Automorphisms of Hermitian generalized algebraic geometry codes (Q899567)

Generalized algebraic geometry (GAG) codes were proposed by Xing, Neiderreiter and Lam [\textit{C. Xing} et al., IEEE Trans. Inf. Theory 45, No. 7, 2498--2501 (1999; Zbl 0956.94023)] with the purpose of improving the performances of the algebraic geometry (Goppa) codes [\textit{V. D. Goppa}, Sov. Math., Dokl. 24, 170--172 (1981; Zbl 0489.94014); translation from Dokl. Akad. Nauk SSSR 259, 1289--1290 (1981)]. In a previous paper the author of the present work and \textit{A. G. Spera} [Discrete Math. 309, No. 2, 328--340 (2009; Zbl 1166.94013)] studied the automorphisms of certain GAG codes (associated with rational, elliptic and hyperelliptic function fields). Now the author studies automorphisms for other GAG codes. Section 2 recalls the concept and parameters of GAG codes (while Goppa codes use rational places of a function field, GAG codes make use of places of a (fixed) degree \(n>1\)) and Section 3 gives the definition of \(n\)-automorphism of a GAG code and recalls some necessary facts proved in [Zbl 1166.94013]. Section 4 shows results for the group of \(n\)-automorphisms of GAG codes associated with the so-called \textit{admissible} function fields, whose better well-known example is the Hermitian function field. The main result (Theorem 4.5) proves that, under suitable conditions, that group of \(n\)-automorphisms is isomorphic to a subgroup of the automorphisms of the corresponding function field.