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

Yaron Shany, Amit Berman

TL;DR

This work resolves a conjecture on the minimum distance of primitive binary BCH codes with designed distance $d=2^{m-2}+1$ by proving that the generating idempotent weight equals the Bose distance $d_B$ for all $m\ge4$, with $d_B=\frac{2^m+1}{3}$ when $m$ is odd and $d_B=\frac{2^m-1}{3}$ when $m$ is even. The authors connect code-theoretic weight to the root-count of cyclic fibbinary polynomials via Fourier-analytic and cyclotomic-coset techniques, and they establish that $f^{(\mathrm{cyc})}_m= X u_m f_{m-3}^4$ with $\deg(u_m)=d_B$, and that all roots of the fibbinary polynomials lie in the splitting field $\mathbb{F}_{2^{m+2}}$, forcing the root count to equal $d_B$. The proof combines recurrences for fibbinary polynomials, a precise factorization, and a splitting-field analysis to bound and achieve the required weight, thus proving the conjecture and extending the result to even $m$ where the minimum distance was previously known only indirectly. These methods illuminate the structure of BCH codes and yield potential generalizations to other designed distances and $q$-ary BCH codes, with practical implications for code design and decoding strategies.

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.

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

TL;DR

This work resolves a conjecture on the minimum distance of primitive binary BCH codes with designed distance by proving that the generating idempotent weight equals the Bose distance for all , with when is odd and when is even. The authors connect code-theoretic weight to the root-count of cyclic fibbinary polynomials via Fourier-analytic and cyclotomic-coset techniques, and they establish that with , and that all roots of the fibbinary polynomials lie in the splitting field , forcing the root count to equal . The proof combines recurrences for fibbinary polynomials, a precise factorization, and a splitting-field analysis to bound and achieve the required weight, thus proving the conjecture and extending the result to even where the minimum distance was previously known only indirectly. These methods illuminate the structure of BCH codes and yield potential generalizations to other designed distances and -ary BCH codes, with practical implications for code design and decoding strategies.

Abstract

In a paper from 2015, Ding et al. (IEEE Trans. IT, May 2015) conjectured that for odd , the minimum distance of the binary BCH code of length and designed distance 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 . 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.
Paper Structure (14 sections, 7 theorems, 23 equations)

This paper contains 14 sections, 7 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Proof technique
  3. Paper outline
  4. Preliminaries
  5. Cyclic codes
  6. Cyclotomic cosets
  7. Fourier transforms on $\mathbb{F}_{2^{m}}^*$
  8. The generating idempotent of a binary cyclic code
  9. Proof of Theorem \ref{['thm:main']}
  10. Finding a polynomial whose number of roots equals the weight of the idempotent
  11. Recursions for $f_m(X),f^{(\mathrm{cyc})}_m(X)$, and a factorization of $f^{(\mathrm{cyc})}_m(X)$
  12. The splitting field of the fibbinary polynomials
  13. Completing the proof of Theorem \ref{['thm:main']}
  14. Concluding remarks and open questions

Key Result

Theorem 1.2

For all integer $m\geq 4$ (both odd and even), the weight $w$ of the generating idempotent of $\mathrm{BCH}(m,{2^{m-2}+1})$ is equal to the Bose distance found in DDZ15, that is,

Theorems & Definitions (18)

  • Conjecture 1.1: Ding et al., Conjecture 1 of DDZ15
  • Theorem 1.2
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Definition 3.3
  • Proposition 3.4
  • proof
  • Proposition 3.5
  • proof
  • ...and 8 more