Optimal Thresholds for GMD Decoding with (L+1)/L-extended Bounded Distance Decoders

TL;DR

This paper develops an optimal threshold formula for multi-trial decoding of concatenated codes with an (L+1)/L-extended bounded distance outer decoder, improving error correction by minimizing residual error probability.

Contribution

It introduces a threshold location formula for optimal erasure of unreliable inner decoding results in multi-trial decoding, generalizing Forney's GMD decoding for any L.

Findings

Derived a formula for optimal threshold placement.

Estimated residual error probability for multiple decoding trials.

Applicable to decoding of L-Interleaved Reed-Solomon codes.

Abstract

We investigate threshold-based multi-trial decoding of concatenated codes with an inner Maximum-Likelihood decoder and an outer error/erasure (L+1)/L-extended Bounded Distance decoder, i.e. a decoder which corrects e errors and t erasures if e(L+1)/L + t <= d - 1, where d is the minimum distance of the outer code and L is a positive integer. This is a generalization of Forney's GMD decoding, which was considered only for L = 1, i.e. outer Bounded Minimum Distance decoding. One important example for (L+1)/L-extended Bounded Distance decoders is decoding of L-Interleaved Reed-Solomon codes. Our main contribution is a threshold location formula, which allows to optimally erase unreliable inner decoding results, for a given number of decoding trials and parameter L. Thereby, the term optimal means that the residual codeword error probability of the concatenated code is minimized. We give an estimation of this probability for any number of decoding trials.