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Register a new userEmpirical distribution of good channel codes with non-vanishing error probability (extended version)
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Added by
Yury Polyanskiy
Publication date
2013
fields
Informatics Engineering
and research's language is
English
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This paper studies several properties of channel codes that approach the fundamental limits of a given (discrete or Gaussian) memoryless channel with a non-vanishing probability of error. The output distribution induced by an $epsilon$-capacity-achieving code is shown to be close in a strong sense to the capacity achieving output distribution. Relying on the concentration of measure (isoperimetry) property enjoyed by the latter, it is shown that regular (Lipschitz) functions of channel outputs can be precisely estimated and turn out to be essentially non-random and independent of the actual code. It is also shown that the output distribution of a good code and the capacity achieving one cannot be distinguished with exponential reliability. The random process produced at the output of the channel is shown to satisfy the asymptotic equipartition property. Using related methods it is shown that quadratic forms and sums of $q$-th powers when evaluated at codewords of good AWGN codes approach the values obtained from a randomly generated Gaussian codeword.
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In this paper we consider regular low-density parity-check codes over a binary-symmetric channel in the decoding regime. We prove that up to a certain noise threshold the bit-error probability of the bit-sampling decoder converges in mean to zero over the code ensemble and the channel realizations. To arrive at this result we show that the bit-error probability of the sampling decoder is equal to the derivative of a Bethe free entropy. The method that we developed is new and is based on convexity of the free entropy and loop calculus. Convexity is needed to exchange limit and derivative and the loop series enables us to express the difference between the bit-error probability and the Bethe free entropy. We control the loop series using combinatorial techniques and a first moment method. We stress that our method is versatile and we believe that it can be generalized for LDPC codes with general degree distributions and for asymmetric channels.
The problem of finding good linear codes for joint source-channel coding (JSCC) is investigated in this paper. By the code-spectrum approach, it has been proved in the authors previous paper that a good linear code for the authors JSCC scheme is a code with a good joint spectrum, so the main task in this paper is to construct linear codes with good joint spectra. First, the code-spectrum approach is developed further to facilitate the calculation of spectra. Second, some general principles for constructing good linear codes are presented. Finally, we propose an explicit construction of linear codes with good joint spectra based on low density parity check (LDPC) codes and low density generator matrix (LDGM) codes.
Using tools developed in a recent work by Shen and the second author, in this paper we carry out an in-depth study on the average decoding error probability of the random matrix ensemble over the erasure channel under three decoding principles, namely unambiguous decoding, maximum likelihood decoding and list decoding. We obtain explicit formulas for the average decoding error probabilities of the random matrix ensemble under these three decoding principles and compute the error exponents. Moreover, for unambiguous decoding, we compute the variance of the decoding error probability of the random matrix ensemble and the error exponent of the variance, which imply a strong concentration result, that is, roughly speaking, the ratio of the decoding error probability of a random code in the ensemble and the average decoding error probability of the ensemble converges to 1 with high probability when the code length goes to infinity.
Polar codes are a class of linear block codes that provably achieves channel capacity, and have been selected as a coding scheme for $5^{rm th}$ generation wireless communication standards. Successive-cancellation (SC) decoding of polar codes has mediocre error-correction performance on short to moderate codeword lengths: the SC-Flip decoding algorithm is one of the solutions that have been proposed to overcome this issue. On the other hand, SC-Flip has a higher implementation complexity compared to SC due to the required log-likelihood ratio (LLR) selection and sorting process. Moreover, it requires a high number of iterations to reach good error-correction performance. In this work, we propose two techniques to improve the SC-Flip decoding algorithm for low-rate codes, based on the observation of channel-induced error distributions. The first one is a fixed index selection (FIS) scheme to avoid the substantial implementation cost of LLR selection and sorting with no cost on error-correction performance. The second is an enhanced index selection (EIS) criterion to improve the error-correction performance of SC-Flip decoding. A reduction of $24.6%$ in the implementation cost of logic elements is estimated with the FIS approach, while simulation results show that EIS leads to an improvement on error-correction performance improvement up to $0.42$ dB at a target FER of $10^{-4}$.
A locally recoverable code is an error-correcting code such that any erasure in a coordinate of a codeword can be recovered from a set of other few coordinates. In this article we introduce a model of local recoverable codes that also includes local error detection. The cases of the Reed-Solomon and Locally Recoverable Reed-Solomon codes are treated in some detail.
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