The Generating Idempotent Is a Minimum-Weight Codeword for Some Binary BCH Codes

In a paper from 2015, Ding et al. (IEEE Trans. IT, May 2015) conjectured that for odd \(m\), the minimum distance of the binary BCH code of length \(2^m-1\) and designed distance \(2^{m-2}+1\) is equal to the Bose distance calculated in the same paper. In this paper, we prove the conjecture. In fact, we prove a stronger result suggested by Ding et al.: the weight of the generating idempotent is equal to the Bose distance for both odd and even \(m\). Our main tools are some new properties of the so-called fibbinary integers, in particular, the splitting field of related polynomials, and the relation of these polynomials to the idempotent of the BCH code.