Decoding Hermitian codes - an engineering approach

Abstract

This thesis introduces and discusses a new algorithm for solving the key equation for Hermitian codes, that belong to the class of algebraic-geometric (AG) codes. First, the most important concepts of channel coding are recalled and the popular Reed-Solomon (RS) codes are used to illustrate them. The decoding of RS codes with the extended Euclidean algorithm is used to illustrate the basic idea of algebraic decoding and also to motivate the new decoding algorithm. After that, some elementary results from algebraic geometry are given: only those basics that are (directly or indirectly) used in the definition of AG codes are introduced. From the definition of general AG codes, RS codes and Hermitian codes are derived as special cases. An alternative definition of Hermitian codes that uses almost no algebraic geometry is also given. After the introduction of the codes, the key equation for Hermitian codes is presented. For a limited error weight, there is a unique solution of minimal degree to the key equation, and the error pattern can be reconstructed from this solution. An algorithm that finds this solution of minimal degree is given. Unfortunately, this algorithm is not capable of decoding all error patterns with weight up to half the minimum distance a bound up to which unique decoding is guaranteed by the properties of general linear codes that include Hermitian codes. An extension that achieves this decoding radius is discussed afterwards, and the complexity of both the algorithm and its extension are estimated. A modification of the algorithm that allows to find a basis for decoding beyond half the minimum distance is given in the last chapter. However, decoding up to this increased radius without (significantly) increasing the complexity is not always possible. A bound on the error weight that allows such decoding for interleaved Hermitian codes with high probability is derived, as well as the probability that decoding fails. Finally, the idea of virtual extension to an interleaved code is described. This principle works only for codes with low rates, and therefore the rate bound is given.