Decoding Generalized Concatenated Codes Using Interleaved Reed-Solomon Codes

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 pre-calculated 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 distance d can be grouped into one single Interleaved Reed-Solomon code which can be decoded beyond d/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.Comment: Proceedings of the 2008 IEEE International Symposium on Information Theory, Toronto, ON, Canada, July 6 - 11, 2008. 5 pages, 2 figure

On the Construction and Decoding of Concatenated Polar Codes

A scheme for concatenating the recently invented polar codes with interleaved block codes is considered. By concatenating binary polar codes with interleaved Reed-Solomon codes, we prove that the proposed concatenation scheme captures the capacity-achieving property of polar codes, while having a significantly better error-decay rate. We show that for any $\epsilon > 0$, and total frame length $N$, the parameters of the scheme can be set such that the frame error probability is less than $2^{-N^{1-\epsilon}}$, while the scheme is still capacity achieving. This improves upon 2^{-N^{0.5-\eps}}, the frame error probability of Arikan's polar codes. We also propose decoding algorithms for concatenated polar codes, which significantly improve the error-rate performance at finite block lengths while preserving the low decoding complexity

Efficient Multi-Point Local Decoding of Reed-Muller Codes via Interleaved Codex

Reed-Muller codes are among the most important classes of locally correctable codes. Currently local decoding of Reed-Muller codes is based on decoding on lines or quadratic curves to recover one single coordinate. To recover multiple coordinates simultaneously, the naive way is to repeat the local decoding for recovery of a single coordinate. This decoding algorithm might be more expensive, i.e., require higher query complexity. In this paper, we focus on Reed-Muller codes with usual parameter regime, namely, the total degree of evaluation polynomials is $d=\Theta({q})$, where $q$ is the code alphabet size (in fact, $d$ can be as big as $q/4$ in our setting). By introducing a novel variation of codex, i.e., interleaved codex (the concept of codex has been used for arithmetic secret sharing \cite{C11,CCX12}), we are able to locally recover arbitrarily large number $k$ of coordinates of a Reed-Muller code simultaneously at the cost of querying $O(q^2k)$ coordinates. It turns out that our local decoding of Reed-Muller codes shows ({\it perhaps surprisingly}) that accessing $k$ locations is in fact cheaper than repeating the procedure for accessing a single location for $k$ times. Our estimation of success error probability is based on error probability bound for $t$-wise linearly independent variables given in \cite{BR94}

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 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.Comment: Accepted for the 2010 IEEE International Symposium on Information Theory, Austin, TX, USA, June 13 - 18, 2010. 5 pages, 2 figure

Optimal Threshold-Based Multi-Trial Error/Erasure Decoding with the Guruswami-Sudan Algorithm

Traditionally, multi-trial error/erasure decoding of Reed-Solomon (RS) codes is based on Bounded Minimum Distance (BMD) decoders with an erasure option. Such decoders have error/erasure tradeoff factor L=2, which means that an error is twice as expensive as an erasure in terms of the code's minimum distance. The Guruswami-Sudan (GS) list decoder can be considered as state of the art in algebraic decoding of RS codes. Besides an erasure option, it allows to adjust L to values in the range 1<L<=2. Based on previous work, we provide formulae which allow to optimally (in terms of residual codeword error probability) exploit the erasure option of decoders with arbitrary L, if the decoder can be used z>=1 times. We show that BMD decoders with z_BMD decoding trials can result in lower residual codeword error probability than GS decoders with z_GS trials, if z_BMD is only slightly larger than z_GS. This is of practical interest since BMD decoders generally have lower computational complexity than GS decoders.Comment: Accepted for the 2011 IEEE International Symposium on Information Theory, St. Petersburg, Russia, July 31 - August 05, 2011. 5 pages, 2 figure

VLSI single-chip (255,223) Reed-Solomon encoder with interleaver

The invention relates to a concatenated Reed-Solomon/convolutional encoding system consisting of a Reed-Solomon outer code and a convolutional inner code for downlink telemetry in space missions, and more particularly to a Reed-Solomon encoder with programmable interleaving of the information symbols and code correction symbols to combat error bursts in the Viterbi decoder