Existence of binary BCH codes with simultaneous dimension and distance guarantees

Determine whether there exist integers \(\delta\) and \(b\) such that the binary BCH code \(\mathcal{C}_{(2,n,\delta,b)}\) satisfies both \(\dim(\mathcal{C}_{(2,n,\delta,b)})\ge (n-1)/2\) and \(d(\mathcal{C}_{(2,n,\delta,b)})\ge \sqrt{n}/2\) for the length families \(n=(2^{2s}+1)(2^s-1)\), \(n=2^{2s}+2^s+1\), and \(n=(2^s-1)/\lambda\), where \(\lambda>1\) is a constant divisor of \(2^s-1\).

Background

Chen, Xie, and Ding’s Open Problem 8.4 asks for binary BCH codes whose dimension is at least half the length (up to the $(n-1)/2$ threshold) while their minimum distance is at least $\sqrt n/2$. The paper establishes affirmative answers for the first two length families by choosing a designed distance of approximately $\sqrt n/2$ and applying cyclotomic-coset estimates together with the BCH bound.

For the third family, $n=(2^s-1)/\lambda$, the paper proves the desired bounds under the sufficient condition $4n\ge s^2$, but gives a counterexample at $(s,\lambda)=(18,13797)$. Thus the unrestricted assertion for the third family is false, and the general characterization of the admissible parameter pairs remains unresolved; the original open problem is included because the authors explicitly identify it as Open Problem 8.4.