Ternary isodual codes and 3-designs

IF 1.4 2区数学 Q3 COMPUTER SCIENCE, THEORY & METHODS

Designs, Codes and Cryptography Pub Date : 2025-01-06 DOI:10.1007/s10623-024-01558-9

Minjia Shi, Ruowen Liu, Dean Crnković, Patrick Solé, Andrea Švob

引用次数: 0

Abstract

Ternary isodual codes and their duals are shown to support 3-designs under mild symmetry conditions. These designs are held invariant by a double cover of the permutation part of the automorphism group of the code. Examples of interest include extended quadratic residues (QR) codes of lengths 14 and 38 whose automorphism groups are PSL(2, 13) and PSL(2, 37), respectively. We also consider Generalized Quadratic Residue (GQR) codes in the sense of Lint and MacWiliams (IEEE Trans Inf Theory 24(6): 730-737,1978). These codes are the abelian generalizations of the Quadratic Residue (QR) codes which are cyclic. We construct them as row span of a Jacobsthal matrix. In lengths 50 and 26 we obtain 3-designs invariant under a double cover of \(P{\Sigma }L(2,49),\) and \(P{\Sigma }L(2,25),\) respectively. In addition, from block orbits of these 3-designs we construct a number of other 3-designs and 2-designs. Finally, we apply the same construction to the binary extended GQR code of length 82.

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三进制单码和3-设计

在温和对称条件下，证明了三元等偶码及其对偶码支持3-设计。这些设计通过代码的自同构群的置换部分的双重覆盖保持不变。我们感兴趣的例子包括长度为14和38的扩展二次残码，它们的自同态群分别是PSL（2,13）和PSL（2,37）。我们还考虑了Lint和macwilliams意义上的广义二次残差（GQR）码（IEEE Trans Inf Theory 24(6): 730-737,1978）。这些码是循环二次残数码的阿贝尔推广。我们把它们构造成雅各布矩阵的行张成的空间。在长度50和26中，我们分别在\(P{\Sigma }L(2,49),\)和\(P{\Sigma }L(2,25),\)的双盖下获得了3种设计不变。此外，从这些3-设计的块轨道中，我们构建了许多其他3-设计和2-设计。最后，我们将相同的结构应用于长度为82的二进制扩展GQR码。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Designs, Codes and Cryptography 工程技术-计算机：理论方法

CiteScore

2.80

自引率

12.50%

发文量

157

审稿时长

16.5 months

期刊介绍： Designs, Codes and Cryptography is an archival peer-reviewed technical journal publishing original research papers in the designated areas. There is a great deal of activity in design theory, coding theory and cryptography, including a substantial amount of research which brings together more than one of the subjects. While many journals exist for each of the individual areas, few encourage the interaction of the disciplines. The journal was founded to meet the needs of mathematicians, engineers and computer scientists working in these areas, whose interests extend beyond the bounds of any one of the individual disciplines. The journal provides a forum for high quality research in its three areas, with papers touching more than one of the areas especially welcome. The journal also considers high quality submissions in the closely related areas of finite fields and finite geometries, which provide important tools for both the construction and the actual application of designs, codes and cryptographic systems. In particular, it includes (mostly theoretical) papers on computational aspects of finite fields. It also considers topics in sequence design, which frequently admit equivalent formulations in the journal’s main areas. Designs, Codes and Cryptography is mathematically oriented, emphasizing the algebraic and geometric aspects of the areas it covers. The journal considers high quality papers of both a theoretical and a practical nature, provided they contain a substantial amount of mathematics.