Abstract
We give a classification of four-circulant singly even self-dual [60, 30, d] codes for \(d=10\) and 12. These codes are used to construct extremal singly even self-dual [60, 30, 12] codes with weight enumerator for which no extremal singly even self-dual code was previously known to exist. From extremal singly even self-dual [60, 30, 12] codes, we also construct optimal singly even self-dual [58, 29, 10] codes with weight enumerator for which no optimal singly even self-dual code was previously known to exist. Finally, we give some restriction on the possible weight enumerators of certain singly even self-dual codes with shadow of minimum weight 1.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bosma W., Cannon J., Playoust C.: The Magma algebra system I: The user language. J. Symb. Comput. 24, 235–265 (1997).
Bouyuklieva S., Russeva R., Yankov N.: On the structure of binary self-dual codes having an automorphism of order a square of an odd prime. IEEE Trans. Inf. Theory 51, 3678–3686 (2005).
Conway J.H., Sloane N.J.A.: A new upper bound on the minimal distance of self-dual codes. IEEE Trans. Inf. Theory 36, 1319–1333 (1990).
Chigira N., Harada M., Kitazume M.: Extremal self-dual codes of length \(64\) through neighbors and covering radii. Des. Codes Cryptogr. 42, 93–101 (2007).
Dontcheva R., Harada M.: Some extremal self-dual codes with an automorphism of order 7. Appl. Algebra Eng. Commun. Comput. 14, 75–79 (2003).
Dougherty S.T., Gulliver T.A., Harada M.: Extremal binary self-dual codes. IEEE Trans. Inf. Theory 43, 2036–2047 (1997).
Gulliver T.A., Harada M.: Weight enumerators of extremal singly-even \([60,30,12]\) codes. IEEE Trans. Inf. Theory 42, 658–659 (1996).
Harada M., Munemasa A.: Some restrictions on weight enumerators of singly even self-dual codes. IEEE Trans. Inf. Theory 52, 1266–1269 (2006).
Karadeniz S., Aksoy R.: Self-dual \(R_k\) lifts of binary self-dual codes. Finite Fields Appl. 34, 317–326 (2015).
Kaya A., Yildiz B., Siap I.: New extremal binary self-dual codes from \({\mathbb{F}}_4+u{\mathbb{F}}_4\)-lifts of quadratic circulant codes over \({\mathbb{F}}_4\). Finite Fields Appl. 35, 318–329 (2015).
Rains E.M.: Shadow bounds for self-dual codes. IEEE Trans. Inf. Theory 44, 134–139 (1998).
Tsai H.-P.: Existence of certain extremal self-dual codes. IEEE Trans. Inf. Theory 38, 501–504 (1992).
Tsai H.-P., Jiang Y.J.: Some new extremal self-dual \([58,29,10]\) codes. IEEE Trans. Inf. Theory 44, 813–814 (1998).
Yankov N., Lee M.H.: New binary self-dual codes of lengths 50–60. Des. Codes Cryptogr. 73, 983–996 (2014).
Acknowledgements
This work was supported by JSPS KAKENHI Grant Number 15H03633. The author would like to thank the anonymous reviewers for the useful comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by P. Charpin.
Rights and permissions
About this article
Cite this article
Harada, M. Binary extremal self-dual codes of length 60 and related codes. Des. Codes Cryptogr. 86, 1085–1094 (2018). https://doi.org/10.1007/s10623-017-0380-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-017-0380-2