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

A rate R=5/20 hypergraph-based woven convolutional code with free distance 120

A rate R=5/20 hypergraph-based woven convolu- tional code with overall constraint length 67 and constituent con- volutional codes is presented. It is based on a 3-partite, 3-uniform, 4-regular hypergraph and contains rate R=3/4 constituent convolutional codes with overall constraint length 5. Although the code construction is based on low-complexity codes, the free distance of this construction, computed with the BEAST algorithm, is dfree=120, which is remarkably large

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

Another look at the exact bit error probability for Viterbi decoding of convolutional codes

In 1995, Best et al. published a formula for the exact bit error probability for Viterbi decoding of the rate R=1/2, memory m=1 (2-state) convolutional encoder with generator matrix G(D)=(1 1+D) when used to communicate over the binary symmetric channel. Their method was later extended to the rate R=1/2, memory m=2 (4-state) generator matrix G(D)=(1+D^2 1+D+D^2) by Lentmaier et al. In this paper, we shall use a different approach to derive the exact bit error probability. We derive and solve a general matrix recurrent equation connecting the average information weights at the current and previous steps of the Viterbi decoding. A closed form expression for the exact bit error probability is given. Our general solution yields the expressions for the exact bit error probability obtained by Best et al. (m=1) and Lentmaier et al. (m=2) as special cases. The exact bit error probability for the binary symmetric channel is determined for various 8 and 16 states encoders including both polynomial and rational generator matrices for rates R=1/2 and R=2/3. Finally, the exact bit error probability is calculated for communication over the quantized additive white Gaussian noise channel

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

Searching for tailbiting codes with large minimum distances

Tailbiting trellis representations of linear block codes with an arbitrary sectionalization of the time axis are studied. A new lower bound on the maximal state complexity of an arbitrary tailbiting code is derived. The asymptotic behavior of the derived bound is investigated. Some new tailbiting representations for linear block codes of rates R=1/c, c=2,3,4 are presente

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