  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 $h=2$.
