Binary $[n,(n\pm 1)\mathrm{/}2]$ cyclic codes with good minimum distances from sequences

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