In this note we answer some questions on LCD\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgr...
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In this note we answer some questions on LCD\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text{ LCD }$$\end{document} group codes posed in de la Cruz and Willems (Des codes Cryptogr 86:2065-2073, 2018) and (Vietnam J Math 51:721-729, 2023). Furthermore, over prime fields we determine completely the p-part of the divisor of an LCD\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text{ LCD }$$\end{document} group code. In addition we present a natural construction of nearly LCD\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text{ LCD }$$\end{document} codes.
In this paper, we consider essential idempotents in the finite semisimple group algebra of a nilpotent group, studying conditions for their existence and other implications. Also, we discuss conditions for nilpotent g...
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In this paper, we consider essential idempotents in the finite semisimple group algebra of a nilpotent group, studying conditions for their existence and other implications. Also, we discuss conditions for nilpotent group codes to be equivalent to Abelian ones.
Our aim is the design of an efficient decoding algorithm in group codes. The algorithm is inspired by the well known syndrome decoding algorithm for linear codes and uses the decomposition of a semisimple group algebr...
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Our aim is the design of an efficient decoding algorithm in group codes. The algorithm is inspired by the well known syndrome decoding algorithm for linear codes and uses the decomposition of a semisimple group algebra KG as a direct sum of two-sided ideals, each of them generated by a central idempotent of KG. When G is an abelian group the algorithm can be modified to make it very simple and efficient. Some illustrative examples are presented.(c) 2023 Elsevier Inc. All rights reserved.
group codes are right or left ideals in a group algebra of a finite group over a finite field. Following the ideas of a paper on binary group codes by Bazzi and Mitter in 2006, we prove that group codes over finite fi...
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group codes are right or left ideals in a group algebra of a finite group over a finite field. Following the ideas of a paper on binary group codes by Bazzi and Mitter in 2006, we prove that group codes over finite fields of any characteristic are asymptotically good. (C) 2020 Elsevier Inc. All rights reserved.
We investigate the properties of binary linear codes of even length whose permutation automorphism group is a cyclic group generated by an involution. Up to dimension or co-dimension 4, we show that there is no quasi-...
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We investigate the properties of binary linear codes of even length whose permutation automorphism group is a cyclic group generated by an involution. Up to dimension or co-dimension 4, we show that there is no quasi-group code whose permutation automorphism group is isomorphic to C-2. By generalizing the method we use to prove this result, we obtain results on the structure of putative extremal self-dual [16, 36, 72] and [20, 48, 96] codes in the presence of an involutory permutation automorphism.
Let p be an odd prime number, q = p(m) for a positive integer m, let F(q )be the finite field with q elements and omega be a primitive element of Fq. We first give an orthogonal decomposition of the ring R = F-q + nu ...
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Let p be an odd prime number, q = p(m) for a positive integer m, let F(q )be the finite field with q elements and omega be a primitive element of Fq. We first give an orthogonal decomposition of the ring R = F-q + nu F-q, where nu(2) = a(3), and a = omega(2l )for a fixed integer l. In addition, Galois dual of a linear code over R is discussed. Meanwhile, constacyclic codes and cyclic codes over the ring R are investigated as well. Remarkably, we obtain that if linear codes C and D are a complementary pair, then the code C and the dual code D-&updatedExpOTTOM;E of D are equivalent to each other.
In this short note we clarify some questions on the greatest common divisor of all weights of a group code. In particular we discuss Ward's condition (E) in Ward, H. (Q. J. Math. Oxford 34, 115-128, 1983) and exte...
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In this short note we clarify some questions on the greatest common divisor of all weights of a group code. In particular we discuss Ward's condition (E) in Ward, H. (Q. J. Math. Oxford 34, 115-128, 1983) and extend a result of Damgard and Landrock on the principal block to self-dual blocks. Furthermore, we give an upper bound for the dimension of a group code in terms of its monomial kernel.
Let F be a finite field and let G be a finite group. We show that if C is a G-code over F with dim(F)(C) 2 (see the examples in [1]), we conclude that the smallest dimension of a non-abelian group code over a finite ...
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Let F be a finite field and let G be a finite group. We show that if C is a G-code over F with dim(F)(C) <= 3 then C is an abelian group code. Since there exist non-abelian group codes of dimension 4 when char F > 2 (see the examples in [1]), we conclude that the smallest dimension of a non-abelian group code over a finite field is 4. Published by Elsevier Inc.
It is well-known that each left ideal in a matrix rings over a finite field is generated by an idempotent matrix. In this work we compute the number of left ideals in these rings, the number of different idempotents g...
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It is well-known that each left ideal in a matrix rings over a finite field is generated by an idempotent matrix. In this work we compute the number of left ideals in these rings, the number of different idempotents generating each left ideal, and give explicitly a set of idempotent generators of all left ideals of a given rank. We then apply these results to give examples of left group codes that have best possible minimum weight.
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