On True Minimum Distances of Goppa Codes

Abstract

Binary Goppa codes are important in coding theory and cryptography. It is well-known that the minimum distance of a binary Goppa code defined by a squarefree polynomial of degree $r$ is at least $2r+1$. Though from the classical results of C. J. Moreno, O. Moreno, F. Levy-dit-Vehel and S. Litsyn that the true minimum distances of binary Goppa codes are $2r+1$, when the extension field degree $m$ is larger than $(r-1)^{4r}$. There have only very few families of binary Goppa codes with determined true minimum distances. In this paper, we give necessary and sufficient conditions for binary Goppa codes such that their minimum distances are $2r+1$ and $2r+2$. Infinitely many infinite families of binary Goppa codes with minimum distance $2r+1$ are given. In particular, we construct such codes for every Goppa degree $r$ in the range $2^{h-1}<r<2^h$ over $\mathbb F_{2^{2h}}$. For every $2\le r\le\lfloor2^m/3\rfloor$, we also obtain at least $2^m-3r+1$ distinct monic squarefree Goppa polynomials of degree $r$ over $\mathbb F_{2^m}$, such that their related binary Goppa codes have minimum distance $2r+1$. We extend a criterion for $q$-ary Goppa codes to split auxiliary polynomials and obtain some infinite families of $q$-ary Goppa codes attaining their designed distances $r+1$. We also construct infinite families of binary Goppa codes with minimum distance exactly $2r+2$.