Improved Bounds on the Parity-Check Density and Achievable Rates of Binary Linear Block Codes with Applications to LDPC Codes

We consider transmission over a general memoryless channel, with bounded decoding complexity per bit under message passing decoding. We show that the achievable rate is bounded below capacity if there is a finite success in the decoding in a specified number of operations per bit at the decoder for some codes on graphs. These codes include LDPC and LDGM codes. Good performance with low decoding complexity suggests strong local structures in the graphs of these codes, which ar...

This paper applies error-exponent and dispersion-style analyses to derive finite-blocklength achievability bounds for low-density parity-check (LDPC) codes over the point-to-point channel (PPC) and multiple access channel (MAC). The error-exponent analysis applies Gallager's error exponent to bound achievable symmetrical and asymmetrical rates in the MAC. The dispersion-style analysis begins with a generalization of the random coding union (RCU) bound from random code ensembl...

A recent line of work has focused on the use of low-density generator matrix (LDGM) codes for lossy source coding. In this paper, wedevelop a generic technique for deriving lower bounds on the rate-distortion functions of binary linear codes, with particular interest on the effect of bounded degrees. The underlying ideas can be viewing as the source coding analog of the classical result of Gallager, providing bounds for channel coding over the binary symmetric channel using b...

We consider communication over binary-input memoryless output-symmetric channels using low-density parity-check codes and message-passing decoding. The asymptotic (in the length) performance of such a combination for a fixed number of iterations is given by density evolution. Letting the number of iterations tend to infinity we get the density evolution threshold, the largest channel parameter so that the bit error probability tends to zero as a function of the iterations. ...

We study the stability of low-density parity-check (LDPC) codes under blockwise or bitwise maximum $\textit{a posteriori}$ (MAP) decoding, where transmission takes place over a binary-input memoryless output-symmetric channel. Our study stems from the consideration of constructing universal capacity-achieving codes under low-complexity decoding algorithms, where universality refers to the fact that we are considering a family of channels with equal capacity. Consider, e.g., t...

An irregular LDGM-LDPC code is studied as a sub-code of an LDPC code with some randomly \emph{punctured} output-bits. It is shown that the LDGM-LDPC codes achieve rates arbitrarily close to the channel-capacity of the binary-input symmetric-output memoryless (BISOM) channel with bounded \emph{complexity}. The measure of complexity is the average-degree (per information-bit) of the check-nodes for the factor-graph of the code. A lower-bound on the average degree of the check-n...

Low-Density Parity-Check (LDPC) codes received much attention recently due to their capacity-approaching performance. The iterative message-passing algorithm is a widely adopted decoding algorithm for LDPC codes \cite{Kschischang01}. An important design issue for LDPC codes is designing codes with fast decoding speed while maintaining capacity-approaching performance. In another words, it is desirable that the code can be successfully decoded in few number of decoding iterati...

In this paper, we investigate the error floors of non-binary low-density parity-check (LDPC) codes transmitted over the memoryless binary-input output-symmetric (MBIOS) channels. We provide a necessary and sufficient condition for successful decoding of zigzag cycle codes over the MBIOS channel by the belief propagation decoder. We consider an expurgated ensemble of non-binary LDPC codes by using the above necessary and sufficient condition, and hence exhibit lower error floo...

In this paper, we design the optimal rate capacity approaching irregular Low-Density Parity-Check code ensemble over Binary Erasure Channel, by using practical Semi-Definite Programming approach. Our method does not use any relaxation or any approximate solution unlike previous works. Our simulation results include two parts; first, we present some codes and their degree distribution functions that their rates are close to the capacity. Second, the maximum achievable rate beh...

Since the classical work of Berlekamp, McEliece and van Tilborg, it is well known that the problem of exact maximum-likelihood (ML) decoding of general linear codes is NP-hard. In this paper, we show that exact ML decoding of a classs of asymptotically good error correcting codes--expander codes, a special case of low density parity check (LDPC) codes--over binary symmetric channels (BSCs) is possible with an expected polynomial complexity. More precisely, for any bit-flippin...