Binary $[n,(n\pm1)/2]$ cyclic codes with good minimum distances from sequences

TL;DR

This paper constructs new binary cyclic codes with parameters close to the square-root bound for minimum distance, using sequence-based methods, and provides explicit parameters for specific cases.

Contribution

It introduces four classes of binary cyclic codes with good minimum distances derived from sequence constructions, expanding the known code families with near optimal parameters.

Findings

Codes achieve minimum distances close to the square-root bound.

Explicit parameters for codes when m ≡ 2 mod 4.

New code constructions with parameters [n,(n±1)/2,≥√n].

Abstract

Recently, binary cyclic codes with parameters $[n,(n\pm1)/2,\geq \sqrt{n}]$ have been a hot topic since their minimum distances have a square-root bound. In this paper, we construct four classes of binary cyclic codes $\mathcal{C}_{\mathcal{S},0}$, $\mathcal{C}_{\mathcal{S},1}$ and $\mathcal{C}_{\mathcal{D},0}$, $\mathcal{C}_{\mathcal{D},1}$ by using two families of sequences, and obtain some codes with parameters $[n,(n\pm1)/2,\geq \sqrt{n}]$. For $m\equiv2\pmod4$, the code $\mathcal{C}_{\mathcal{S},0}$ has parameters $[2^m-1,2^{m-1},\geq2^{\frac{m}{2}}+2]$, and the code $\mathcal{C}_{\mathcal{D},0}$ has parameters $[2^m-1,2^{m-1},\geq2^{\frac{m}{2}}+2]$ if $h=1$ and $[2^m-1,2^{m-1},\geq2^{\frac{m}{2}}]$ if $h=2$.