Introducing locality in some generalized AG codes

In 1999, Xing, Niederreiter and Lam introduced a generalization of AG codes (GAG codes) using the evaluation at non-rational places of a function field. In this paper, we show that one can obtain a locality parameter r in such codes by using only non-rational places of degree at most r. This is, up to the author’s knowledge, a new way to construct locally recoverable codes (LRCs). We give an example of such a code reaching the Singleton-like bound for LRCs, and show the parameters obtained for some longer codes over \(\mathbb F_3\). We then investigate similarities with some concatenated codes. Contrary to previous methods, our construction allows one to obtain directly codes whose dimension is not a multiple of the locality. Finally, we give an asymptotic study using the Garcia–Stichtenoth tower of function fields, for both our construction with GAG codes and a construction of concatenated codes. We give explicit infinite families of LRCs with locality 2 over any finite field of cardinality greater than 3 following our approach with GAG codes.