Decoding of (Interleaved) Generalized Goppa Codes

02/04/2021

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by   Hedongliang Liu, et al.

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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 parity-check 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.