A code decomposition approach for decoding cyclic and algebraic-geometric codes

Abstract

An approach based on code decomposition and partial transform for constructing algorithms for decoding cyclic codes and algebraic-geometric (AG) codes is introduced. A general decoding procedure applicable to arbitrary linear codes and their subfield subcodes is then developed. In particular, we have developed algorithms for error-and-erasure decoding of cyclic codes up to their actual minimum distance, error-and-erasure decoding of AG codes up to their designed minimum distance, and decoding of subfield subcodes of AG codes (geometric BCH codes). Moreover, a generalization of geometric BCH codes is introduced. It is shown by an example that for a simple class of AG codes, the so-called one-point codes, this generalization brings a better estimate of their minimum distance. It is also shown that the algorithms developed in this paper can be applied to decode these codes up to their estimated minimum distance.