Lots of attention has recently been focused on isodual (resp. formally self-dual (FSD)) codes and linear complementary dual (LCD) codes for their theoretical and practical importance. In this paper, we consider a gene...
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Lots of attention has recently been focused on isodual (resp. formally self-dual (FSD)) codes and linear complementary dual (LCD) codes for their theoretical and practical importance. In this paper, we consider a general construction, namely the class of four Toeplitz (FT) codes and prove that they arbitrarily approach the asymptotic Gilbert-Varshamov bound on the relative distance with asymptotic rate 12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\fracisodualisodual code$$\end{document} over arbitrary finite fields. Then we study a special family of isodual and further FSD FT codes whose asymptotic behavior is the same as that of FT codes. In terms of eigenvalues, we further provide a sufficient and necessary condition for a family of FT codes to be LCD. In addition, we introduce bordered FT codes, which can be effectively tested for constructing isodual codes with longer lengths. Finally, we obtain many improved isodual LCD and further FSD LCD codes compared to the latest results reported in the literature.
Blackford (Finite Fields Appl. 24, 29-44 2013) introduced Type I constacyclic duadic codes over the finite field F-q, where q is an odd prime power, and obtained isodual codes from them. In this paper, we generalize t...
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Blackford (Finite Fields Appl. 24, 29-44 2013) introduced Type I constacyclic duadic codes over the finite field F-q, where q is an odd prime power, and obtained isodual codes from them. In this paper, we generalize this idea and present Type II q-splitting of some special natural numbers n over F-q. By using it, we construct isodual codes of length n + r over F-q for some r, where r is some divisor of n and q - 1, and provide some examples of optimal isodual codes.
In 2020, Cao et al. proved that any repeated-root constacyclic code is monomially equivalent to a matrix product code of simple-root constacyclic codes. In this paper, we study a family of matrix product codes with wo...
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In 2020, Cao et al. proved that any repeated-root constacyclic code is monomially equivalent to a matrix product code of simple-root constacyclic codes. In this paper, we study a family of matrix product codes with wonderful properties, which is a generalization of linear codes obtained from the [u+v|u-v]-construction and [u+v|lambda(-1)u-lambda(-1)v]-construction. Then we show that any lambda-constacyclic code (not necessary repeated-root lambda-constacyclic code) of length N over the finite field F-q with gcd(q-1/ord(lambda), N) >= 2, where ord(lambda) is the order of lambda in the cyclic group F-q* = F-q\isodual, is a matrix product code of some constacyclic codes. It is a highly interesting question that the existence of sequences {C-1, C-2, C-3, ...} of Euclidean (or Hermitian) self-dual codes with square-root-like minimum Hamming distances, i.e., C-i is an [n(C-i), k(Ci), d(C-i)](q)-linear code such that lim(i ->+infinity) n(C-i) = +infinity and lim(i ->+infinity)d(C-i)/root n(C-i) > 0. Based on the [u+v|lambda(-1)u-lambda(-1)v]-construction, we construct several families of Euclidean (or Hermitian) self-dual codes with square-root-like minimum Hamming distances by using Reed-Muller codes, projective Reed-Muller codes. And we construct some new Euclidean isodual lambda-constacyclic codes with square-root-like minimum Hamming distances from Euclidean self-dual cyclic codes and Euclidean self-dual negacyclic codes by monomial equivalences.
We develop a construction method of isodual codes over GF(q), where q is a prime power;we construct isodual codes over GF(q) of length 2n + 2 from isodual codes over GF(q) of length 2n. Using this method, we find some...
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We develop a construction method of isodual codes over GF(q), where q is a prime power;we construct isodual codes over GF(q) of length 2n + 2 from isodual codes over GF(q) of length 2n. Using this method, we find some isodual codes over GF(q), where q = 2, 3 and 5. In more detail, we obtain binary isodual codes of lengths 32, 34, 36, 38, and 40, where all these codes of lengths 32, 34, and 36 are optimal and some codes of length 38 are optimal. We note that all these binary isodual codes are not self-dual codes, and in particular, in the case of length 38 all their weight enumerators are different from those of binary self-dual codes of the same length;in fact, four binary isodual codes of length 38 are formally self dual even codes. We construct isodual codes over GF(3) and GF(5) of lengths 4, 6, and 8 as well. (C) 2017 Elsevier Inc. All rights reserved.
In this paper, we investigate free Hermitian self-dual codes whose generator matrices are of the form [I, A + vB] over the ring F-2 + vF(2) = {0, 1, v, 1 + v} with v(2) = v. We use the double-circulant, the bordered d...
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In this paper, we investigate free Hermitian self-dual codes whose generator matrices are of the form [I, A + vB] over the ring F-2 + vF(2) = {0, 1, v, 1 + v} with v(2) = v. We use the double-circulant, the bordered double-circulant and the symmetric construction methods to obtain free Hermitian self-dual codes of even length. By describing a new shortening method over this ring, we are able to obtain Hermitian self-dual codes of odd length. Using these methods, we also obtain a number of extremal codes. We tabulate the Hermitian self-dual codes with the highest minimum weights of lengths up to 50. (C) 2019 Elsevier B.V. All rights reserved.
We classify all binary linear [30, 15, 8] codes with dual distance 8 using the software package QEXTNEWEDITION. There are exactly 42 such codes and all of them are formally self-dual even codes. This result closes the...
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ISBN:
(纸本)9781665402873
We classify all binary linear [30, 15, 8] codes with dual distance 8 using the software package QEXTNEWEDITION. There are exactly 42 such codes and all of them are formally self-dual even codes. This result closes the long standing open problem about the existence of a formally self-dual odd code with these parameters.
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