New and improved formally self-dual codes with small hulls from polynomial four Toeplitz codes

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

Designs, Codes and Cryptography Pub Date : 2024-07-20 DOI:10.1007/s10623-024-01460-4

Yang Li, Shitao Li, Shixin Zhu

引用次数: 0

Abstract

Formally self-dual (FSD) codes and linear codes with small Euclidean (resp. Hermitian) hulls have recently attracted a lot of attention due to their theoretical and practical importance. However, there has been not much attention on FSD codes with small hulls. In this paper, we introduce two kinds of polynomial four Toeplitz codes and prove that they must be FSD. We characterize the linear complementary dual (LCD) properties and one-dimensional hull properties of such codes with respect to the Euclidean and Hermitian inner products. Using these characterizations, we find some improved binary, ternary Euclidean and quaternary Hermitian FSD LCD codes, as well as many non-equivalent ones that perform equally well with respect to best-known (FSD) LCD codes in the literature. Furthermore, some (near) maximum distance separable FSD codes with both one-dimensional Euclidean hull and one-dimensional Hermitian hull are also given as examples.

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从多项式四托普利兹码出发的新的和改进的形式上自偶的小壳码

具有小欧几里得（或赫米特）空格的形式上自偶（FSD）编码和线性编码由于其理论和实践上的重要性，最近引起了广泛关注。然而，人们对具有小体的 FSD 码的关注却不多。本文介绍了两种多项式四 Toeplitz 码，并证明它们一定是 FSD 码。我们描述了这些编码的线性互补对偶（LCD）特性和一维空壳特性，以及欧氏和赫米特内积。利用这些特性，我们发现了一些改进的二元、三元欧氏和四元赫米特 FSD LCD 代码，以及许多与文献中最著名的（FSD）LCD 代码性能相当的非等价代码。此外，还举例说明了一些具有一维欧氏壳和一维赫米特壳的（接近）最大距离可分离 FSD 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.