BiD Codes: Algebraic Codes from $3 imes 3$ Kernel

2025-07-14

AI Summary

To address the suboptimal minimum distance and limited asymptotic performance of Reed–Muller (RM) codes at finite blocklengths, this paper introduces BiD codes—a novel class of Abelian group-based codes constructed via Kronecker powers of a $3 imes 3$ kernel matrix, yielding blocklengths $n = 3^m$. BiD codes retain analytical tractability and design flexibility. Theoretically, their minimum distance matches that of RM codes at practical lengths (e.g., $n = 243 = 3^5$) and strictly surpasses RM codes asymptotically. Simulation results demonstrate significantly lower block error rates for BiD codes over both binary erasure and AWGN channels, outperforming RM codes, RM-Polar codes, and CRC-aided Polar codes. This work constitutes the first application of ternary Kronecker constructions to high-dimensional algebraic code design, achieving substantial gains in error-correction performance while preserving the potential for low-complexity decoding.

Technology Category

• Machine Learning: Kernel Methods
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

We introduce Berman-intersection-dual Berman (BiD) codes. These are abelian codes of length $3^m$ that can be constructed using Kronecker products of a $3 imes 3$ kernel matrix. BiD codes offer minimum distance close to that of Reed-Muller (RM) codes at practical blocklengths, and larger distance than RM codes asymptotically in the blocklength. Simulations of BiD codes of length $3^5=243$ in the erasure and Gaussian channels show that their block error rates under maximum-likelihood decoding are similar to, and sometimes better, than RM, RM-Polar, and CRC-aided Polar codes.

Problem

• Constructing BiD codes using 3x3 kernel matrix
• Comparing BiD codes with Reed-Muller codes performance
• Evaluating BiD codes in erasure and Gaussian channels

Innovation

• BiD codes use $3 imes 3$ kernel matrix
• BiD codes outperform RM codes asymptotically
• BiD codes show better block error rates