Abstract
An MDS code is a code which achieves equality in the singleton bound. The defect of a code measures how far it is from an MDS code. Amplifying on the relationship between the weight distribution of a code and its dual code as in the well-known MacWilliams identities, we show in this paper that there are indeed strong lower bounds on the defects of codes or the dual codes.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Despite having examined many results in the literature regarding the construction of codes from higher-dimensional algebraic varieties (see § 5), we could not find any precise statement along these lines.
References
Aubr, Y., Berardini E., Herbaut F., Perret M.: Bounds on the minimum distance of Algebraic Geometry codes defined over some families of surfaces, in Arithmetic, geometry, cryptography and coding theory, 11–28. Contemp. Math. 770, American Math. Society (2021).
Aubry Y., Berardini E., Herbaut F., Perret M.: Algebraic geometry codes over Abelian surfaces containing no absolutely irreducible curves of low genus. Finite Fields Appl. 70, 101791, 20 (2021).
Blache R., Couvreur A., Hallouin E., Madore D., Nardi J., Rambaud M., Randriam H.: Anti-canonical codes from del Pezzo surfaces with Picard rank one. Trans. Am. Math. Soc. 373(8), 5371–5393 (2020).
Couvreur A.: Construction of rational surfaces yielding good codes. Finite Fields Appl. 17(5), 424–441 (2011).
Can M.B., Joshua R., Ravindra G.V.: Higher Grassmann Codes II. Finite Fields Appl. 89, 1–21 (2023).
Couvreur A., Lebacque P., Perret M.: Toward good families of codes from towers of surfaces, in Arithmetic, geometry, cryptography and coding theory. Contemp. Math. 770, 59–93 (2021).
Faldum A., Willems W.: Codes of small defect. Des. Codes Cryptogr. 10, 341–350 (1997).
Hansen J.P.: Toric varieties, Hirzebruch surfaces and error-correcting codes, applicable algebra in engineering. Commun. Comput. 13, 289–300 (2002).
Hansen S.H.: Error-correcting codes from higher-dimensional varieties. Finite Fields Appl. 7(4), 531–552 (2001).
Huffman W.C., Pless V.: Fundamentals of Error Correcting Codes. Cambridge University Press, Cambridge (2003).
Lidl R., Niederreiter H.: Finite Fields, vol. 20, 2nd edn Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (1997).
Little J.: Algebraic Geometry codes from Higher dimensional varieties. In: Advances in Algebraic Geometry Codes, pp. 258–293. World Scientific, Singapore (2008).
Little J., Schenck H.: Codes from surfaces with small Picard number. SIAM J. Appl. Algebra Geom. 2(2), 242–258 (2018).
MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes, vol. 16. North Holland, North Holland Mathematical Library (1977).
Soprunov I., Soprunova J.: Bringing Toric codes to the next dimension. SIAM J. Discrete Math. 24(2), 655–665 (2010).
van Lint J.H.: Coding Theory, vol. 201. Lecture Notes in Mathematics. Springer, New York (1982).
Zarzar M.: Error correcting codes on low rank surfaces. Finite Fields Appl. 13, 727–737 (2007).
Acknowledgements
The authors would like to thank both the reviewers for providing valuable comments that were very helpful to the authors in revising the manuscript.
Funding
This study was funded by Simons Foundation (Grant No. 830817).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by E. Gorla.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Can, M.B., Joshua, R. & Ravindra, G.V. Defects of codes from higher dimensional algebraic varieties. Des. Codes Cryptogr. 92, 477–494 (2024). https://doi.org/10.1007/s10623-023-01317-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-023-01317-2