Abstract
A condition on the weight of a codeword of a linear code is obtained using polynomials over the \(p\)-adic numbers. This condition is obtained by proving a bound on the size of a \(t\)-fold blocking set of hyperplanes in a finite affine space.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ball S.: On intersection sets in desarguesian affine spaces. Eur. J. Comb. 21, 441–446 (2000).
Ball S.: The polynomial method in Galois geometries. In: Current Research Topics in Galois Geometry, Chapter 5. Nova Science Publishers, New York (2012).
Ball S., Blokhuis A.: A bound for the maximum weight of a linear code. SIAM J. Discret. Math. 27, 575–583 (2013).
Blokhuis A.: On multiple nuclei and a conjecture of Lunelli and Sce. Bull. Belg. Math. Soc. 3, 349–353 (1994).
Blokhuis A., Brouwer A.E., Szőnyi T.: Covering all points except one. J. Algebr. Comb. 32, 59–66 (2010).
Brouwer A.E., Schrijver A.: The blocking number of an affine space. J. Comb. Theory Ser. A 24, 251–253 (1978).
Bruen A.A.: Polynomial multiplicities over finite fields and intersection sets. J. Comb. Theory Ser. A 60, 19–33 (1992).
Bruen A.A.: Applications of finite fields to combinatorics and finite geometries. Acta Appl. Math. 93, 179–196 (2006).
Jamison R.: Covering finite fields with cosets of subspaces. J. Comb. Theory Ser. A 22, 253–266 (1977).
Koblitz N.: \(p\)-Adic Numbers, \(p\)-Adic Analysis and Zeta Functions, 2nd ed. Springer-Verlag, New York (1984).
Landjev I., Rousseva A.: On the sharpness of Bruen’s bound for intersection sets in Desarguesian affine spaces, Des. Codes Cryptogr. 2012, 1–8 (2012).
Singleton R.C.: Maximum distance \(q\)-nary codes. IEEE Trans. Inf. Theory 10, 116–118 (1964).
Acknowledgments
The author acknowledges the support of the project MTM2008-06620-C03-01 of the Spanish Ministry of Science and Education and the project 2009-SGR-01387 of the Catalan Research Council. The author would like to thank the anonymous referees for their comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
This is one of several papers published in Designs, Codes and Cryptography comprising the special topic on “Finite Geometries: A special issue in honor of Frank De Clerck”
Rights and permissions
About this article
Cite this article
Ball, S. A \(p\)-adic condition on the weight of a codeword of a linear code. Des. Codes Cryptogr. 72, 177–183 (2014). https://doi.org/10.1007/s10623-013-9863-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-013-9863-y