New Identities Relating Wild Goppa Codes

Abstract

For a given support $L\in {\mathbb{F}}_{q^m}^n$ and a polynomial $g\in {\mathbb{F}}_{q^m}[x]$ with no roots in ${\mathbb{F}}_{q^m}$, we prove equality between the $q$-ary Goppa codes ${\Gamma }_q(L,N(g))={\Gamma }_q(L,N(g)/g)$ where $N(g)$ denotes the norm of $g$, that is $g^{q^{m-1}+\cdots +q+1}.$ In particular, for $m=2$, that is, for a quadratic extension, we get ${\Gamma }_q(L,g^q)={\Gamma }_q(L,g^{q+1})$. If $g$ has roots in ${\mathbb{F}}_{q^m}$, then we do not necessarily have equality and we prove that the difference of the dimensions of the two codes is bounded above by the number of distinct roots of $g$ in ${\mathbb{F}}_{q^m}$. These identities provide numerous code equivalences and improved designed parameters for some families of classical Goppa codes.

14 pages

References in corpus (1)

• Cumulative-Separable Codes