Constructing rate 1/p systematic binary quasi-cyclic codes based on the matroid theory
Designs, Codes and Cryptography
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A new class of linear block codes, called self-orthogonal quasi-cyclic codes, is defined. It is shown that the problem of designing these codes is equivalent to the problem of designing disjoint difference sets. As a result, several classes of optimal and near-optimal codes can be constructed analytically and other codes can be found by a computer-aided search procedure. A list of codes is given for practical values of minimum distance and efficiency. Two easily implemented decoding algorithms are described, and a Monte Carlo evaluation of the performance of several codes on the binary symmetric channel is presented. This evaluation shows that, when decoded with the better of the two algorithms, these codes perform nearly as well as the Bose-Chaudhuri-Hocquenghem (BCH) codes with the same minimum distance and efficiency in the cases examined. Although these codes must be long relative to the BCH codes, the low cost and lack of complexity of the equipment required to correct large numbers of errors should make them competitive for practical systems.