Linear codes with optimal parameters have a wide range of practical applications in fields such as cryptography, quantum theory and distributed storage. This paper generalizes the work of (Heng et al. Designs, codes a...
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Linear codes with optimal parameters have a wide range of practical applications in fields such as cryptography, quantum theory and distributed storage. This paper generalizes the work of (Heng et al. Designs, codes and Cryptography 91:3953-3976, 2023) to the general case utilizing weakly regular plateaued functions. To begin with, a new class of self-orthogonal codes is constructed by using weakly regular plateaued functions. Then, by using the properties of exponential sums tools over finite fields, we determine the related parameters as well as weight distributions of this codes. Moreover, the minimum distance of the dual of these augmented codes are shown to be 3. Finally, we study the constructed self-orthogonal codes and gain some new p-ary almost optimal or optimal LCD codes.
Multithreshold decoders (MTD) of self-orthogonal codes (SOC) for channels with errors as well as erasure channels are discussed. The work offers analytical estimations of MTD efficiency considering SOC structure as we...
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ISBN:
(纸本)9781728169491
Multithreshold decoders (MTD) of self-orthogonal codes (SOC) for channels with errors as well as erasure channels are discussed. The work offers analytical estimations of MTD efficiency considering SOC structure as well as decoder parameters. These estimations are shown to correspond well with computer simulation results allowing us to estimate the area of MTD efficient operation in the conditions of high noise level.
self-orthogonal codes have a wide range of applications in various fields, especially communication and cryptography. In this paper, we construct four families of linear codes over finite fields via a defining-set con...
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self-orthogonal codes have a wide range of applications in various fields, especially communication and cryptography. In this paper, we construct four families of linear codes over finite fields via a defining-set construction. The weight distributions of these codes are determined. We show that most of these codes are self-orthogonal and reach the Grismer bound. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
The construction of self-orthogonal codes is an interesting topic due to their wide applications in communication and cryptography. In this paper, we construct several families of self-orthogonal cyclic codes with len...
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The construction of self-orthogonal codes is an interesting topic due to their wide applications in communication and cryptography. In this paper, we construct several families of self-orthogonal cyclic codes with length n = q(m)-1/lambda , where lambda | q - 1 and m >= 3 is odd. It is proved that there exist q-ary self-orthogonal cyclic codes with parameters [ n, n - 1/2 >= d] for even prime power q , and [ n, n - 1/2, >= d ] or [ n, n - 1/2, >= d ] for odd prime power q , where d is significantly better than the square-root bound. These several families of self-orthogonal cyclic codes contain some optimal linear codes.
This paper presents four constructions of general self-orthogonal matrix-product codes associated with Toeplitz matrices. The first one relies on the dual of a known general dual-containing matrix-product code;the sec...
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This paper presents four constructions of general self-orthogonal matrix-product codes associated with Toeplitz matrices. The first one relies on the dual of a known general dual-containing matrix-product code;the second one is founded on a specific family of matrices, where we provide an efficient algorithm for generating them based on Toeplitz matrices and it has an interesting application in producing new non-singular by columns quasi-unitary matrices;and the last two ones are based on the utilization of certain special Toeplitz matrices. Concrete examples and detailed comparisons are provided. As a byproduct, we also find an application of Toeplitz matrices, closely related to the constructions of quantum codes.
We prove that there is a Hermitian self-orthogonal k-dimensional truncated generalised Reed-Solomon code of length n <= q(2) over F-q2 if and only if there is a polynomial g is an element of F-q2 of degree at most ...
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We prove that there is a Hermitian self-orthogonal k-dimensional truncated generalised Reed-Solomon code of length n <= q(2) over F-q2 if and only if there is a polynomial g is an element of F-q2 of degree at most (q - k)q - 1 such that g + g(q) has q(2) - n distinct zeros. This allows us to determine the smallest n for which there is a Hermitian self-orthogonal k-dimensional truncated generalised Reed-Solomon code of length n over F-q2, verifying a conjecture of Grassi and Riitteler. We also provide examples of Hermitian self-orthogonal k-dimensional generalised Reed-Solomon codes of length q(2) + 1 over F-q2, for k = q - 1 and q an odd power of two.
The self-orthogonality and divisibility are two important properties of linear codes. It is interesting to establish relationship between them. By the well-known Gleason-Pierce-Ward Theorem, all self-dual divisible co...
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The self-orthogonality and divisibility are two important properties of linear codes. It is interesting to establish relationship between them. By the well-known Gleason-Pierce-Ward Theorem, all self-dual divisible codes have been totally classified. However, the relationship between the self-orthogonality and divisibility of a q-ary linear codes is known only for q = 2, 3 by Huffman and Pless in 2003. It has remained open for more than 20 years to consider other cases. The purpose of this paper is to settle this open problem under certain conditions and construct new families of self-orthogonal codes. Let q be a power of an odd prime p. Firstly, we prove that any pdivisible code containing the all-1 vector over the finite field F-q is self-orthogonal. More generally, it is concluded that any p-divisible [n, k] linear code over F-q containing codewords of weight n is monomially equivalent to an [n, k] self-orthogonal code over F-q. This result provides a very efficient way to find self-orthogonal codes from p-divisible codes. Secondly, we apply this result to construct self-orthogonal codes with excellent parameters or nice applications. For one thing, we use this result to study the self-orthogonality of generalized Reed-Muller codes, certain projective two-weight codes, and Griesmer codes. For another thing, by this useful result as well as the extending and augmentation techniques for linear codes, we construct eight new families of self-orthogonal divisible codes. These self-orthogonal codes and their duals contain many optimal or almost optimal codes. Besides, some self-orthogonal codes support combinatorial designs and some of them are proved to be optimal or almost optimal locally recoverable codes.
self-orthogonal codes with dual distance three and quantum codes with distance three constructed from self-orthogonal codes over F-5 are discussed in this paper. Firstly, for given code length n >= 5, a [n, k](5) s...
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self-orthogonal codes with dual distance three and quantum codes with distance three constructed from self-orthogonal codes over F-5 are discussed in this paper. Firstly, for given code length n >= 5, a [n, k](5) self-orthogonal code with minimal dimension k and dual distance three is constructed. Secondly, for each n >= 5, two nested self-orthogonal codes with dual distance two and three are constructed, and consequently quantum code of length n and distance three is constructed via Steane construction. All of these quantum codes constructed via Steane construction are optimal or near optimal according to the quantum Hamming bound.
By means of a construction method outlined by Harada and Tonchev, we determine some non-binary self-orthogonal codes obtained from the row span of orbit matrices of Bush-type Hadamard matrices that admit a fixed-point...
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By means of a construction method outlined by Harada and Tonchev, we determine some non-binary self-orthogonal codes obtained from the row span of orbit matrices of Bush-type Hadamard matrices that admit a fixed-point-free and fixed-block-free automorphism of prime order. We show that the code [20, 15, 4](5) obtained from a (100, 45, 20) design is optimal, and those with parameters [36, 21, 6](3) and [20, 14, 4](5) obtained from a (36, 15, 6) and a (100, 45, 20) design respectively, are near-optimal for the given length and dimension. Furthermore, we obtained a conjecturally optimal self-dual doubly-even [72, 36, 12](2) code, and examined the code of an orbit matrix of a putative (676, 325, 156) design.
We give a method of constructing self-orthogonal codes from equitable partitions of association schemes. By applying this method, we construct self-orthogonal codes from some distance-regular graphs. Some of the obtai...
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We give a method of constructing self-orthogonal codes from equitable partitions of association schemes. By applying this method, we construct self-orthogonal codes from some distance-regular graphs. Some of the obtained codes are optimal. Further, we introduce a notion of self-orthogonal subspace codes. We show that under some conditions equitable partitions of association schemes yield such self-orthogonal subspace codes and we give some examples from distance-regular graphs.
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