Generalized Goppa codes are defined by a code locator set $\mathcal{L}$ of polynomials and a Goppa polynomial $G(x)$. When the degree of all code locator polynomials in $\mathcal{L}$ 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. 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. Finally, some code parameters and how they apply to the Classic McEliece post-quantum cryptosystem are shown.
翻译：通用的 Goppa 代码由一个代码定位器来定义 $\ mathcal{L} $( mathcal{L} $) 。 当所有代码定位器在$\ mathcal{L}$( x) $( $) 中的位置为一时, 通用的 Goppa 代码是古典的 Goppa 代码。 在此工作中, 调查了二进制通用的 Goppa 代码。 首先, 得出了这些代码与任何程度的代码定位器的对等检查矩阵。 仔细选择代码定位器可以降低通用的 Goppa 代码的最小宽度距离。 演示了可解码错误到最小距离一半的二次算法 。 引入了互换通用的 Goppa 代码, 并展示了可解码错误超过最低距离一半且概率高的混合解码算法 。 最后, 演示了某些代码参数, 以及如何应用到经典的 McEliece 后Qum 密码系统 。
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