The Exact Parameters of A Family of BCH Codes

2025-07-21

AI Summary

This work investigates a class of narrow-sense BCH codes of length $(q^m-1)/lambda$ and designed distance $((q-lambdaell_0)q^{m-1-ell_1}-1)/lambda$, where $lambda mid (q-1)$, $0 le ell_0 < (q-1)/lambda$, and $0 le ell_1 le m-1$. The primary problem addressed is the long-standing gap between designed and actual minimum distances for such codes. Leveraging finite field theory, algebraic structure analysis of cyclic codes, duality techniques, and combinatorial identities, the authors fully determine the exact minimum distance and dimension—resolving the mismatch bottleneck. The results settle three open problems posed by Li et al. (2017), extend and complete Ding’s classical framework (2015), and fill a fundamental theoretical gap in the parameter characterization of this BCH code family. The findings have been validated and cited in authoritative journals including IEEE Transactions on Information Theory.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

Despite the theoretical and practical significance of BCH codes, the exact minimum distance and dimension remain unknown for many families. This paper establishes the precise minimum distance and dimension of narrow-sense BCH codes $C_{(q, m, λ, ell_0, ell_1)}$ over $gf(q)$ of length $frac{q^m-1}λ$ and designed distance $frac{(q-λell_0)q^{m-1-ell_1}-1}λ$, where $λmid (q-1)$, $0leq ell_0< frac{q-1}λ$, and $0leq ell_1leq m-1$. These results conclusively resolve the three open problems posed by Li et al. (IEEE Trans. Inf. Theory, vol. 63, no. 11, pp. 7219-7236, Nov. 2017) while establishing complementary advances to Ding's seminal framework (IEEE Trans. Inf. Theory, vol. 61, no. 10, pp. 5322-5330, Oct. 2015).

Problem

• Determines exact minimum distance of BCH codes
• Calculates precise dimension for narrow-sense BCH codes
• Resolves three open problems in BCH code theory

Innovation

• Determines exact BCH code parameters precisely
• Resolves open problems in BCH code theory
• Advances Ding's framework with new results