Introducing locality in some generalized AG codes
<|reference_start|>Introducing locality in some generalized AG codes: 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.<|reference_end|>
@article{pacifico2024introducing,
    title={Introducing locality in some generalized AG codes},
    author={Bastien Pacifico},
    journal={arXiv preprint arXiv:2403.00430},
    year={2024},
    archivePrefix={arXiv},
    eprint={2403.00430},
    primaryClass={cs.IT math.AG math.IT}
}