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Repeated-Root Cyclic Codes with Optimal Parameters or Best Parameters Known

Hao Chen, Conghui Xie, Cunsheng Ding

TL;DR

This work advances repeated-root cyclic codes by constructing several infinite families with distance-optimal or best-known parameters across even and binary fields, leveraging BCH building blocks and the generalized van Lint framework. Central to the approach is the Plotkin sum construction, which yields repeated-root codes ${f C}(g_1,g_2)$ with minimum distance $d= ext{min}igrace 2d_1, d_2 igrace$ and explicit dimension formulas, enabling systematic generation of codes with $d=3,4,6,8$ or higher. The paper demonstrates distance-optimality via sphere-packing bounds and related criteria, provides extensive instances (including 27 codes of length up to $254$ with optimal or best-known parameters), and shows that repeated-root cyclic codes can rival or surpass many simple-root constructions. Overall, the results indicate that repeated-root cyclic codes are a fruitful area with promising infinite families, particularly for even $q$ and binary fields, and they invite further exploration into higher-distance regimes and smaller alphabets.

Abstract

Cyclic codes are the most studied subclass of linear codes and widely used in data storage and communication systems. Many cyclic codes have optimal parameters or the best parameters known. They are divided into simple-root cyclic codes and repeated-root cyclic codes. Although there are a huge number of references on cyclic codes, few of them are on repeated-root cyclic codes. Hence, repeated-root cyclic codes are rarely studied. There are a few families of distance-optimal repeated-root binary and $p$-ary cyclic codes for odd prime $p$ in the literature. However, it is open whether there exists an infinite family of distance-optimal repeated-root cyclic codes over $\bF_q$ for each even $q \geq 4$. In this paper, three infinite families of distance-optimal repeated-root cyclic codes with minimum distance 3 or 4 are constructed; two other infinite families of repeated-root cyclic codes with minimum distance 3 or 4 are developed; seven infinite families of repeated-root cyclic codes with minimum distance 6 or 8 or 10 are presented; and two infinite families of repeated-root binary cyclic codes with parameters $[2n, k, d \geq (n-1)/\log_2 n]$, where $n=2^m-1$ and $k \geq n$, are constructed. In addition, 27 repeated-root cyclic codes of length up to $254$ over $\bF_q$ for $q \in \{2, 4, 8\}$ with optimal parameters or best parameters known are obtained in this paper. The results of this paper show that repeated-root cyclic codes could be very attractive and are worth of further investigation.

Repeated-Root Cyclic Codes with Optimal Parameters or Best Parameters Known

TL;DR

This work advances repeated-root cyclic codes by constructing several infinite families with distance-optimal or best-known parameters across even and binary fields, leveraging BCH building blocks and the generalized van Lint framework. Central to the approach is the Plotkin sum construction, which yields repeated-root codes with minimum distance and explicit dimension formulas, enabling systematic generation of codes with or higher. The paper demonstrates distance-optimality via sphere-packing bounds and related criteria, provides extensive instances (including 27 codes of length up to with optimal or best-known parameters), and shows that repeated-root cyclic codes can rival or surpass many simple-root constructions. Overall, the results indicate that repeated-root cyclic codes are a fruitful area with promising infinite families, particularly for even and binary fields, and they invite further exploration into higher-distance regimes and smaller alphabets.

Abstract

Cyclic codes are the most studied subclass of linear codes and widely used in data storage and communication systems. Many cyclic codes have optimal parameters or the best parameters known. They are divided into simple-root cyclic codes and repeated-root cyclic codes. Although there are a huge number of references on cyclic codes, few of them are on repeated-root cyclic codes. Hence, repeated-root cyclic codes are rarely studied. There are a few families of distance-optimal repeated-root binary and -ary cyclic codes for odd prime in the literature. However, it is open whether there exists an infinite family of distance-optimal repeated-root cyclic codes over for each even . In this paper, three infinite families of distance-optimal repeated-root cyclic codes with minimum distance 3 or 4 are constructed; two other infinite families of repeated-root cyclic codes with minimum distance 3 or 4 are developed; seven infinite families of repeated-root cyclic codes with minimum distance 6 or 8 or 10 are presented; and two infinite families of repeated-root binary cyclic codes with parameters , where and , are constructed. In addition, 27 repeated-root cyclic codes of length up to over for with optimal parameters or best parameters known are obtained in this paper. The results of this paper show that repeated-root cyclic codes could be very attractive and are worth of further investigation.
Paper Structure (12 sections, 18 theorems, 55 equations)

This paper contains 12 sections, 18 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Cyclic codes and repeated-root cyclic codes
  3. BCH cyclic codes
  4. Motivations and objectives of this paper
  5. The organisation of this paper
  6. The generalized van Lint theorem
  7. Infinite families of repeated-root cyclic codes with minimum distance four or three
  8. Infinite families of repeated-root cyclic codes over ${\mathbf{{F}}}_2$ with best parameters known
  9. Infinite families of repeated-root cyclic codes over ${\mathbf{{F}}}_2$ with minimum distance $8$ or at least $10$
  10. Two infinite families of repeated-root cyclic codes over ${\mathbf{{F}}}_q$ with minimum distance 6 for $q \geq 4$
  11. Infinite families of repeated-root binary cyclic codes with large dimensions and large minimum distances
  12. Summary of contributions and concluding remarks

Key Result

Theorem 2.1

Let $q$ be a power of $2$ and $n$ be an odd positive integer. Let ${\bf C}_1 \subseteq {\bf F}_q^n$ be a cyclic code with generator polynomial $g_1(x) \in {\bf F}_q[x]$ and ${\bf C}_2 \subseteq {\bf F}_q^n$ be a cyclic code generated by the polynomial $g_1(x)g_2(x) \in {\bf F}_q[x]$, where $g_2(x)$ dimension and minimum distance here and hereafter $d({\mathbf{{C}}})$ denotes the minimum distanc

Theorems & Definitions (37)

  • Theorem 2.1: The generalized van Lint theorem ChenDing
  • Lemma 3.1
  • Theorem 3.1
  • Example 3.1
  • Theorem 3.2
  • Example 3.2
  • Example 3.3
  • Example 3.4
  • Theorem 3.3
  • Example 3.5
  • ...and 27 more