Binary $[n,(n\pm1)/2]$ cyclic codes with good minimum distances from sequences
Xianhong Xie, Yaxin Zhao, Zhonghua Sun, Xiaobo Zhou
TL;DR
The paper develops four binary cyclic code families, $\mathcal{C}_{\mathcal{S},0}$, $\mathcal{C}_{\mathcal{S},1}$, $\mathcal{C}_{\mathcal{D},0}$, and $\mathcal{C}_{\mathcal{D},1}$, of length $n=2^m-1$ by leveraging two sequence constructions and links them to the cyclotomic-coset framework. By analyzing Si-Ding and Ding-Zhou sequences, the authors obtain subcodes of Tang et al.’s binary cyclic codes and derive new lower bounds on the minimum distance, showing cases where $d^2\ge n$ for $m$ even and significant $d$ values for several $m$ odd. The results include explicit bounds such as $d(\mathcal{C}_{\mathcal{S},1})\ge 2^{m/2}+2$ (and higher in some regimes) and $d(\mathcal{C}_{\mathcal{D},1})\ge 2^{m/2}+4$ (with parallel bounds for $\mathcal{C}_{\mathcal{D},0}$), as well as concrete small-$m$ optimal codes. These constructions advance the search for near-optimal $[n,(n\pm1)/2,d]$ binary cyclic codes by exploiting defining-set analysis, BCH-type distance bounds, and coset-structure arguments that tie to well-studied code families.
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 $h=2$.