The Minimum Distance of Some Narrow-Sense Primitive BCH Codes

TL;DR

This paper establishes new theoretical results on the minimum distance of narrow-sense primitive BCH codes with specific Bose distances, using advanced algebraic and combinatorial methods over finite fields.

Contribution

It provides explicit minimum distance formulas for certain narrow-sense primitive BCH codes with special Bose distances, a problem largely unresolved for decades.

Findings

Minimum distance equals Bose distance for specified BCH codes.

Uses association schemes and quadratic forms over finite fields.

Advances theoretical understanding of BCH code parameters.

Abstract

Due to wide applications of BCH codes, the determination of their minimum distance is of great interest. However, this is a very challenging problem for which few theoretical results have been reported in the last four decades. Even for the narrow-sense primitive BCH codes, which form the most well-studied subclass of BCH codes, there are very few theoretical results on the minimum distance. In this paper, we present new results on the minimum distance of narrow-sense primitive BCH codes with special Bose distance. We prove that for a prime power $q$, the $q$-ary narrow-sense primitive BCH code with length $q^m-1$ and Bose distance $q^m-q^{m-1}-q^i-1$, where $\frac{m-2}{2} \le i \le m-\lfloor \frac{m}{3} \rfloor-1$, has minimum distance $q^m-q^{m-1}-q^i-1$. This is achieved by employing the beautiful theory of sets of quadratic forms, symmetric bilinear forms and alternating bilinear forms over finite fields, which can be best described using the framework of association schemes.