BiD Codes: Algebraic Codes from $3 \times 3$ Kernel

TL;DR

BiD codes are a new class of algebraic codes constructed from a 3x3 kernel that achieve near-RM minimum distance at practical lengths and outperform RM codes asymptotically, with promising simulation results in erasure and Gaussian channels.

Contribution

Introduction of BiD codes, a novel algebraic code family based on Kronecker products of a 3x3 kernel, with competitive distance properties and practical decoding performance.

Findings

BiD codes have minimum distance close to Reed-Muller codes at practical lengths.

BiD codes outperform Reed-Muller codes asymptotically in blocklength.

Simulation results show competitive error rates in erasure and Gaussian channels.

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 $3 \times 3$ 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.