This paper investigates the maximal channel coding rate achievable at a given blocklength and error probability. For general classes of channels new achievability and converse bounds are given, which are tighter than existing bounds for wide ranges of parameters of interest, and lead to tight approximations of the maximal achievable rate for blocklengths n as short as 100. It is also shown analytically that the maximal rate achievable with error probability ? isclosely approximated by C - ?(V/n) Q-1(?) where C is the capacity, V is a characteristic of the channel referred to as channel dispersion , and Q is the complementary Gaussian cumulative distribution function.

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Channel Coding Rate in the Finite Blocklength Regime

This paper exhibits a class of universal Raptor codes: for a given integer k, and any real /spl epsiv/>0, Raptor codes in this class produce a potentially infinite stream of symbols such that any subset of symbols of size k(1 + /spl epsiv/) is sufficient to recover the original k symbols, with high probability. Each output symbol is generated using O(log(1//spl epsiv/)) operations, and the original symbols are recovered from the collected ones with O(klog(1//spl epsiv/)) operations.

Raptor codes

We propose a general method for constructing Tanner graphs having a large girth by establishing edges or connections between symbol and check nodes in an edge-by-edge manner, called progressive edge-growth (PEG) algorithm. Lower bounds on the girth of PEG Tanner graphs and on the minimum distance of the resulting low-density parity-check (LDPC) codes are derived in terms of parameters of the graphs. Simple variations of the PEG algorithm can also be applied to generate linear-time encodeable LDPC codes. Regular and irregular LDPC codes using PEG Tanner graphs and allowing symbol nodes to take values over GF(q) (q>2) are investigated. Simulation results show that the PEG algorithm is a powerful algorithm to generate good short-block-length LDPC codes.

Regular and irregular progressive edge-growth tanner graphs

LT-codes are a new class of codes introduced by Luby for the purpose of scalable and fault-tolerant distribution of data over computer networks. In this paper, we introduce Raptor codes, an extension of LT-codes with linear time encoding and decoding. We will exhibit a class of universal Raptor codes: for a given integer k and any real epsiv>0, Raptor codes in this class produce a potentially infinite stream of symbols such that any subset of symbols of size k(1+epsiv) is sufficient to recover the original k symbols with high probability. Each output symbol is generated using O(log(1/epsiv)) operations, and the original symbols are recovered from the collected ones with O(klog(1/epsiv)) operations. We will also introduce novel techniques for the analysis of the error probability of the decoder for finite length Raptor codes. Moreover, we will introduce and analyze systematic versions of Raptor codes, i.e., versions in which the first output elements of the coding system coincide with the original k elements

Raptor Codes

We introduce a computational technique that accurately predicts performance for a given LDPC code in the error floor region. We present some results obtained by applying the technique and describe certain aspects of it.

Error Floors of LDPC Codes

In this paper, we are concerned with the finite-length analysis of low-density parity-check (LDPC) codes when used over the binary erasure channel (BEC). The main result is an expression for the exact average bit and block erasure probability for a given regular ensemble of LDPC codes when decoded iteratively. We also give expressions for upper bounds on the average bit and block erasure probability for regular LDPC ensembles and the standard random ensemble under maximum-likelihood (ML) decoding. Finally, we present what we consider to be the most important open problems in this area.

/pdf/finite-length-analysis-of-low-density-parity-check-codes-on-4jld71f17q.pdf

D. Proietti

Papers

Finite-length analysis of low-density parity-check codes on the binary erasure channel