关于码的广义填充半径与覆盖半径

中文摘要 AI 辅助

最小距离和覆盖半径是码的两个基本性质。两者均已被推广：前者推广为广义汉明重量层级，后者推广为广义覆盖半径层级。在这两种情况下，层级的最低层分别对应于经典的最小距离和覆盖半径。从几何角度看，码的最小距离决定了填充半径，而填充半径以覆盖半径为上界。曾有猜想认为这一关系可推广到层级的所有其他阶，即同阶的广义填充半径以广义覆盖半径为上界。本文证明了该猜想对二阶半径成立。我们还证明了当码率至多为$3/5$时，该猜想对所有阶成立。最后，我们证明对于$(0,1)$中的任意码率，对所有足够长的码，该猜想对所有阶均成立。

英文摘要

The minimum distance and the covering radius are two fundamental properties of the code. Both have been extended: the former to the generalized Hamming weights hierarchy, and the latter to the generalized covering radii hierarchy. In both cases, the lowest level of the hierarchies corresponds to the classical minimum distance and covering radius, respectively. From a geometric point of view, the minimum distance of the code determines the packing radius, which is upper bounded by the covering radius. It was conjectured this relation extends to all other orders of the hierarchy, namely, that the generalized packing radii are upper bounded by the generalized covering radii of the same order. In this paper we prove this conjecture is true for the second order radii. We also prove the conjecture holds for all orders when the code rate is at most $3/5$. Finally, we show that for any code rate in $(0,1)$, for all sufficiently long codes the conjecture holds for all orders.

发表机构

• Institute of Mathematics and Interdisciplinary Sciences, Xidian University（西安电子科技大学数学与交叉科学研究院）
• Department of Electrical and Computer Engineering, McMaster University（麦克马斯特大学电气与计算机工程系）

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