Florian Hug

Searching for Voltage Graph-Based LDPC Tailbiting Codes With Large Girth

The relation between parity-check matrices of quasi-cyclic (QC) low-density parity-check (LDPC) codes and biadjacency matrices of bipartite graphs supports searching for powerful LDPC block codes. Using the principle of tailbiting, compact representations of bipartite graphs based on convolutional codes can be found. Bounds on the girth and the minimum distance of LDPC block codes constructed in such a way are discussed. Algorithms for searching iteratively for LDPC block codes with large girth and for determining their minimum distance are presented. Constructions based on all-one matrices, Steiner Triple Systems, and QC block codes are introduced. Finally, new QC regular LDPC block codes with girth up to 24 are given.

Searching for high-rate convolutional codes via binary syndrome trellises

Rate R = (c-1)/c convolutional codes of constraint length ν can be represented by conventional syndrome trellises with a state complexity of s = ν or by binary syndrome trellises with a state complexity of s = ν or s = ν + 1, which corresponds to at most 2s 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 = ν. 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.

New low-density parity-check codes with large girth based on hypergraphs

The relation between low-density parity-check (LDPC) codes and hypergraphs supports searching for powerful LDPC codes based on hypergraphs. On the other hand, coding theory methods can be used in searching for hypergraphs with large girth. Moreover, compact representations of hypergraphs based on convolutional codes can be found. Algorithms for iteratively constructing LDPC codes with large girth and for determining their minimum distance are introduced. New quasi-cyclic (QC) LDPC codes are presented, some having both optimal girth and optimal minimum distance.

A Closed-Form Expression for the Exact Bit Error Probability for Viterbi Decoding of Convolutional Codes

In 1995, Best etal. published a formula for the exact bit error probability for Viterbi decoding of the rate R =1/2, memory m = 1 (two-state) convolutional encoder with generator matrix G(D) = (1 1 + D) when used to communicate over the binary symmetric channel. Their formula was later extended to the rate R = 1/2, memory m = 2 (four-state) convolutional encoder with generator matrix G(D) = (1 + D2 1 + D + D2) by Lentmaier et al. In this paper, a different approach to derive the exact bit error probability is described. A general recurrent matrix equation, connecting the average information weight at the current and previous states of a trellis section of the Viterbi decoder, is derived and solved. The general solution of this matrix equation yields a closed-form expression for the exact bit error probability. As special cases, the expressions obtained by Best et al. for the two-state encoder and by Lentmaier et al. for a four-state encoder are used. The closed-form expression derived in this paper is evaluated for various realizations of encoders, including rate R = 1/2 and R = 2/3 encoders, of as many as 16 states. Moreover, it is shown that it is straightforward to extend the approach to communication over the quantized additive white Gaussian noise channel.

Some voltage graph-based LDPC tailbiting codes with large girth

The relation between the parity-check matrices of quasi-cyclic (QC) low-density parity-check (LDPC) codes and the biadjacency matrices of bipartite graphs supports searching for powerful LDPC block codes. Algorithms for searching iteratively for LDPC block codes with large girth are presented and constructions based on Steiner Triple Systems and short QC block codes are introduced, leading to new QC regular LDPC block codes with girth up to 24.

A greedy search for improved QC LDPC codes with good girth profile and degree distribution

The girth profile is introduced and search algorithms for regular and irregular quasi-cyclic LDPC block codes with both good girth profile and good degree distribution are presented. New QC LDPC block codes of various code rates are obtained and their bit error rate performance is compared with that of the corresponding LDPC block codes defined in the IEEE 802.16 WiMAX standard of the same block length and code rate.

High-rate QC LDPC codes of short and moderate length with good girth profile

Irregular QC LDPC codes with parity-check matrices having different degree distributions are studied. A new algorithm for finding regular and irregular QC LDPC codes with a good girth profile as well as a good sliding-window girth is presented. As examples, simulation results for QC LDPC codes with good girth profile, rate R=4/5, and lengths about 1000, 2000, and 4000, constructed from base matrices with proper degree distributions are given. Their simulated BER and FER performances for belief propagation decoding are compared with the best previously known irregular QC LDPC codes of the same rate and length. It is shown that the constructed codes outperform the best previously known codes of same rate and lengths.

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.

On weight enumerators and MacWilliams identity for convolutional codes

Convolutional codes are defined to be equivalent if their code symbols differ only in how they are ordered and two generator matrices are defined to be weakly equivalent (WE) if they encode equivalent convolutional codes. It is shown that tailbiting convolutional codes encoded by WE minimal-basic generator matrices have the same spectra. Shearer and McEliece showed that MacWilliams identity does not hold for convolutional codes. However, for the spectra of truncated convolutional codes and their duals, MacWilliams identity clearly holds. It is shown that the dual of a truncated convolutional code is not a truncation of a convolutional code but its reversal is. Finally, a recursion for the spectra of truncated convolutional codes is given and the spectral components are related to those for the corresponding dual codes.

A Rate

A rate R=5/20 hypergraph-based woven convolutional code with overall constraint length 67 and constituent convolutional codes is presented. It is based on a 3-partite, 3-uniform, 4-regular hypergraph and contains rate Rc=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.