# New identities relating wild Goppa codes

1 GRACE - Geometry, arithmetic, algorithms, codes and encryption Inria Saclay - Ile de France 2 LIX - Laboratoire d-informatique de l-École polytechnique Palaiseau 3 LITIS - Laboratoire d-Informatique, de Traitement de l-Information et des Systèmes 4 SECRET - Security, Cryptology and Transmissions Inria Paris-Rocquencourt

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 $\Gamma qL,Ng = \Gamma qL,Ng-g$ where $Ng$ 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 $\Gamma qL,g^q = \Gamma qL,g^{q+1}$. 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.

Keywords : Norms and Traces Goppa codes Wild Goppa codes BCH codes Norms and Traces.

Author: Alain Couvreur - Ayoub Otmani - Jean-Pierre Tillich -

Source: https://hal.archives-ouvertes.fr/