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IEICE Technical Report; IEICE Tech. Rep.

Reliability Fanction, Asymptotic Distance Ratio, and Probability of Undetected Error for Concatenated Codes

Toshihisa Nishijima, Kin‐ichiroh Tokiwa

Published Jul 16, 2009• 0 citations
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By utilizing the method to get the upper bound on the average probability of undetected error for all binary linear systematic block codes, we newly get each upper bound on the average probability of undetected error for the ensembles of all binary product codes and all binary expansions of concatenated codes which include the constructive algebraic codes satisfying Shannon's channel coding theorem or being the asymptotically good. By comparing those upper bounds, we can clearly show that the average capability of concatenation structure is stronger than that of product codes, although the bound for concatenated codes is below that for the ensemble of all binary linear systematic block codes.
This paper shows that bit-level reliability can be converted into block-level reliability for binary codes over BMS channels through Forney's concatenation scheme. The construction concatenates an inner code of rate R-o(1) and vanishing bit-error probability εwith an outer code of rate 1-o(1) equipped with a bounded-distance decoder. We show that, when the outer correction radius exceeds εand a suitable large-deviation condition holds, the concatenated code has rate R-o(1) and vanishing block-error probability. Hence, a family with vanishing bit-error probability for rates below capacity can be converted into a capacity-achieving concatenated-code family. We further show that, when binary BCH codes are used as outer codes, inner codes with bit-error probability ε=o(1/\log\log n) can be converted into a capacity-achieving concatenated family.