The Full Rank Condition for Sparse Random Matrices

We derive a sufficient condition for a sparse random matrix with given numbers of non-zero entries in the rows and columns having full row rank. Inspired by low-density parity check codes, the family of random matrices that we investigate is very general and encompasses both matrices over finite fields and {0,1}-matrices over the rationals. The proof combines statistical physics-inspired coupling techniques with local limit arguments

Density Evolution for Asymmetric Memoryless Channels

Density evolution is one of the most powerful analytical tools for low-density parity-check (LDPC) codes and graph codes with message passing decoding algorithms. With channel symmetry as one of its fundamental assumptions, density evolution (DE) has been widely and successfully applied to different channels, including binary erasure channels, binary symmetric channels, binary additive white Gaussian noise channels, etc. This paper generalizes density evolution for non-symmetric memoryless channels, which in turn broadens the applications to general memoryless channels, e.g. z-channels, composite white Gaussian noise channels, etc. The central theorem underpinning this generalization is the convergence to perfect projection for any fixed size supporting tree. A new iterative formula of the same complexity is then presented and the necessary theorems for the performance concentration theorems are developed. Several properties of the new density evolution method are explored, including stability results for general asymmetric memoryless channels. Simulations, code optimizations, and possible new applications suggested by this new density evolution method are also provided. This result is also used to prove the typicality of linear LDPC codes among the coset code ensemble when the minimum check node degree is sufficiently large. It is shown that the convergence to perfect projection is essential to the belief propagation algorithm even when only symmetric channels are considered. Hence the proof of the convergence to perfect projection serves also as a completion of the theory of classical density evolution for symmetric memoryless channels.Comment: To appear in the IEEE Transactions on Information Theor

On Universal Properties of Capacity-Approaching LDPC Ensembles

This paper is focused on the derivation of some universal properties of capacity-approaching low-density parity-check (LDPC) code ensembles whose transmission takes place over memoryless binary-input output-symmetric (MBIOS) channels. Properties of the degree distributions, graphical complexity and the number of fundamental cycles in the bipartite graphs are considered via the derivation of information-theoretic bounds. These bounds are expressed in terms of the target block/ bit error probability and the gap (in rate) to capacity. Most of the bounds are general for any decoding algorithm, and some others are proved under belief propagation (BP) decoding. Proving these bounds under a certain decoding algorithm, validates them automatically also under any sub-optimal decoding algorithm. A proper modification of these bounds makes them universal for the set of all MBIOS channels which exhibit a given capacity. Bounds on the degree distributions and graphical complexity apply to finite-length LDPC codes and to the asymptotic case of an infinite block length. The bounds are compared with capacity-approaching LDPC code ensembles under BP decoding, and they are shown to be informative and are easy to calculate. Finally, some interesting open problems are considered.Comment: Published in the IEEE Trans. on Information Theory, vol. 55, no. 7, pp. 2956 - 2990, July 200

FPGA Implementation of Low Density Party Check Codes Decoder

Reliable communication over the noisy channel has become one of the major concerns in the field of digital wireless communications. The low density parity check codes (LDPC) has gained lot of attention recently because of their excellent error-correcting capacity. It was first proposed by Robert G. Gallager in 1960. LDPC codes belong to the class of linear block codes. Near capacity performance is achievable on a large collection of data transmission and storage.In my thesis I have focused on hardware implementation of (3, 6) - regular LDPC codes. A fully parallel decoder will require too high complexity of hardware realization. Partly parallel decoder has the advantage of effective compromise between decoding throughput and high hardware complexity. The decoding of the codeword follows the belief propagation alias probability propagation algorithm in log domain. A 9216 bit, (3, 6) regular LDPC code with code rate ½ was implemented on FPGA targeting Xilinx Virtex 4 XC4VLX80 device with package FF1148. This decoder achieves a maximum throughput of 82 Mbps. The entire model was designed in VHDL in the Xilinx ISE 9.2 environment

Statistical physics methods for sparse graph codes

This thesis deals with the asymptotic analysis of coding systems based on sparse graph codes. The goal of this work is to analyze the decoder performance when transmitting over a general binary-input memoryless symmetric-output (BMS) channel. We consider the two most fundamental decoders, the optimal maximum a posteriori (MAP) decoder and the sub-optimal belief propagation (BP) decoder. The BP decoder has low-complexity and its performance analysis is, hence, of great interest. The MAP decoder, on the other hand, is computationally expensive. However, the MAP decoder analysis provides fundamental limits on the code performance. As a result, the MAP-decoding analysis is important in designing codes which achieve the ultimate Shannon limit. It would be fair to say that, over the binary erasure channel (BEC), the performance of the MAP and BP decoder has been thoroughly understood. However, much less is known in the case of transmission over general BMS channels. The combinatorial methods used for analyzing the case of BEC do not extend easily to the general case. The main goal of this thesis is to advance the analysis in the case of transmission over general BMS channels. To do this, we use the recent convergence of statistical physics and coding theory. Sparse graph codes can be mapped into appropriate statistical physics spin-glass models. This allows us to use sophisticated methods from rigorous statistical mechanics like the correlation inequalities, interpolation method and cluster expansions for the purpose of our analysis. One of the main results of this thesis is that in some regimes of noise, the BP decoder is optimal for a typical code in an ensemble of codes. This result is a pleasing extension of the same result for the case of BEC. An important consequence of our results is that the heuristic predictions of the replica and cavity methods of spin-glass theory are correct in the realm of sparse graph codes