The number of nonzero weights of a linear code is essential in coding theory as it unveils salient properties of the code, such as its covering radius. In this paper, we establish two upper bounds on the number of non...
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The number of nonzero weights of a linear code is essential in coding theory as it unveils salient properties of the code, such as its covering radius. In this paper, we establish two upper bounds on the number of nonzero weights of a linear code with prescribed automorphism. Our bounds are applicable for almost all linear codes and tighter than previously known bounds. Examples confirm that our bounds are sharp on numerous occasions. In addition, we give an infinite family of linear codes that attain our bounds with equality.
Applied in communication, data storage system, secret sharing schemes, authentication codes and association schemes, linear codes attract much attention. In this paper, a class of three-weight linear codes is obtained...
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Applied in communication, data storage system, secret sharing schemes, authentication codes and association schemes, linear codes attract much attention. In this paper, a class of three-weight linear codes is obtained by the defining sets over finite fields of odd characteristic. The parameters and weight distributions of linear codes are determined by the additive characters, multiplicative characters and Gauss sums. Further, most of linear codes obtained are minimal, which can be used to construct secret sharing schemes.
We classify all q-ary ?-divisible linear codes which are spanned by codewords of weight ?. The basic building blocks are the simplex codes, and for q = 2 additionally the first order Reed-Muller codes and the parity c...
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We classify all q-ary ?-divisible linear codes which are spanned by codewords of weight ?. The basic building blocks are the simplex codes, and for q = 2 additionally the first order Reed-Muller codes and the parity check codes. This generalizes a result of Pless and Sloane, where the binary self-orthogonal codes spanned by codewords of weight 4 have been classified, which is the case q = 2 and ? = 4 of our classification. As an application, we give an alternative proof of a theorem of Liu on binary ?-divisible codes of length 4? in the projective case.
Let V be an n-dimensional vector space over the finite field consisting of q elements and let Gamma(k)(V) be the Grassmann graph formed by k-dimensional subspaces of V, 1 < k < n-1. Denote by Gamma(n, k)(q) the ...
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Let V be an n-dimensional vector space over the finite field consisting of q elements and let Gamma(k)(V) be the Grassmann graph formed by k-dimensional subspaces of V, 1 < k < n-1. Denote by Gamma(n, k)(q) the restriction of Gamma(k)(V) to the set of all non-degenerate linear [n, k](q), codes. We show that for any two codes the distance in Gamma(n, k)(q) coincides with the distance in Gamma(k)(V) only in the case when n < (q + 1)(2) + k - 2, i.e. if n is sufficiently large then for some pairs of codes the distances in the graphs Gamma(k)(V) and Gamma(n, k)(q) are distinct. We describe one class of such pairs. (C) 2016 Elsevier Inc. All rights reserved.
The weight hierarchy of a linear code have been an important research topic in coding theory since Wei's original work in 1991. In this paper, choosing D ={(x,y) is an element of (F-ps1 x F-ps2)\{(0,0)} : f(x) + T...
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The weight hierarchy of a linear code have been an important research topic in coding theory since Wei's original work in 1991. In this paper, choosing D ={(x,y) is an element of (F-ps1 x F-ps2)\{(0,0)} : f(x) + Tr-1(s2)(alpha y) = 0} as a defining set, where alpha is an element of F-ps2(*), f(x) is a quadratic form over F-ps1 with values in F-p and f(x) can be non-degenerate or not, we construct a family of three-weight p-ary linear codes and determine their weight distributions and weight hierarchies completely. Most of the codes can be used in secret sharing schemes.
A fascinating topic of combinatorics is the study of t-designs, which has a very long history. The incidence matrix of a t-design generates a linear code over GF(q) for any prime power q, which is called the linear co...
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A fascinating topic of combinatorics is the study of t-designs, which has a very long history. The incidence matrix of a t-design generates a linear code over GF(q) for any prime power q, which is called the linear code of the t-design over GF(q). On the other hand, some linear codes hold t-designs with t >= 1. The purpose of this paper is to study the linear codes of t-designs held in the Reed-Muller and Simplex codes. Some general theory for the linear codes of t-designs held in linear codes is presented. Several open problems are also presented.
The binary Hamming codes with parameters [2(m) - 1, 2(m) - 1- m, 3] are perfect. Their extended codes have parameters [2(m), 2(m) - 1- m, 4] and are distance-optimal. The first objective of this paper is to construct ...
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The binary Hamming codes with parameters [2(m) - 1, 2(m) - 1- m, 3] are perfect. Their extended codes have parameters [2(m), 2(m) - 1- m, 4] and are distance-optimal. The first objective of this paper is to construct a class of binary linear codes with parameters [2(m+s) + 2(s) - 2(m), 2(m+s) + 2(s) - 2(m) - 2m- 2, 4], which have better information rates than the class of extended binary Hamming codes, and are also distance-optimal. The second objective is to construct a class of distance-optimal binary codes with parameters [2(m) +2, 2(m)- 2m, 6]. Both classes of binary linear codes have new parameters.
This paper presents four classes of linear codes from coset representatives of subgroups and cyclotomic coset families of certain finite field, and determines their weight enumerators. These linear codes may have appl...
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This paper presents four classes of linear codes from coset representatives of subgroups and cyclotomic coset families of certain finite field, and determines their weight enumerators. These linear codes may have applications in consumer electronics, communications and secret sharing schemes.
Properties of the weight distribution of low-dimensional generalized Reed-Muller codes are used to obtain restrictions on the weight distribution of linear codes over arbitrary fields. These restrictions are used in n...
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Properties of the weight distribution of low-dimensional generalized Reed-Muller codes are used to obtain restrictions on the weight distribution of linear codes over arbitrary fields. These restrictions are used in non-existence proofs for ternary linear code with parameters [74, 10, 44] [82, 6, 53] and [96, 6, 62]. (C) 2001 Elsevier Science B.V. All rights reserved.
An error-erasure channel is a simple noise model that introduces both errors and erasures. While the two types of errors can be corrected simultaneously with error-correcting codes, it is also known that any linear co...
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An error-erasure channel is a simple noise model that introduces both errors and erasures. While the two types of errors can be corrected simultaneously with error-correcting codes, it is also known that any linear code allows for first correcting errors and then erasures in two-step decoding. In particular, a carefully designed parity-check matrix not only allows for separating erasures from errors but also makes it possible to efficiently correct erasures. The separating redundancy of a linear code is the number of parity-check equations in a smallest parity-check matrix that has the required property for this error-erasure separation. In a sense, it is a parameter of a linear code that represents the minimum overhead for efficiently separating erasures from errors. While several bounds on separating redundancy are known, there still remains a wide gap between upper and lower bounds except for a few limited cases. In this paper, using probabilistic combinatorics and design theory, we improve both upper and lower bounds on separating redundancy. We also show a relation between parity-check matrices for error-erasure separation and special matrices, called X-codes, for data compaction circuits in VLSI testing. This leads to an exponentially improved bound on the size of an optimal X-code.
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