Hulls of projective Reed–Muller codes

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

Designs, Codes and Cryptography Pub Date : 2024-12-14 DOI:10.1007/s10623-024-01543-2

Nathan Kaplan, Jon-Lark Kim

引用次数: 0

Abstract

Projective Reed–Muller codes are constructed from the family of projective hypersurfaces of a fixed degree over a finite field \(\mathbb {F}_q\). We consider the relationship between projective Reed–Muller codes and their duals. We determine when these codes are self-dual, when they are self-orthogonal, and when they are LCD. We then show that when q is sufficiently large, the dimension of the hull of a projective Reed–Muller code is 1 less than the dimension of the code. We determine the dimension of the hull for a wider range of parameters and describe how this leads to a new proof of a recent result of Ruano and San-José.

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射影里德-穆勒码的船体

射影Reed-Muller码是由有限域上的一组固定度的射影超曲面构成的\(\mathbb {F}_q\)。研究了投影Reed-Muller码及其对偶之间的关系。我们确定了这些码什么时候是自对偶的，什么时候是自正交的，什么时候是LCD的。然后，我们证明了当q足够大时，一个投影Reed-Muller码的外壳维数比码的维数小1。我们确定了更广泛参数范围的船体尺寸，并描述了这如何导致Ruano和san - jos最近结果的新证明。

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.