Top

Designs, Codes and Cryptography

Published in:

23-02-2021

New families of self-dual codes

Author: Lin Sok

Published in: Designs, Codes and Cryptography | Issue 5/2021

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

Recently, the author has constructed families of MDS Euclidean self-dual codes from genus zero algebraic geometry (AG) codes. In the present correspondence, more families of optimal Euclidean self-dual codes from AG codes are explored. New families of MDS Euclidean self-dual codes of odd characteristic and those of almost MDS Euclidean self-dual codes are constructed explicitly from genus zero and genus one curves, respectively. More families of Euclidean self-dual codes are constructed from algebraic curves of higher genus.

previous article Some constructions of quantum MDS codes

next article A note on Assmus–Mattson type theorems

Ball S.: On sets of vectors of a finite space in which every subset of basis size is a basis. J. Eur. Soc. 14, 733–748 (2012).MathSciNetCrossRef

Betsumiya K., Georgiou S., Gulliver T.A., Harada M., Koukouvinos C.: On self-dual codes over some prime fields. Discrete Math. 262, 37–58 (2003).MathSciNetCrossRef

Bosma W., Cannon J.: Handbook of Magma Functions, Sydney (1995).

Cramer R., Daza V., Gracia I., Urroz J.J., Leander G., Marti-Farre J., Padro C.: On codes, matroids, and secure multiparty computation from linear secret-sharing schemes. IEEE Trans. Inform. Theory 54(6), 2647–2657 (2008).MathSciNetCrossRef

Database of best known linear codes, http://www.codetables.de.

de Boer M.A.: Almost MDS codes. Des. Codes Cryptogr. 9, 143–155 (1996).MathSciNetCrossRef

Dodunekov S.M., Landjev I.N.: Near-MDS codes over some small fields. Discrete Math. 213, 55–65 (2000).MathSciNetCrossRef

Dougherty S.T., Mesnager S., Solé P.: Secret-sharing schemes based on self-dual codes. In: Proc. Inf. Theory Workshop, May, pp. 338–342 (2008).

Driencourt Y., Stichtenoth H.: A criterion for self-duality of geometric codes. Commun. Algebra 17(4), 885–898 (1989).MathSciNetCrossRef

10.

Fang W., Fu F.: New constructions of MDS Euclidean self-dual codes from GRS codes and extended GRS codes. IEEE Trans. Inform. Theory 65(9), 5574–5579 (2019).MathSciNetCrossRef

11.

Georgiou S., Koukouvinos C.: MDS self-dual codes over large prime fields. Finite Fields Appl. 8, 455–470 (2002).MathSciNetCrossRef

12.

Goppa V.D.: Algebraico-geometric codes. Math. USSR-lvz. 21(1), 75–91 (1983).CrossRef

13.

Grassl M., Gulliver T.A.: On self-dual MDS codes, ISIT 2008, Toronto, Canada, July 6 –11 (2008).

14.

Guenda K.: New MDS self-dual codes over finite fields. Des. Codes Cryptogr. 62, 31–42 (2012).MathSciNetCrossRef

15.

Huffman W.C., Pless V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003).CrossRef

16.

Jin L.F., Xing C.P.: New MDS self-dual codes from generalized Reed-Solomon codes. IEEE Trans. Inform. Theory 63(3), 1434–1438 (2017).MathSciNetCrossRef

17.

Kim J.-L., Lee Y.: Construction of MDS self-dual codes over Galois rings. Des. Codes Cryptogr. 45, 247–258 (2007).MathSciNetCrossRef

18.

Kim J.-L., Lee Y.: Euclidean and Hermitian self-dual MDS codes over large finite fields. J. Comb. Theory Ser. A 105, 79–95 (2004).MathSciNetCrossRef

19.

MacWilliams F.J., Sloane N.J.A., Thompson J.G.: Good self-dual codes exist. Discrete Math. 3, 153–162 (1972).MathSciNetCrossRef

20.

MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. North Holland, Amsterdam (1977).MATH

21.

Massey J.: Some applications of coding theory in cryptography. In: Proc. 4th IMA Conf. Cryptogr. Coding, pp. 33–47 (1995).

22.

Sok L.: Explicit constructions of MDS self-dual codes. IEEE Trans. Inform. Theory 66(6), 3603–3615 (2020).MathSciNetCrossRef

23.

Sok, L.: On Euclidean self-dual codes and isometry codes. Applicable Algebra in Engineering, Communication and Computing. https://doi.org/10.1007/s00200-020-00434-y.

24.

Stichtenoth H.: Self-dual Goopa codes. J. Pure Appl. Algebra 55, 199–211 (1988).MathSciNetCrossRef

25.

Stichtenoth H.: Algebraic Function Fields and Codes. Springer, New York (2008).MATH

26.

Tsfasman M.A., Vlǎduţ S.G.: Algebraic-geometric codes. Kluwer Academic Publication, Mathematics and Its Applications, vol. 58 (1991).

27.

Tong H., Wang X.: New MDS Euclidean and Hermitian self-dual codes over finite fields. Adv. Pure Math. 7, 325–333 (2017).CrossRef

28.

Yan H.: A note on the constructions of MDS self-dual codes. Cryptogr. Commun. 11, 259–268 (2019).MathSciNetCrossRef

Title: New families of self-dual codes
Author: Lin Sok
Publication date: 23-02-2021
Publisher: Springer US
Published in: Designs, Codes and Cryptography / Issue 5/2021
Print ISSN: 0925-1022
Electronic ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-021-00847-x

Springer Professional

New families of self-dual codes

Abstract

Premium Partner

Springer Professional

Abstract

Please log in to get access to your license.

Other articles of this Issue 5/2021

Lattice-based zero-knowledge arguments for additive and multiplicative relations

Constructions of doubly resolvable Steiner quadruple systems

Near-MDS codes from elliptic curves

Some constructions of quantum MDS codes

A scalable post-quantum hash-based group signature

(In)security of concrete instantiation of Lin17’s functional encryption scheme from noisy multilinear maps

Premium Partner