New Perfect and Distance-Optimal Sum-Rank Codes

arXiv - CS - Information Theory Pub Date : 2024-01-20 DOI:arxiv-2401.11160

Hao Chen

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Abstract

Constructions of infinite families of distance-optimal codes in the Hamming metric and the sum-rank metric are challenging problems and have attracted many attentions. In this paper, we give the following three results. 1) If $\lambda|q^{sm}-1$ and $\lambda <\sqrt{\frac{(q^s-1)}{2(q-1)^2(1+\epsilon)}}$, an infinite family of distance-optimal $q$-ary cyclic sum-rank codes with the block length $t=\frac{q^{sm}-1}{\lambda}$, the matrix size $s \times s$, the cardinality $q^{s^2t-s(2m+3)}$ and the minimum sum-rank distance four is constructed. 2) Block length $q^4-1$ and the matrix size $2 \times 2$ distance-optimal sum-rank codes with the minimum sum-rank distance four and the Singleton defect four are constructed. These sum-rank codes are close to the sphere packing bound , the Singleton-like bound and have much larger block length $q^4-1>>q-1$. 3) For given positive integers $n$ and $m$ satisfying $m

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新的完美代码和距离最优和值代码

构建汉明度量和和秩度量中的无限距离最优码族是一个具有挑战性的问题，吸引了许多人的关注。本文给出了以下三个结果。1) 若 $\lambda|q^{sm}-1$ 且 $\lambda>q-1$.3) 对于给定的满足 $m

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