Introducing locality in some generalized AG codes
Abstract: 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.
- Codes with hierarchical locality from covering maps of curves. IEEE Transactions on Information Theory, 65(10):6056–6071, 2019.
- Families of curves over any finite field attaining the generalized Drinfeld-Vlăduţ bound. Publications Mathématiques de Besançon, Algèbre et Théorie des Nombres, pages 5–18, 2011.
- On the tensor rank of multiplication in finite extensions of finite fields and related issues in algebraic geometry. Uspekhi Mathematichskikh Nauk, 76:1(457), 31–94, 2021.
- Locally recoverable codes from algebraic curves and surfaces. In Everett W. Howe, Kristin E. Lauter, and Judy L. Walker, editors, Algebraic Geometry for Coding Theory and Cryptography, pages 95–127, Cham, 2017. Springer International Publishing.
- A family of optimal locally recoverable codes. IEEE Transactions on Information Theory, 60(8):4661–4676, 2014.
- Locally recoverable codes on algebraic curves. IEEE Transactions on Information Theory, 63(8):4928–4939, 2017.
- The Magma algebra system. I. The user language. Journal of Symbolic Computation, 24(3-4):235–265, 1997. Computational algebra and number theory (London, 1993).
- Bounds on the size of locally recoverable codes. IEEE Transactions on Information Theory, 61(11):5787–5794, 2015.
- Asymptotic bound for multiplication complexity in the extensions of small finite fields. IEEE Transactions on Information Theory, 58(7):4930–4935, 2012.
- Long optimal and small-defect lrc codes with unbounded minimum distances. IEEE Transactions on Information Theory, 67(5):2786–2792, 2021.
- Optimal and asymptotically good locally repairable codes via propagation rules. IEEE Transactions on Communications, 71(10):5623–5632, 2023.
- Algebraic complexities and algebraic curves over finite fields. Journal of Complexity, 4:285–316, 1988.
- G.D. Forney. Concatenated Codes. M.I.T. Press research monographs. M.I.T. Press, 1966.
- A tower of Artin-Schreier extensions of function fields attaining the Drinfeld-Vlăduţ bound. Inventiones Mathematicae, 121:211–222, 1995.
- On the locality of codeword symbols. IEEE Transactions on Information Theory, 58(11):6925–6934, 2012.
- V. D. Goppa. Codes on algebraic curves. Dokl. Akad. Nauk SSSR, 259(6):1289–1290, 1981.
- How long can optimal locally repairable codes be? IEEE Transactions on Information Theory, 65(6):3662–3670, 2019.
- Pyramid codes: Flexible schemes to trade space for access efficiency in reliable data storage systems. In Sixth IEEE International Symposium on Network Computing and Applications (NCA 2007), pages 79–86, 2007.
- Constructions of optimal binary locally recoverable codes via a general construction of linear codes. IEEE Transactions on Communications, 69(8):4987–4997, 2021.
- Constructive asymptotic bounds of locally repairable codes via function fields. IEEE Transactions on Information Theory, 66(9):5395–5403, 2020.
- Locally recoverable codes on surfaces. IEEE Transactions on Information Theory, 67(9):5765–5777, 2021.
- Curves with many points and multiplication in finite fields. In H. Stichtenoth and M.A. Tsfasman, editors, Coding Theory and Algebraic Geometry, number 1518 in Lectures Notes in Mathematics, pages 145–169, Berlin, 1992. Springer-Verlag. Proceedings of AGCT-3 conference, June 17-21, 1991, Luminy.
- Optimal locally repairable codes via rank-metric codes. In 2013 IEEE International Symposium on Information Theory, pages 1819–1823, 2013.
- Henning Stichtenoth. Algebraic Function Fields and Codes. Number 254 in Graduate Texts in Mathematics. Springer-Verlag, second edition, 2008.
- Bounds on the parameters of locally recoverable codes. IEEE Transactions on Information Theory, 62(6):3070–3083, 2016.
- Optimal locally repairable codes and connections to matroid theory. In 2013 IEEE International Symposium on Information Theory, pages 1814–1818, 2013.
- The minimum locality of linear codes. Des. Codes Cryptography, 91(1):83–114, 2023.
- A generalization of algebraic-geometry codes. IEEE Trans. Inf. Theory, 45(7):2498–2501, 1999.
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.