Minjia Shi, Tingting Tong, Thomas Honold, Patrick Solé
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引用次数: 0
Abstract
The problem of determining the largest possible number of distinct Hamming weights in several classes of codes over finite fields was studied recently in several papers (Shi et al. in Des Codes Cryptogr 87(1):87–95, 2019, in IEEE Trans Inf Theory 66(11):6855–6862, 2020; Chen et al. in IEEE Trans Inf Theory 69(2):995–1004, 2022). A further problem is to find the minimum length of codes meeting those bounds with equality. These two questions are extended here to linear codes over chain rings for the homogeneous weight. An explicit upper bound is given for codes of given type and arbitrary length as a function of the residue field size. This bound is then shown to be tight by an argument based on Hjemslev geometries. The second question is studied for chain rings with residue field of order two.
最近有几篇论文研究了确定有限域上几类编码中最大可能数目的不同汉明权重的问题（Shi 等，载于 Des Codes Cryptogr 87(1):87-95, 2019；IEEE Trans Inf Theory 66(11):6855-6862, 2020；Chen 等，载于 IEEE Trans Inf Theory 69(2):995-1004, 2022）。另一个问题是找到满足这些等价界限的编码的最小长度。这两个问题在这里被扩展到同权重链环上的线性编码。对于给定类型和任意长度的编码，给出了一个明确的上界，它是残差域大小的函数。然后，通过基于赫耶姆斯列夫几何的论证，证明了这一约束的严密性。第二个问题是研究具有二阶残差域的链环。
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