Decoding of (Interleaved) Generalized Goppa Codes

Abstract

Generalized Goppa codes are defined by a code locator set L of polynomials and a Goppa polynomial G(x). When the degree of all code locator polynomials in L is one, generalized Goppa codes are classical Goppa codes. In this work, binary generalized Goppa codes are investigated. First, a paritycheck matrix for these codes with code locators of any degree is derived. A careful selection of the code locators leads to a lower bound on the minimum Hamming distance of generalized Goppa codes which improves upon previously known bounds. A quadratic-time decoding algorithm is presented which can decode errors up to half of the minimum distance. Moreover, interleaved generalized Goppa codes are introduced and a joint decoding algorithm is presented which can decode errors beyond half the minimum distance with high probability.