Algebraic hierarchical locally recoverable codes with nested affine subspace recovery

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

Designs, Codes and Cryptography Pub Date : 2024-10-24 DOI:10.1007/s10623-024-01510-x

Kathryn Haymaker, Beth Malmskog, Gretchen Matthews

引用次数: 0

Abstract

Codes with locality, also known as locally recoverable codes, allow for recovery of erasures using proper subsets of other coordinates. These subsets are typically of small cardinality to promote recovery using limited network traffic and other resources. Hierarchical locally recoverable codes allow for recovery of erasures using sets of other symbols whose sizes increase as needed to allow for recovery of more symbols. In this paper, we describe a hierarchical recovery structure arising from geometry in Reed–Muller codes and codes with availability from fiber products of curves. We demonstrate how the fiber product hierarchical codes can be viewed as punctured subcodes of Reed–Muller codes, uniting the two constructions. This point of view provides natural structures for local recovery with availability at each level in the hierarchy.

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具有嵌套仿射子空间恢复功能的代数分层局部可恢复编码

具有局部性的代码，也称为局部可恢复代码，允许使用其他坐标的适当子集来恢复擦除。这些子集通常具有较小的卡度，以促进利用有限的网络流量和其他资源进行恢复。分层局部可恢复代码允许使用其他符号集恢复擦除，这些符号集的大小会根据需要增加，以便恢复更多符号。在本文中，我们描述了一种分层恢复结构，这种结构源于里德-穆勒编码中的几何结构，以及由曲线的纤维乘积产生的可用性编码。我们演示了如何将纤维积分层码视为里德-穆勒码的穿刺子码，从而将这两种结构结合起来。这种观点为在层次结构中的每一级提供了具有可用性的局部恢复的自然结构。

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.