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Binary duadic codes and their related codes with a square-root-like lower bound

Tingting Wu, Lanqiang Li, Xiuyu Zhang, Shixin Zhu

Abstract

Binary cyclic codes have been a hot topic for many years, and significant progress has been made in the study of this types of codes. As is well known, it is hard to construct infinite families of binary cyclic codes [n, n+1/2] with good minimum distance. In this paper, by using the BCH bound on cyclic codes, one of the open problems proposed by Liu et al. about binary cyclic codes (Finite Field Appl 91:102270, 2023) is settled. Specially, we present several families of binary duadic codes with length 2^m-1 and dimension 2^(m-1), and the minimum distances have a square-root-like lower bound. As a by-product, the parameters of their dual codes and extended codes are provided, where the latter are self-dual and doubly-even.

Binary duadic codes and their related codes with a square-root-like lower bound

Abstract

Binary cyclic codes have been a hot topic for many years, and significant progress has been made in the study of this types of codes. As is well known, it is hard to construct infinite families of binary cyclic codes [n, n+1/2] with good minimum distance. In this paper, by using the BCH bound on cyclic codes, one of the open problems proposed by Liu et al. about binary cyclic codes (Finite Field Appl 91:102270, 2023) is settled. Specially, we present several families of binary duadic codes with length 2^m-1 and dimension 2^(m-1), and the minimum distances have a square-root-like lower bound. As a by-product, the parameters of their dual codes and extended codes are provided, where the latter are self-dual and doubly-even.
Paper Structure (19 theorems, 49 equations)

This paper contains 19 theorems, 49 equations.

Key Result

Theorem 1

(Square root bound) Let $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ be a pair of odd-like duadic codes of length $n$ over $\mathbb{F}_{2}$. Let $d_0$ be their (common) minimum odd weight. Then the following hold: (1) $d_{0}^{2}\geq n$. (2) If the splitting defining the duadic codes is given by $\mu=-1$,

Theorems & Definitions (25)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • proof
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • ...and 15 more