Applying the Gale transform on certain linear and non-linear geometrical objects, and studying the orbits under the action of the associated automorphism groups in the higher-dimensional space, we construct some famil...
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Applying the Gale transform on certain linear and non-linear geometrical objects, and studying the orbits under the action of the associated automorphism groups in the higher-dimensional space, we construct some families of cap codes and other structures admitting the same automorphism groups. (C) 2011 Elsevier B.V. All rights reserved.
The concept of minimal linear codes was introduced by Ashikhmin and Barg in 1998, leading to the development of various methods for constructing these codes over finite fields. In this context, minimality is defined a...
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A non-zero F-linear map from a finite-dimensional commutative F-algebra to the field Fis called an F-valued trace if its kernel does not contain any non-zero ideals. In this article, we utilize an F2-valued trace of t...
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A non-zero F-linear map from a finite-dimensional commutative F-algebra to the field Fis called an F-valued trace if its kernel does not contain any non-zero ideals. In this article, we utilize an F2-valued trace of the F2-algebra R2:= F2[x]/x3- x to study binary subfield code C(2) Dof CD:={(x center dot d) d.D: x. Rm2} for each defining set Dderived from a certain simplicial complex. For m. Nand X.{1, 2,..., m}, define X:={v. Fm2: Supp(v). X} and D :=(1 + u2) D1+ u2D2+(u + u2)D3, a subset of Rm2, where u = x + x3- x, D1.{L, cL}, D2.{ M, cM} and D3.{N, cN}, for L, M, N.{1, 2,..., m}. The parameters and the Hamming weight distribution of the binary subfield code C(2)Dof CDare determined for each D. These binary subfield codes are minimal under certain mild conditions on the cardinalities of L, Mand N. Moreover, most of these codes are distanceoptimal. Consequently, we obtain a few infinite families of minimal, self-orthogonal and distance-optimal binary linear codes that are either 2-weight or 4-weight. It is worth mentioning that we have obtained several new distance-optimal binary linear codes. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
In this paper, several classes of two-weight or three-weight linear codes over Fp from quadratic or non-quadratic functions are constructed and their weight distributions are determined. From the constructed codes, we...
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In this paper, several classes of two-weight or three-weight linear codes over Fp from quadratic or non-quadratic functions are constructed and their weight distributions are determined. From the constructed codes, we obtain some optimal linear codes with respect to the Singleton bound and the Griesmer bound. These two- or three-weight linear codes may have applications in secret sharing, authentication codes, association schemes and strongly regular graphs.
linear codes play a key role in widespread applications. In this paper, we propose three new constructions of linear codes. We give some sufficient conditions for the constructed linear codes to be optimal or distance...
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linear codes play a key role in widespread applications. In this paper, we propose three new constructions of linear codes. We give some sufficient conditions for the constructed linear codes to be optimal or distance-optimal in terms of the Griesmer bound. Three classes of distance-optimal linear codes with new parameters are presented. Under some constraints, we show that some of the presented linear codes have few weights.
Let F-2m be the finite field with 2(m) elements, where m is a positive integer. Recently, Heng and Ding in (Finite Fields Appl. 56:308-331, 2019) studied the subfield codes of two families of hyperovel codes and deter...
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Let F-2m be the finite field with 2(m) elements, where m is a positive integer. Recently, Heng and Ding in (Finite Fields Appl. 56:308-331, 2019) studied the subfield codes of two families of hyperovel codes and determined the weight distribution of the linear code C-a,C-b = {((Tr-1(m) (af (x) + bx) + c)(x is an element of F2m), Tr-1(m) (a), Tr-1(m) (b)) : a, b is an element of F-2m, c is an element of F-2}, for f (x) = x(2) and f (x) = x(6) with odd m. Let v(2)(.) denote the 2-adic order function. This paper investigates more subfield codes of linear codes and obtains the weight distribution of C-a,C-b for f (x) = x(2i+2j), where i, j are nonnegative integers such that v(2)(m) <= v(2)(i - j) (i >= j). In addition to this, we further investigate the punctured code of C-a,C-b as follows: C-a = {((Tr-1(m)(ax(2i+2j) + bx) + c)(x is an element of F2m), Tr-1(m)(a)) : a, b is an element of F-2m, c is an element of F-2}, and determine its weight distribution for any nonnegative integers i, j. The parameters of these binary linear codes are new in most cases. Some of the codes and their duals obtained are optimal or almost optimal.
Upper bounds on the weight distribution function for codes of minimum distance at least 2 are given. codes, where the bound is met with equality, are characterized. An improved upper bound on the weight distribution f...
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Upper bounds on the weight distribution function for codes of minimum distance at least 2 are given. codes, where the bound is met with equality, are characterized. An improved upper bound on the weight distribution function for codes of minimum distance at least 3 is given. As an application, a sharp upper bound on the probability of undetected error for linear codes with full support is characterized.
Let Gamma(k) (V) be the Grassmann graph formed by k-dimensional subspaces of an n-dimensional vector space over the finite field Fq consisting of q elements and 1 < k < n - 1. Denote by r(n, k)(q) the restrictio...
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Let Gamma(k) (V) be the Grassmann graph formed by k-dimensional subspaces of an n-dimensional vector space over the finite field Fq consisting of q elements and 1 < k < n - 1. Denote by r(n, k)(q) the restriction of the Grassmann graph to the set of all non-degenerate linear [n, k](q) codes. We describe maximal cliques of the graph T(n, k)(q) and show that every automorphism of this graph is induced by a monomial semilinear automorphism of V. (C) 2017 Elsevier Inc. All rights reserved.
The worst case probability of undetected error for a linear [n,k;q] code used on a local binomial channel is studied. For the two most important cases it is determined in terms of the weight hierarchy of the code. The...
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The worst case probability of undetected error for a linear [n,k;q] code used on a local binomial channel is studied. For the two most important cases it is determined in terms of the weight hierarchy of the code. The worst case probability of undetected error is determined explicitly for some classes of codes.
In addition to their applications in data communication and storage, linear codes also have nice applications in combinatorics and cryptography. Minimal linear codes, a special type of linear codes, are preferred in s...
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In addition to their applications in data communication and storage, linear codes also have nice applications in combinatorics and cryptography. Minimal linear codes, a special type of linear codes, are preferred in secret sharing. In this paper, a necessary and sufficient condition for a binary linear code to be minimal is derived. This condition enables us to obtain three infinite families of minimal binary linear codes with w(min)/w(max) <= 1/2 from a generic construction, where w(min) and w(max), respectively, denote the minimum and maximum nonzero weights in a code. The weight distributions of all these minimal binary linear codes are also determined.
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