Cyclic and BCH Codes whose Minimum Distance Equals their Maximum BCH bound
José Joaquín Bernal, Diana H. Bueno-Carreño, Juan Jacobo Simón
TL;DR
This work investigates when the minimum distance $d(C)$ of a cyclic code equals the maximum BCH bound $\Delta(C)$, introducing a framework based on the discrete Fourier transform (Mattson-Solomon transform) and the apparent distance $d^*(\cdot)$. It establishes necessary and sufficient conditions (via $f\in \mathbb L(n)$ with $d^*(f)=d^*(C)$ and $\varphi^{-1}_{\alpha,f}\in C$ for some $\alpha$ in the apparent-distance set) and provides constructive methods that use divisors of the polynomial $x^n-1$ to build such codes. The paper then presents techniques to construct BCH codes with designed distance $\delta$ that satisfy $d(C)=\Delta(C)=\delta$, including permutation-equivalent families and dimensional-extensions, demonstrated through several binary examples at lengths including $15$, $21$, $33$, and $45$, yielding new codes with the desired equality. These results offer a practical pathway to design BCH-like codes with optimal distance properties and provide a bridge between algebraic divisors and distance bounds via the Fourier-analytic view of cyclic codes.
Abstract
In this paper we study the family of cyclic codes such that its minimum distance reaches the maximum of its BCH bounds. We also show a way to construct cyclic codes with that property by means of computations of some divisors of a polynomial of the form X^n-1. We apply our results to the study of those BCH codes C, with designed distance delta, that have minimum distance d(C)= delta. Finally, we present some examples of new binary BCH codes satisfying that condition. To do this, we make use of two related tools: the discrete Fourier transform and the notion of apparent distance of a code, originally defined for multivariate abelian codes.