Woven Graph Codes: Asymptotic Performances and Examples

Constructions of woven graph codes based on constituent block and convolutional codes are studied. It is shown that within the random ensemble of such codes based on $s$-partite, $s$-uniform hypergraphs, where $s$ depends only on the code rate, there exist codes satisfying the Varshamov-Gilbert (VG) and the Costello lower bound on the minimum distance and the free distance, respectively. A connection between regular bipartite graphs and tailbiting codes is shown. Some examples of woven graph codes are presented. Among them an example of a rate $R_{\rm wg}=1/3$ woven graph code with $d_{\rm free}=32$ based on Heawood's bipartite graph and containing $n=7$ constituent rate $R^{c}=2/3$ convolutional codes with overall constraint lengths $\nu^{c}=5$ is given. An encoding procedure for woven graph codes with complexity proportional to the number of constituent codes and their overall constraint length $\nu^{c}$ is presented.Comment: Submitted to IEEE Trans. Inform. Theor

Codes on Graphs and More

Modern communication systems strive to achieve reliable and efficient information transmission and storage with affordable complexity. Hence, efficient low-complexity channel codes providing low probabilities for erroneous receptions are needed. Interpreting codes as graphs and graphs as codes opens new perspectives for constructing such channel codes. Low-density parity-check (LDPC) codes are one of the most recent examples of codes defined on graphs, providing a better bit error probability than other block codes, given the same decoding complexity. After an introduction to coding theory, different graphical representations for channel codes are reviewed. Based on ideas from graph theory, new algorithms are introduced to iteratively search for LDPC block codes with large girth and to determine their minimum distance. In particular, new LDPC block codes of different rates and with girth up to 24 are presented. Woven convolutional codes are introduced as a generalization of graph-based codes and an asymptotic bound on their free distance, namely, the Costello lower bound, is proven. Moreover, promising examples of woven convolutional codes are given, including a rate 5/20 code with overall constraint length 67 and free distance 120. The remaining part of this dissertation focuses on basic properties of convolutional codes. First, a recurrent equation to determine a closed form expression of the exact decoding bit error probability for convolutional codes is presented. The obtained closed form expression is evaluated for various realizations of encoders, including rate 1/2 and 2/3 encoders, of as many as 16 states. Moreover, MacWilliams-type identities are revisited and a recursion for sequences of spectra of truncated as well as tailbitten convolutional codes and their duals is derived. Finally, the dissertation is concluded with exhaustive searches for convolutional codes of various rates with either optimum free distance or optimum distance profile, extending previously published results

Searching for high-rate convolutional codes via binary syndrome trellises

Rate R=(c-1)/c convolutional codes of constraint length nu can be represented by conventional syndrome trellises with a state complexity of s=nu or by binary syndrome trellises with a state complexity of s=nu or s=nu+1, which corresponds to at most 2^s states at each trellis level. It is shown that if the parity-check polynomials fulfill certain conditions, there exist binary syndrome trellises with optimum state complexity s=nu. The BEAST is modified to handle parity-check matrices and used to generate code tables for optimum free distance rate R=(c-1)/c, c=3,4,5, convolutional codes for conventional syndrome trellises and binary syndrome trellises with optimum state complexity. These results show that the loss in distance properties due to the optimum state complexity restriction for binary trellises is typically negligible

Woven convolutional graph codes with large free distances

Constructions of woven graph codes based on constituent convolutional codes are studied and examples of woven convolutional graph codes are presented. The existence of codes, satisfying the Costello lower bound on the free distance, within the random ensemble of woven graph codes based on s-partite, s-uniform hypergraphs, where s depends only on the code rate, is shown. Simulation results for Viterbi decoding of woven graph codes are presented and discussed

Double-Hamming based QC LDPC codes with large minimum distance

A new method using Hamming codes to construct base matrices of (J, K)-regular LDPC convolutional codes with large free distance is presented. By proper labeling the corresponding base matrices and tailbiting these parent convolutional codes to given lengths, a large set of quasi-cyclic (QC) (J, K)-regular LDPC block codes with large minimum distance is obtained. The corresponding Tanner graphs have girth up to 14. This new construction is compared with two previously known constructions of QC (J, K)-regular LDPC block codes with large minimum distance exceeding (J+1)!. Applying all three constructions, new QC (J, K)-regular block LDPC codes with J=3 or 4, shorter codeword lengths and/or better distance properties than those of previously known codes are presented

An improved bound on the list error probability and list distance properties

List decoding of binary block codes for the additive white Gaussian noise channel is considered. The output of a list decoder is a list of the $L$ most likely codewords, that is, the L signal points closest to the received signal in the Euclidean-metric sense. A decoding error occurs when the transmitted codeword is not on this list. It is shown that the list error probability is fully described by the so-called list configuration matrix, which is the Gram matrix obtained from the signal vectors forming the list. The worst-case list configuration matrix determines the minimum list distance of the code, which is a generalization of the minimum distance to the case of list decoding. Some properties of the list configuration matrix are studied and their connections to the list distance are established. These results are further exploited to obtain a new upper bound on the list error probability, which is tighter than the previously known bounds. This bound is derived by combining the techniques for obtaining the tangential union bound with an improved bound on the error probability for a given list. The results are illustrated by examples