor to study the structure of minimal codes and cutting blocking sets in finite geometry, each of which has a particular application. The first approach uses techniques from algebraic combinatorics, describing the supports in a linear code via the Alon--Füredi theorem and the combinatorial Nullstellensatz. The second approach combines methods from coding theory and statistics to compare the mean and variance of the nonzero weights in a minimal code. Finally, the third approach regards minimal codes as cutting blocking sets and studies these using the theory of spreads in finite geometry. By applying and combining these approaches with each other, we derive several new bounds and constraints on the parameters of minimal codes. Moreover, we obtain two new constructions of cutting blocking sets of small cardinality in finite projective spaces. In turn, these allow us to give explicit constructions of minimal codes having short length for the given field and dimension.
A strong blocking set in a finite projective space is a set of points that intersects each hyperplane in a spanning set. We provide a new graph theoretic construction of …
A strong blocking set in a finite projective space is a set of points that intersects each hyperplane in a spanning set. We provide a new graph theoretic construction of such sets: combining constant-degree expanders with asymptotically good codes, we explicitly construct strong blocking sets in the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis k minus 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(k-1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dimensional projective space over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper F Subscript q"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">F</mml:mi> </mml:mrow> <mml:mi>q</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathbb {F}_q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that have size at most <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c q k"> <mml:semantics> <mml:mrow> <mml:mi>c</mml:mi> <mml:mi>q</mml:mi> <mml:mi>k</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">c q k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for some universal constant <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c"> <mml:semantics> <mml:mi>c</mml:mi> <mml:annotation encoding="application/x-tex">c</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Since strong blocking sets have recently been shown to be equivalent to minimal linear codes, our construction gives the first explicit construction of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper F Subscript q"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">F</mml:mi> </mml:mrow> <mml:mi>q</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathbb {F}_q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-linear minimal codes of length <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and dimension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, for every prime power <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, for which <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n less-than-or-equal-to c q k"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>c</mml:mi> <mml:mi>q</mml:mi> <mml:mi>k</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">n \leq c q k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This solves one of the main open problems on minimal codes.
In this paper, we generalize constructions in two recent works of Ding, Heng, and Zhou to any field F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> , q odd, providing infinite families of minimal …
In this paper, we generalize constructions in two recent works of Ding, Heng, and Zhou to any field F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> , q odd, providing infinite families of minimal codes for which the Ashikhmin-Barg bound does not hold.
This book, along with volume I, which appeared previously, presents a survey of the structure and representation theory of semi groups. Volume II goes more deeply than was possible in …
This book, along with volume I, which appeared previously, presents a survey of the structure and representation theory of semi groups. Volume II goes more deeply than was possible in volume I into the theories of minimal ideals in a semi group, inverse semi groups, simple semi groups, congruences on a semi group, and the embedding of a semi group in a group. Among the more important recent developments of which an extended treatment is presented are B. M. Sain's theory of the representations of an arbitrary semi group by partial one-to-one transformations of a set, L. Redei's theory of finitely generated commutative semi groups, J. M. Howie's theory of amalgamated free products of semi groups, and E. J. Tully's theory of representations of a semi group by transformations of a set. Also a full account is given of Malcev's theory of the congruences on a full transformation semi group.