Projective linear codes are a special class of linear codes whose duals have minimum distance at least 3. The columns of the generator matrix of an [n, k] projective code over finite field F-q can be viewed as points ...
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Projective linear codes are a special class of linear codes whose duals have minimum distance at least 3. The columns of the generator matrix of an [n, k] projective code over finite field F-q can be viewed as points in the projective space PG( k - 1, F-q). Projective codes are of interest not only because their duals have good error correcting capability but also because they may be related to interesting combinatorial structures. The objective of this paper is to construct projective linear codes with five families of almost difference sets. To this end, the augmentation and extension techniques for linear codes are used. The parameters and weight distributions of the projective codes are explicitly determined. Several infinite families of optimal or almost optimal codes including MDS codes, near MDS codes, almost MDS odes and Griesmer codes are obtained. Besides, we also give some applications of these codes.
linear codes with large minimum distances perform well in error and erasure corrections. Constructing such linear codes is a main topic in coding theory. In this paper, we propose four families of linear codes which a...
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linear codes with large minimum distances perform well in error and erasure corrections. Constructing such linear codes is a main topic in coding theory. In this paper, we propose four families of linear codes which are optimal or distance-optimal with respect to the Griesmer bound. Using the theory of characters over finite fields, we determine the weight distribution of these linear codes. The results show that these linear codes are two-weight codes. Finally, we analyse the locality of these linear codes and present three families of distance-optimal binary locally repairable code with locality 2 or 3.
linear codes have been an interesting topic in both theory and practice for many years. In this paper, two classes of linear codes over the finite field GF(p) are presented and their weight distributions are also dete...
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linear codes have been an interesting topic in both theory and practice for many years. In this paper, two classes of linear codes over the finite field GF(p) are presented and their weight distributions are also determined, where p is an odd prime. Some of the linear codes obtained are optimal or almost optimal in the sense that their parameters meet certain bound on linear codes.
Projective two-weight linear codes are closely related to finite projective spaces and strongly regular graphs. In this paper, a family of q-ary two-weight linear codes including two subfamilies of projective codes is...
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Projective two-weight linear codes are closely related to finite projective spaces and strongly regular graphs. In this paper, a family of q-ary two-weight linear codes including two subfamilies of projective codes is presented, where q is a prime power. The parameters of both the codes and their duals are excellent. As applications, the codes are used to derive strongly regular graphs with new parameters and secret sharing schemes with interesting access structures.
Let R be the Galois ring of characteristic 4 and cardinality 4(m), where m is a natural number. Let C be a linear code of length n over R and Phi be the Homogeneous Gray map on R-n. In this paper, we show that Phi(C) ...
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Let R be the Galois ring of characteristic 4 and cardinality 4(m), where m is a natural number. Let C be a linear code of length n over R and Phi be the Homogeneous Gray map on R-n. In this paper, we show that Phi(C) is linear if and only if for every X, Y is an element of C, 2(X circle dot Y) is an element of C. Using the generator polynomial of a cyclic code of odd length over R, a necessary and sufficient condition is given which its Gray image is linear.
linear codes with a few weights have important applications in secret sharing, authentication codes, data storage system, association schemes, and strongly regular graphs. Hence, the construction of linear codes with ...
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linear codes with a few weights have important applications in secret sharing, authentication codes, data storage system, association schemes, and strongly regular graphs. Hence, the construction of linear codes with a few weights is an important research topic in coding theory. In this paper, we construct two new classes of linear codes with two and three weights, and determine their complete weight enumerators. Our work generalizes the results of Wang et al. (Discret Math 340(3):467-480, 2017).
Adleman's successful solution of a seven-vertex instance of the NP-complete Hamiltonian directed path problem by a DNA algorithm initiated the field of biomolecular computing. In this correspondence, we describe D...
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Adleman's successful solution of a seven-vertex instance of the NP-complete Hamiltonian directed path problem by a DNA algorithm initiated the field of biomolecular computing. In this correspondence, we describe DNA algorithms based on the sticker model to perform encoding, minimum-distance computation, and maximum-likelihood (ML) decoding of binary linear codes. We also discuss feasibility and limitations of the sticker algorithms.
linear codes have widespread applications in data storage systems. There are two major contributions in this paper. We first propose infinite families of optimal or distance-optimal linear codes over F-p constructed f...
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linear codes have widespread applications in data storage systems. There are two major contributions in this paper. We first propose infinite families of optimal or distance-optimal linear codes over F-p constructed from projective spaces. Moreover, a necessary and sufficient condition for such linear codes to be Griesmer codes is presented. Secondly, as an application in data storage systems, we investigate the locality of the linear codes constructed. Furthermore, we show that these linear codes are alphabet-optimal locally repairable codes with locality 2.
Secret sharing is an important concept in cryptography, however it is a difficult problem to determine the access structure of the secret sharing scheme based on a linear code. In this work, we construct twoweight lin...
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Secret sharing is an important concept in cryptography, however it is a difficult problem to determine the access structure of the secret sharing scheme based on a linear code. In this work, we construct twoweight linear codes over finite field by using linear codes over finite ring. We first study MacDonald codes over the finite ring F2+v F2+v2F2with v3= v. Then we give torsion codes of MacDonald codes of type α and β, which are twoweight linear codes. Finally we give the access structures of secret sharing schemes based on the dual codes of the two-weight codes.
Subfield codes of linear codes over finite fields have recently received much attention since they can produce optimal codes, which may have applications in secret sharing, authentication codes and association schemes...
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Subfield codes of linear codes over finite fields have recently received much attention since they can produce optimal codes, which may have applications in secret sharing, authentication codes and association schemes. In this paper, we first present a construction framework of 3-dimensional linear codes C(f.g )over F-q(m) parameterized by any two functions f, g over F-q(m) , and then study the properties of six types of C-f.g, its punctured code C-f.g*, and their corresponding subfield codes over F-q. The classification of C-f,C-g is based on special choices of f, g as trace function, norm function, almost bent function, Boolean bent function or a combination of these functions. For the first two types of C-f.g, we explicitly determine the weight distributions and dualities of C-f,C-g, C-f.g* and their subfield codes over F-q. The remaining four types of C-f.g are restricted to q = 2, and the weight distributions and dualities of the subfields code C-f,C-g(q) and C-f.g(*(q)) are completely determined. Most of the resultant linear codes (over F-q(m) or over F-q) have few weights. Some of them are optimal and some have the best-known parameters according to the tables maintained at http://***. In fact, 16 infinite families of optimal linear codes are produced in this paper. As a byproduct, a family of [2(4m-2), 2m+1, 2(4m-3)] quaternary Hermitian self-orthogonal codes are obtained with m >= 2. As an application, we present several infinite families of 2-designs or 3-designs with some of the codes presented in this paper.
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