Binary cyclic codes have been a hot topic for a long time, and their study has made significant progress. As is well known, constructing infinite families of binary cyclic codes \([n,\frac{n\pm 1}{2}]\) with good minimum distance remains a challenge. In this paper, we employ the BCH bound for cyclic codes to address an open problem regarding binary cyclic codes proposed by Liu et al (Finite Field Appl 91:102270, 2023). Specifically, we present binary duadic codes with parameters \([2^m-1, 2^{m-1},d]\) , where the minimum distance d exhibits a square-root-like lower bound. Furthermore, we determine the parameters for their duals and extended codes.
