| Based on rational points on algebraic curves over finite fields,Goppa first defined and studied algebraic geometry codes.One of the important properties of algebraic geometry codes is that there exists a squence of algebraic geometry codes that exceeds the Gilbert-Varshamov bound.Additionally,algebraic geometry codes has an application in constructing codes with good properties.For example,algebraic geometry codes are used to construct locally repairable codes,self-dual near maximal distance separable codes(self-dual NMDS codes)and linear complementary dual codes(LCD codes).In 2017,Fan et al.generalized the Euclidean inner products and the Hermitian inner products,and defined the Galois inner product.From this,the concepts of Galois dual codes and Galois LCD codes are naturally introduced.In known results,the Galois dual codes of algebraic gometry codes are not in the form of algebraic geometry codes.The main aim of this article is to representing the Galois dual codes of algebraic gometry codes in the form of algebraic geometry codes.The main results of this article is the following:first,we show that the h-Galois dual code of an algebraic geometry code CL,F(D,G)from function field F/Fpe can be represented as algebraic geometry code qΩ,F’(φh(D),φh(G))from an associated function field with an isomorphism φh:F→F’ satisfying φh(a)=ape-h for all a ∈ Fpe;then we provide an condition on F/Fpe so that the associated function field F’/Fpe can be F/Fpe itself;finally,as an application of these results,we construct a family of Galois LCD MDS codes. |
