The binary quadratic-residue codes and the punctured Reed-Muller codes <inline-formula> <tex-math notation="LaTeX">{\mathcal {R}}_{2}((m-1)/2, m)) </tex-math></inline-formula> are two families of binary cyclic codes with parameters <inline-formula> <tex-math notation="LaTeX">n, (n+1)/2, d \geq \sqrt {n} </tex-math></inline-formula>. These two families of binary cyclic codes are interesting partly due to the fact that their minimum distances have a square-root bound. The objective of this paper is to construct two families of binary cyclic codes of length <inline-formula> <tex-math notation="LaTeX">2^{m}-1 </tex-math></inline-formula> and dimension near <inline-formula> <tex-math notation="LaTeX">2^{m-1} </tex-math></inline-formula> with good minimum distances. When <inline-formula> <tex-math notation="LaTeX">m \geq 3 </tex-math></inline-formula> is odd, the codes become a family of duadic codes with parameters <inline-formula> <tex-math notation="LaTeX">2^{m}-1, 2^{m-1}, d </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">d \geq 2^{(m-1)/2}+1 </tex-math></inline-formula> if <inline-formula> <tex-math notation="LaTeX">m \equiv 3 \pmod {4} </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">d \geq 2^{(m-1)/2}+3 </tex-math></inline-formula> if <inline-formula> <tex-math notation="LaTeX">m \equiv 1 \pmod {4} </tex-math></inline-formula>. The two families of binary cyclic codes contain some optimal binary cyclic codes.