Decoding generalized concatenated codes using Interleaved Reed-Solomon codes

Abstract

Abstract-Generalized Concatenated codes are a code construction consisting of a number of outer codes whose code symbols are protected by an inner code. As outer codes, we assume the most frequently used Reed-Solomon codes; as inner code, we assume some linear block code which can be decoded up to half its minimum distance. Decoding up to half the minimum distance of Generalized Concatenated codes is classically achieved by the Blokh-Zyablov-Dumer algorithm, which iteratively decodes by first using the inner decoder to get an estimate of the outer code words and then using an outer error/erasure decoder with a varying number of erasures determined by a set of precalculated thresholds. In this paper, a modified version of the Blokh-Zyablov-Dumer algorithm is proposed, which exploits the fact that a number of outer Reed-Solomon codes with average minimum distanced can be grouped into one single Interleaved Reed-Solomon code which can be decoded beyondd/2. This allows to skip a number of decoding iterations on the one hand and to reduce the complexity of each decoding iteration significantly -while maintaining the decoding performance -on the other.

Adaptive single-trial error/erasure decoding of binary codes

“…The complexity of this calculation is O(n 4 √ n) using Pankiewiczs' algorithm [20]. The calculation of P σ (τ ) in line 13 involves summation of 2 4 √ n probabilities Pr(Y τ = ε). Using the pre-computed values of the derivatives from lines 11-12, each Pr(Y τ = ε) can be calculated with complexity linear in n, hence P σ (τ ) can be calculated with complexity O(n 4 √ n).…”

“…Note that calculating P σ (τ ) in Algorithm 1 is in O(nd min ). Alltogether, the loop in lines 7-15 is in O(n 4 √ nd min ) and thus the overall complexity of Algorithm 2 is O(n 2 4 √ n).…”

“…Obviously, one can expect the adaptive approach to yield better results than the fixed approach, especially for mediocre channel conditions. Different settings for optimal fixed erasing have been considered in [4]- [6] (radius maximization), [1], [7] (error probability minimization), and [8], [9] (both). Results about adaptive erasing can be found in [6], [10]- [13] (radius maximization).…”

“…Different settings for optimal fixed erasing have been considered in [4]- [6] (radius maximization), [1], [7] (error probability minimization), and [8], [9] (both). Results about adaptive erasing can be found in [6], [10]- [13] (radius maximization).…”

Optimal threshold-based multi-trial error/erasure decoding with the Guruswami-Sudan algorithm

“…For the sake of completeness we should also mention other publications on related topics, e.g. maximization of the decoding radius of concatenated block codes with an outer λdecoder using threshold erasing [10], [11] and symbol erasing [12]- [14]. Outer list decoders have already been considered by Nielsen [15], but with the aim of maximizing the decoding radius of the concatenated code construction.…”