Abstract
We give two formulas involving covering polynomials and use them to study projections \({{\mathcal {C}}}\longrightarrow {{\mathcal {C}}}*v\) for binary self-dual codes \({{\mathcal {C}}}\) and vectors v. The dimension of \({{\mathcal {C}}}*v\) is determined in terms of the weight of v and the number of codewords covered by v. These projections are very useful in the study of self-dual codes. An example is given proving some new properties of a doubly-even self-dual code with parameters [72, 36, 12].
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
The referee pointed out the similarity of these polynomials to those defined by C. Bachoc in [1]
References
Bachoc C.: On harmonic weight enumerators of binary codes. Des. Codes Cryptogr. 18(1–3), 11–28 (1999).
Dontcheva R.: New binary [70,35,12] self-dual and binary [72,36,12] self-dual doubly-even codes. Serdica Math. J. 27(4), 287–302 (2001).
Janusz G.J.: Overlap and covering polynomials with applications to designs and self-dual codes. Siam. J. Discret. Math. 13(2), 154–178 (2000).
MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. North-Holland Mathematical Library, vol. 16. North-Holland Publishing Co, Amsterdam (1977).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by G. Kyureghyan.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Janusz, G.J. Covering polynomials and projections of self-dual codes. Des. Codes Cryptogr. 90, 2481–2489 (2022). https://doi.org/10.1007/s10623-022-01091-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-022-01091-7