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Strong Singleton-Like Bounds, Quasi-Perfect Codes and Distance-Optimal Codes in the Sum-Rank Metric

Chao Liu, Hao Chen, Qinqin Ji, Ziyan Xie, Dabin Zheng

TL;DR

This work advances the theory of sum-rank codes by establishing covering-code based constructions that yield new upper bounds on size and block-lengths, and by proving strong Singleton-like bounds that outperform classical bounds for large block lengths. It introduces infinite families of quasi-perfect sum-rank codes, distance-optimal cyclic codes with minimum sum-rank distance four, and almost MSRD codes with larger block lengths, all tied to concrete cyclic/Hamming constructions and BCH-based arguments. Additionally, the paper develops Plotkin-sums for sum-rank codes to produce further distance-optimal instances, including binary constructions with larger block lengths. Collectively, these results deepen the toolkit for designing robust sum-rank codes in multishot network coding, space-time coding, and distributed storage, with practical impact in achieving tight distance and covering properties over diverse matrix sizes.

Abstract

Codes in the sum-rank metric have received many attentions in recent years, since they have wide applications in the multishot network coding, the space-time coding and the distributed storage. In this paper, by constructing covering codes in the sum-rank metric from covering codes in the Hamming metric, we derive new upper bounds on sizes, the covering radii and the block length functions of codes in the sum-rank metric. As applications, we present several strong Singleton-like bounds that are tighter than the classical Singleton-like bound when block lengths are large. In addition, we give the explicit constructions of the distance-optimal sum-rank codes of matrix sizes $s\times s$ and $2\times 2$ with minimum sum-rank distance four respectively by using cyclic codes in the Hamming metric. More importantly, we present an infinite families of quasi-perfect $q$-ary sum-rank codes with matrix sizes $2\times m$. Furthermore, we construct almost MSRD codes with larger block lengths and demonstrate how the Plotkin sum can be used to give more distance-optimal sum-rank codes.

Strong Singleton-Like Bounds, Quasi-Perfect Codes and Distance-Optimal Codes in the Sum-Rank Metric

TL;DR

This work advances the theory of sum-rank codes by establishing covering-code based constructions that yield new upper bounds on size and block-lengths, and by proving strong Singleton-like bounds that outperform classical bounds for large block lengths. It introduces infinite families of quasi-perfect sum-rank codes, distance-optimal cyclic codes with minimum sum-rank distance four, and almost MSRD codes with larger block lengths, all tied to concrete cyclic/Hamming constructions and BCH-based arguments. Additionally, the paper develops Plotkin-sums for sum-rank codes to produce further distance-optimal instances, including binary constructions with larger block lengths. Collectively, these results deepen the toolkit for designing robust sum-rank codes in multishot network coding, space-time coding, and distributed storage, with practical impact in achieving tight distance and covering properties over diverse matrix sizes.

Abstract

Codes in the sum-rank metric have received many attentions in recent years, since they have wide applications in the multishot network coding, the space-time coding and the distributed storage. In this paper, by constructing covering codes in the sum-rank metric from covering codes in the Hamming metric, we derive new upper bounds on sizes, the covering radii and the block length functions of codes in the sum-rank metric. As applications, we present several strong Singleton-like bounds that are tighter than the classical Singleton-like bound when block lengths are large. In addition, we give the explicit constructions of the distance-optimal sum-rank codes of matrix sizes and with minimum sum-rank distance four respectively by using cyclic codes in the Hamming metric. More importantly, we present an infinite families of quasi-perfect -ary sum-rank codes with matrix sizes . Furthermore, we construct almost MSRD codes with larger block lengths and demonstrate how the Plotkin sum can be used to give more distance-optimal sum-rank codes.
Paper Structure (12 sections, 25 theorems, 84 equations)

This paper contains 12 sections, 25 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Codes in the Hamming metric and the sum-rank metric
  3. Related works
  4. Our contributions
  5. A construction of covering codes in the sum-rank metric
  6. Strong Singleton-like bounds on general sum-rank codes
  7. Constructions of quasi-perfect sum-rank codes
  8. Constructions of distance-optimal $q$-ary sum-rank codes
  9. Distance-optimal $q$-ary codes in the sum-rank metric with matrix size $s\times s$
  10. Distance-optimal $q$-ary sum-rank codes with matrix size $2\times 2$
  11. Plotkin sum of sum-rank codes
  12. Concluding remarks

Key Result

Theorem 1

Let ${\mathcal{C}}_1,{\mathcal{C}}_2,\dots,{\mathcal{C}}_m$ be $m$ general Hamming metric codes in ${\mathbb{F}_{q^m}^t}$ with covering radius $R_1,R_2,\dots,R_m$ respectively. Then for any ${\bf x}\in {\mathbb{F}_q^{(m,m),\dots,(m,m)}}$, there is a codeword ${\bf c}\in SR_{covering}({\mathcal{C}}_1

Theorems & Definitions (51)

  • Definition 1
  • Definition 2
  • Theorem 1
  • proof
  • Corollary 1
  • Lemma 1
  • proof
  • Lemma 2
  • Theorem 2
  • proof
  • ...and 41 more