Goppa codes have some of the largest minimum distances possible for linear codes. The authors use some binary Goppa codes in which four new binary codes are obtained with parameters better than any codes known to date. The authors also present the necessary conditions for a code obtained by shortening a longer code which will have a greater minimum distance than the original code.

We present an efficient construction of extended length Goppa codes. Using this construction, we obtain 78 new nonbinary codes with better minimum distance than the previously known codes with the same length and dimension. The construction is based on the observation that certain Goppa codes can be seen as BCH codes.

The authors show that a popular way of constructing quasi-cyclic low-density parity check (LDPC) codes is a special case of a construction which is common in graph theory and group theory. It is shown that a generalisation of this construction as coset graphs produces (dv, dc)-regular LDPC codes that have an advantage in terms of the minimum stopping set size compared to quasi-cyclic LDPC codes. A (dv, dc)-regular quasi-cyclic LDPC code cannot have minimum stopping set size larger than (dv+1)!. However, by using coset graphs, a (3, 5)-regular LDPC code with minimum stopping set size of 28 and a (3, 4)-regular LDPC code with minimum stopping set size larger than 32 have been obtained. In addition, the idea of coset graphs also provides a compact algebraic way of describing bipartite graph and the associated parity-check matrix of an LDPC code. Simulation results of iterative decoding of the coset graphs LDPC codes over the binary erasure channel show that some of the codes converge well and based on the truncated stopping set distributions of the codes, which are exhaustively and efficiently enumerated, the error-floor of the codes at low probability of erasure is estimated.

It is shown that the generator matrix of a putative linear binary code [36, 10, 14] contains the generator matrix of a [32, 7, 14] or a [33, 8, 14] binary linear code. In a first step, all nonisomorphic [32, 7, 14] and [33, 8, 14] binary codes are enumerated. Then it is shown that none of these codes yields a [35, 10, 13] linear binary code.

Algebraic geometry codes or Goppa codes are defined with places of degree one. In constructing generalised algebraic geometry codes places of higher degree are used. In this paper we present 41 new codes over F16which improve on the best known codes of the same length and rate. The construction method uses places of small degree with a technique originally published over 10 years ago for the construction of generalised algebraic geometry codes.

It is shown that chains of cyclic codes in conjunction with constructions X and XX can be used to efficiently construct sequences of linear codes for application in incremental redundancy communications. In addition to using a CRC for error detection, a novel CRC-less approach to error detection, which is based on the confidence level of the soft decision decoding output, is introduced. This novel approach provides an attractive trade-off between error rate performance and throughput for incremental redundancy communications.

It is shown that the relatively unknown Dorsch decoder may be extended to produce a decoder that is capable of maximum-likelihood decoding. The extension involves a technique for any linear (n, k) code that ensures that n−k less reliable, soft decisions of each received vector may be treated as erasures in determining candidate codewords. These codewords are derived from low information weight codewords and it is shown that an upper bound of this information weight may be calculated from each received vector in order to guarantee that the decoder will achieve maximum-likelihood decoding. Using the cross-correlation function, it is shown that the most likely codeword may be derived from a partial correlation function of these low information weight codewords, which leads to an efficient fast decoder. For a practical implementation, this decoder may be further simplified into a concatenation of a hard-decision decoder and a partial correlation decoder with insignificant performance degradation. Results are presented for some powerful, known codes, including a GF(4) non-binary BCH code. It is shown that maximum-likelihood decoding is realised for a high percentage of decoded codewords and that performance close to the sphere packing bound is attainable for codeword lengths up to 1000 bits.

The number and weight of low-weight codewords of a binary linear code determine the erasure channel performance. Analysis is given of the probability density function of the number of erasures correctable by the code in terms of the weight enumerator polynomial. For finite-length codes, zero erasure decoder error rate is impossible, even with maximum-distance-separable (MDS) codes and maximum-likelihood decoding. However, for codes that have binomial weight spectra, for example BCH, Goppa and double-circulant codes, the erasure correction performance is close to that of MDS codes. One surprising result is that, for many (n, k) codes, the average number of correctable erasures is almost equal to n−k, which is significantly larger than dmin−1. For the class of iteratively decodable codes (LDPC and turbo codes), the erasure performance is poor in comparison to algebraic codes designed for maximum dmin. It is also shown that the turbo codes that have optimised dmin have significantly better performance than LDPC codes. A probabilistic method, which has considerably smaller search space than that of the generator matrix-based methods, is presented to determine the dmin of a linear code using random erasure patterns. Using this approach, it is shown that there are (168, 84, 24) and (216, 108, 24) quadratic double-circulant codes.

A new method of finding binary cyclic codes from the finite-field transform domain is presented. These cyclic codes have sparse parity-check matrix and thus are suitable for iterative decoding. Some interesting properties of the proposed construction method include the knowledge of the minimum distance and the ability to trade the increase in code dimension with a reduction in the parity-check matrix sparsity. By means of simulations, we show that the error correcting performance of the codes under iterative decoding is very close to the sphere-packing-bound constrained for binary transmission.