  Binary cyclic codes have been a hot topic for many years, and significant
progress has been made in the study of this types of codes. As is well known,
it is hard to construct infinite families of binary cyclic codes [n, n+1/2]
with good minimum distance. In this paper, by using the BCH bound on cyclic
codes, one of the open problems proposed by Liu et al. about binary cyclic
codes (Finite Field Appl 91:102270, 2023) is settled. Specially, we present
several families of binary duadic codes with length 2^m-1 and dimension
2^(m-1), and the minimum distances have a square-root-like lower bound. As a
by-product, the parameters of their dual codes and extended codes are provided,
where the latter are self-dual and doubly-even.
