2000 character limit reached
Fractional decoding of algebraic geometry codes over extension fields (2404.07201v1)
Published 10 Apr 2024 in cs.IT, math.AG, and math.IT
Abstract: In this paper, we study algebraic geometry codes from curves over $\mathbb{F}{q\ell}$ through their virtual projections which are algebraic geometric codes over $\mathbb{F}_q$. We use the virtual projections to provide fractional decoding algorithms for the codes over $\mathbb{F}{q\ell}$. Fractional decoding seeks to perform error correction using a smaller fraction of $\mathbb{F}_q$-symbols than a typical decoding algorithm. In one instance, the bound on the number of correctable errors differs from the usual lower bound by the degree of a pole divisor of an annihilator function. In another, we view the virtual projections as interleaved codes to, with high probability, correct more errors than anticipated.
- Improved decoding of interleaved AG codes. In Nigel P. Smart, editor, Cryptography and Coding, pages 37–46, Berlin, Heidelberg, 2005. Springer Berlin Heidelberg.
- Network coding for distributed storage systems. IEEE Transactions on Information Theory, 56(9):4539–4551, 2010.
- Hulls of generalized Reed-Solomon codes via Goppa codes and their applications to quantum codes. IEEE Transactions on Information Theory, 67(10):6619–6626, 2021.
- Repairing Reed-Solomon codes. IEEE Transactions on Information Theory, 63(9):5684–5698, 2017.
- List decoding Reed-Solomon, algebraic-geometric, and Gabidulin subcodes up to the Singleton bound. In Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing, STOC 13, pages 843–852, New York, NY, USA, 2013. Association for Computing Machinery.
- Multivariate Goppa codes. IEEE Transactions on Information Theory, 69(1):126–137, 2023.
- Fractional decoding of codes from Hermitian curves. In 2021 IEEE International Symposium on Information Theory (ISIT), pages 515–520, 2021.
- Fractional decoding of r-Hermitian codes. Finite Fields and Their Applications, 92:102278, 2023.
- Improved power decoding of interleaved one-point Hermitian codes. Designs, Codes and Cryptography, 87:589–607, 2019.
- Improved power decoding of algebraic geometry codes. In 2021 IEEE International Symposium on Information Theory (ISIT), pages 509–514, 2021.
- Welington Santos. On fractional decoding of Reed-Solomon codes. In 2019 IEEE International Symposium on Information Theory (ISIT), pages 1552–1556, 2019.
- Henning Stichtenoth. Algebraic Function Fields and Codes. Springer Publishing Company, Incorporated, 2nd edition, 2008.
- Fractional decoding: Error correction from partial information. In 2017 IEEE International Symposium on Information Theory (ISIT), pages 998–1002, 2017.
- Explicit constructions of MDS array codes and RS codes with optimal repair bandwidth. In 2016 IEEE International Symposium on Information Theory (ISIT), pages 1202–1206, 2016.