In 1999, Xing, Niederreiter and Lam introduced a generalization of AG 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 degrees 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 certain 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 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 new approach.

翻译：1999年，邢、Niederreiter和Lam通过使用函数域非有理点的求值引入了AG码的推广形式。本文证明，通过仅使用次数不超过$r$的非有理点，可以在该类码中获取局部性参数$r$。据作者所知，这是构造局部可恢复码（LRCs）的一种新方法。我们给出一个达到LRC单形界（Singleton-like bound）的码实例，并展示在$\mathbb F_3$上某些较长码的参数。随后研究其与某些级联码的相似性。不同于先前方法，我们的构造可直接获得维数非局部分数倍的码。最后，利用Garcia-Stichtenoth函数域塔，对我们的构造和级联码构造进行渐近分析。基于新方法，我们在任意大于3的有限域上给出了局部性为2的LRCs的显式无穷族。