Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. Constructing binary cyclic codes with parameters \([n, \frac{n+1}{2}, d \ge \sqrt{n}]\) is an interesting topic in coding theory, as their minimum distances have a square-root bound. Let \(n=2^\lambda -1\) , where \(\lambda \) has three forms: \(p^2, p_1p_2, 2p_2\) for odd primes \(p, p_1, p_2\) . In this paper, we mainly construct several classes of binary cyclic codes with parameters \([2^\lambda -1, k \ge 2^{\lambda -1}, d \ge \sqrt{n}]\) . Specifically, the binary cyclic codes \({\mathcal {C}}_{(1, p^2)}\) , \({\mathcal {C}}_{(1, 2p_2)}\) , \({\mathcal {C}}_{(2, 2p_2)}\) , and \({\mathcal {C}}_{(1, p_1p_2)}\) have minimum distance \(d \ge \sqrt{n}\) though their dimensions satisfy \(k > \frac{n+1}{2}\) . Moreover, two classes of binary cyclic codes \({\mathcal {C}}_{(2, p^2)}\) and \({\mathcal {C}}_{(2, p_1p_2)}\) with dimension \(k= \frac{n+1}{2}\) and minimum distance d much exceeding the square-root bound are presented, which extends the results given by Sun, Li, and Ding [30]. In fact, the rate of these two classes of binary cyclic codes are around \(\frac{1}{2}\) and the lower bounds on their minimum distances are close to \(\frac{n}{\log _2 n}\) . In addition, their extended codes are also investigated.
