Binary [n,(n ± 1)/2] cyclic codes with good minimum distances from sequences

Abstract

Recently, binary cyclic codes with parameters $[n,(n\pm 1)/2,\ge \sqrt{n}]$ 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 $C_{S,0}$, $C_{S,1}$ and $C_{D,0}$, $C_{D,1}$ by using two families of sequences, and obtain some codes with parameters $[n,(n\pm 1)/2,\ge \sqrt{n}]$. For $m\equiv 2\left(\mathrm{mod}4\right)$, the code $C_{S,0}$ has parameters $[2^m-1,2^{m-1},\ge 2^{\frac{m}{2}}+2]$, and the code $C_{D,0}$ has parameters $[2^m-1,2^{m-1},\ge 2^{\frac{m}{2}}+2]$ if $h=1$ and $[2^m-1,2^{m-1},\ge 2^{\frac{m}{2}}]$ if $h=2$.