Error-Building Decoding of Linear Block Codes

TL;DR

This paper introduces an error-building decoding framework for linear block codes that achieves maximum-likelihood performance using only the parity-check matrix, with optimized complexity especially for extended Hamming codes.

Contribution

It presents a novel ML soft-decision decoding method called error-building decoding (EBD) that simplifies implementation and reduces complexity without sacrificing performance.

Findings

EBD achieves near-ML decoding performance.

Complexity is significantly reduced for extended Hamming codes.

EBD requires fewer floating-point operations than Viterbi decoding.

Abstract

This paper proposes a novel maximum-likelihood (ML) soft-decision decoding framework for linear block codes, termed error-building decoding (EBD). The complete decoding process can be performed using only the parity-check matrix, without requiring any other pre-constructed information (such as trellis diagrams or error-pattern lists), and it can also be customized by exploiting the algebraic properties of the code. We formally define error-building blocks, and derive a recursive theorem that allows efficient construction of larger locally optimal blocks from smaller ones, thereby effectively searching for the block associated with the most likely error pattern. The EBD framework is further optimized for extended Hamming codes as an example, through offline and online exclusion mechanisms, leading to a substantial complexity reduction without loss of ML performance. Complexity analysis shows that, for extended Hamming codes of lengths 64, 128, and 256, the fully optimized EBD requires approximately an order of magnitude fewer floating-point operations on average than minimum-edge trellis Viterbi decoding at a frame error rate of $10^{-3}$.