Optimal thresholds for GMD decoding with ℓ+1/ℓ-extended Bounded Distance decoders

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 ε errors and τ erasures if ℓ+1/ℓ-ε+τ ≤ d° - 1, where d° is the minimum distance of the outer code and ℓ ∈ ℕ\{0}. 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.