On a new class of error control codes and symmetric functions

A general key equation based on elementary symmetric functions is developed for decoding some binary error control codes. Here, the syndrome is obtained by computing the elementary symmetric functions (instead of the power-sums) of the received word. A new class of codes is introduced in this paper which can correct up to t₀ 0 rarr 1 errors and, simultaneously, up to t₁ 1 rarr 0 errors. The new key equation can be used to decode this new class of codes and some known codes such as some t-asymmetric error correcting (t-AEC) codes, the t-symmetric error correcting (t-SEC) BCH codes and Goppa codes. Some generalizations to the non binary case are also given.