Note on the dimension of Goppa codes

Let \(\Gamma (L, g)\) be a Goppa code over \({\mathbb {F}}_q\), where \(L\subset \mathbb {F}_{q^{m}}\) is a support and \(g(x)\in \mathbb {F}_{q^{m}}[x]\) is a polynomial with s distinct roots in \({\mathbb {F}}_{q^m}\). In [Couvreur A, Otmani A, Tillich JP (2014) New identities relating wild Goppa codes. Finite Field Appl 29: 178–197.], Couvreur at al. gave the bound: \(\dim _{{\mathbb {F}}_{q}}\Gamma (L,g^e)-\dim _{{\mathbb {F}}_{q}}\Gamma (L,g^{e+1})\le s,\) where \(e=q^{m-1}+q^{m-2}+\cdots +q\). In this paper, we give the conditions such that \(\dim _{{\mathbb {F}}_{q}}\Gamma (L,g^e)=\dim _{{\mathbb {F}}_{q}}\Gamma (L,g^{e+1})\).