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Cyclic and Negacyclic Sum-Rank Codes

Hao Chen, Cunsheng Ding, Zhiqiang Cheng, Conghui Xie

TL;DR

This work introduces cyclic, negacyclic, and constacyclic sum-rank codes (CSR/NSR/λ-constacyclic) and provides a direct bridge from cyclic/Hamming-metric codes to sum-rank constructions. It extends the CSC framework and derives BCH and Hartmann-Tzeng type bounds for CSR codes, enabling explicit distance and dimension guarantees. The authors present several families of CSR/NSR/λ-constacyclic sum-rank codes with known dimensions and controllable minimum sum-rank distances, including an infinite family of distance-optimal binary cyclic sum-rank codes with $d_{sr}=4$. They also show decoding reductions to underlying component codes, broadening practical applicability for multishot network coding and distributed storage. Overall, the paper advances systematic construction and analysis of sum-rank codes with strong distance properties via cyclic-algebraic methods.

Abstract

Sum-rank codes have known applications in the multishot network coding, the distributed storage and the construction of space-time codes. U. Martínez-Peñas introduced the cyclic-skew-cyclic sum-rank codes and proposed the BCH bound on the cyclic-skew-cyclic sum-rank codes in his paper published in IEEE Trans. Inf. Theory, vol. 67, no. 8, 2021. Afterwards, many sum-rank BCH codes with lower bounds on their dimensions and minimum sum-rank distances were constructed. Sum-rank Hartmann-Tzeng bound and sum-rank Roos bound on cyclic-skew-cyclic codes were proposed and proved by G. N. Alfarano, F. J. Lobillo, A. Neri, and A. Wachter-Zeh in 2022. In this paper, cyclic, negacyclic and constacyclic sum-rank codes are introduced and a direct construction of cyclic, negacyclic and constacyclic sum-rank codes of the matrix size $m \times m$ from cyclic, negacyclic and constacyclic codes over ${\bf F}_{q^m}$ in the Hamming metric is proposed. The cyclic-skew-cylic sum-rank codes are special cyclic sum-rank codes. In addition, BCH and Hartmann-Tzeng bounds for a type of cyclic sum-rank codes are developed. Specific constructions of cyclic, negacyclic and constacyclic sum-rank codes with known dimensions and controllable minimum sum-rank distances are proposed. Moreover, many distance-optimal binary sum-rank codes and an infinite family of distance-optimal binary cyclic sum-rank codes with minimum sum-rank distance four are constructed. This is the first infinite family of distance-optimal sum-rank codes with minimum sum-rank distance four in the literature.

Cyclic and Negacyclic Sum-Rank Codes

TL;DR

This work introduces cyclic, negacyclic, and constacyclic sum-rank codes (CSR/NSR/λ-constacyclic) and provides a direct bridge from cyclic/Hamming-metric codes to sum-rank constructions. It extends the CSC framework and derives BCH and Hartmann-Tzeng type bounds for CSR codes, enabling explicit distance and dimension guarantees. The authors present several families of CSR/NSR/λ-constacyclic sum-rank codes with known dimensions and controllable minimum sum-rank distances, including an infinite family of distance-optimal binary cyclic sum-rank codes with . They also show decoding reductions to underlying component codes, broadening practical applicability for multishot network coding and distributed storage. Overall, the paper advances systematic construction and analysis of sum-rank codes with strong distance properties via cyclic-algebraic methods.

Abstract

Sum-rank codes have known applications in the multishot network coding, the distributed storage and the construction of space-time codes. U. Martínez-Peñas introduced the cyclic-skew-cyclic sum-rank codes and proposed the BCH bound on the cyclic-skew-cyclic sum-rank codes in his paper published in IEEE Trans. Inf. Theory, vol. 67, no. 8, 2021. Afterwards, many sum-rank BCH codes with lower bounds on their dimensions and minimum sum-rank distances were constructed. Sum-rank Hartmann-Tzeng bound and sum-rank Roos bound on cyclic-skew-cyclic codes were proposed and proved by G. N. Alfarano, F. J. Lobillo, A. Neri, and A. Wachter-Zeh in 2022. In this paper, cyclic, negacyclic and constacyclic sum-rank codes are introduced and a direct construction of cyclic, negacyclic and constacyclic sum-rank codes of the matrix size from cyclic, negacyclic and constacyclic codes over in the Hamming metric is proposed. The cyclic-skew-cylic sum-rank codes are special cyclic sum-rank codes. In addition, BCH and Hartmann-Tzeng bounds for a type of cyclic sum-rank codes are developed. Specific constructions of cyclic, negacyclic and constacyclic sum-rank codes with known dimensions and controllable minimum sum-rank distances are proposed. Moreover, many distance-optimal binary sum-rank codes and an infinite family of distance-optimal binary cyclic sum-rank codes with minimum sum-rank distance four are constructed. This is the first infinite family of distance-optimal sum-rank codes with minimum sum-rank distance four in the literature.
Paper Structure (15 sections, 17 theorems, 50 equations)

This paper contains 15 sections, 17 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Cyclic-skew-cyclic sum-rank codes and their bounds
  3. Cyclic and negacyclic sum-rank codes
  4. The classical constacyclic codes in the Hamming metric
  5. Lower bounds on cyclic codes in the Hamming metric
  6. Cyclic and negacyclic sum-rank codes
  7. Cyclic sum-rank codes with the matrix size $m \times m$
  8. Specific families of cyclic sum-rank codes
  9. Cyclic sum-rank codes from BCH cyclic codes
  10. Cyclic sum-rank codes from other cyclic codes
  11. Negacyclic sum-rank codes
  12. Constacyclic sum-rank codes
  13. Distance-optimal binary cyclic sum-rank codes
  14. Decoding of the cyclic or negacyclic sum-rank codes $\mathrm{SR}({\bf C}_0, \ldots, {\bf C}_{m-1})$
  15. Concluding remarks

Key Result

Theorem 2.1

If the defining set ${\bf T_C}$ of the cyclic-skew-cyclic code ${\bf C}$ is the smallest that contains $\{(a^{b+i},\sigma^{i}(\beta)) \in {\bf F}_{q^s}\times {\bf F}_{q^{ns}}^*:0\leq i\leq \delta-2\}$, where $b\geq 0$ and $2\leq\delta\leq mt$, then $d_{sr}\geq\delta$, $\dim_{{\bf F}_{q}}({\bf C})\ge

Theorems & Definitions (32)

  • Theorem 2.1: Sum-rank BCH bound, MP21
  • Theorem 2.2: Sum-rank HT bound, ALNWZ
  • Theorem 2.3: Sum-rank Roos bound, ALNWZ
  • Definition 3.1: Cyclic sum-rank code
  • Definition 3.2: Negacyclic sum-rank code
  • Theorem 3.1
  • Proposition 3.1
  • Proposition 3.2
  • Example 3.1
  • Theorem 3.2: BCH bound for cyclic sum-rank codes $\mathrm{SR}({\mathbf{{C}}}_0, \ldots, {\mathbf{{C}}}_{m-1})$
  • ...and 22 more