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

2024/08/15 by Shany, Yaron, Berman, Amit · 2 citations
#FOS: Computer and information sciences #Information Theory (cs.IT)

paper · doi:10.48550/arxiv.2408.08218

Abstract

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.

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