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New constructions of MDS symbol-pair codes via simple-root cyclic codes

Rongxing Qiu, Weijun Fang

TL;DR

The paper addresses the problem of constructing MDS symbol-pair codes with large pair distance using simple-root cyclic codes. The authors develop three infinite families by selecting generator roots and applying the decomposition of cyclic codes together with dual-orthogonality arguments, including equation-solving over finite fields for two cases and a decomposition-based approach for the third. The main contributions are explicit families $(n,d_P)_q=(4q+4,7)_q$ with $q\equiv1\pmod4$, $(4q-4,8)_q$ with $q\equiv3\pmod4$, and $(2q+2,9)_q$ for odd prime power $q$, achieving the longest known code lengths in several regimes and demonstrating the effectiveness of simple-root cyclic-code techniques for symbol-pair optimization. These results advance the systematic construction of MDS symbol-pair codes and offer a framework that could extend to larger pair distances and lengths.

Abstract

In modern storage technologies, symbol-pair codes have emerged as a crucial framework for addressing errors in channels where symbols are read in overlapping pairs to guard against pair errors. A symbol-pair code that meets the Singleton-type bound is called a maximum distance separable (MDS) symbol-pair code. MDS symbol-pair codes are optimal in the sense that they have the highest pair error-correcting capability. In this paper, we focus on new constructions of MDS symbol-pair codes using simple-root cyclic codes. Specifically, three new infinite families of $(n, d_P)_q$-MDS symbol-pair codes are obtained: (1) $(n=4q+4,d_P=7)_q$ for $q\equiv 1\pmod 4$; (2) $(n=4q-4,d_P=8)_q$ for $q\equiv 3\pmod 4$; (3) $(n=2q+2,d_P=9)_q$ for $q$ being an odd prime power. The first two constructions are based on analyzing the solutions of certain equations over finite fields. The third construction arises from the decomposition of cyclic codes, where we utilize the orthogonal relationships between component codes and their duals to rigorously exclude the presence of specific codewords. It is worth noting that for the pair distance $d_P=7$ or $8$, our $q$-ary MDS symbol-pair codes achieve the longest known code length when $q$ is not a prime. Furthermore, for $d_P=9$, our codes attain the longest code length regardless of whether $q$ is prime or not.

New constructions of MDS symbol-pair codes via simple-root cyclic codes

TL;DR

The paper addresses the problem of constructing MDS symbol-pair codes with large pair distance using simple-root cyclic codes. The authors develop three infinite families by selecting generator roots and applying the decomposition of cyclic codes together with dual-orthogonality arguments, including equation-solving over finite fields for two cases and a decomposition-based approach for the third. The main contributions are explicit families with , with , and for odd prime power , achieving the longest known code lengths in several regimes and demonstrating the effectiveness of simple-root cyclic-code techniques for symbol-pair optimization. These results advance the systematic construction of MDS symbol-pair codes and offer a framework that could extend to larger pair distances and lengths.

Abstract

In modern storage technologies, symbol-pair codes have emerged as a crucial framework for addressing errors in channels where symbols are read in overlapping pairs to guard against pair errors. A symbol-pair code that meets the Singleton-type bound is called a maximum distance separable (MDS) symbol-pair code. MDS symbol-pair codes are optimal in the sense that they have the highest pair error-correcting capability. In this paper, we focus on new constructions of MDS symbol-pair codes using simple-root cyclic codes. Specifically, three new infinite families of -MDS symbol-pair codes are obtained: (1) for ; (2) for ; (3) for being an odd prime power. The first two constructions are based on analyzing the solutions of certain equations over finite fields. The third construction arises from the decomposition of cyclic codes, where we utilize the orthogonal relationships between component codes and their duals to rigorously exclude the presence of specific codewords. It is worth noting that for the pair distance or , our -ary MDS symbol-pair codes achieve the longest known code length when is not a prime. Furthermore, for , our codes attain the longest code length regardless of whether is prime or not.

Paper Structure

This paper contains 9 sections, 20 theorems, 38 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Constacyclic Codes
  4. The Decomposition of Cyclic Codes
  5. Constructions of MDS symbol-pair codes
  6. MDS symbol-pair codes with $d_P=7$ and $n=4q+4$
  7. MDS symbol-pair codes with $d_P=8$ and $n=4q-4$
  8. MDS symbol-pair codes with $d_P = 9$ and $n=2q+2$
  9. Conclusion

Key Result

Lemma 1

Assume that $\gcd(n,q)=1$. Let $\mathcal{C}$ be a $\lambda$-constacyclic code over $\mathbb{F}_q$ of length $n$ with defining set $T$. Let ord$(\lambda)=r$. If $\left\{1+ri: 0 \leq i \leq \delta -2 \right\} \subseteq T$, then the minimum Hamming distance of $\mathcal{C}$ is not less than $\delta$.

Theorems & Definitions (38)

  • Lemma 1: BCH bound for constacyclic codes Krishna1990
  • Lemma 2: Hartmann–Tzeng bound for cyclic codes Huffman2010
  • Lemma 3: Chen2017
  • Lemma 4: Hughes2000
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • ...and 28 more