Abstract
Given a constant weight linear code, we investigate its weight hierarchy and the Stanley–Reisner resolution of its associated matroid regarded as a simplicial complex. We also exhibit conditions on the higher weights sufficient to conclude that the code is of constant weight.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Björner A.: The homology and shellability of matroids and geometric lattices. In: Matroid Applications. Encyclopedia of Mathematics Application, vol. 40, pp. 226–283. Cambridge University Press, Cambridge (1992).
Eagon J.A., Reiner V.: Resolutions of Stanley-Reisner rings and Alexander duality. J. Pure Appl. Algebra 130(3), 265–275 (1998).
Helleseth T., Kløve T., Mykkeltveit J.: The weight distribution of irreducible cyclic codes with block lengths \(n_1((q^l-1)/N)\). Discr. Math. 18, 179–211 (1977).
Johnsen T., Verdure H.: Hamming weights and Betti numbers of Stanley-Reisner rings associated to matroids. AAECC, to appear.
Konvalina J.: A unified interpretation of the binomial coefficients, the Stirling numbers, and the Gaussian coefficients. Am. Math. Month. 107(10), 901–910 (2000).
Liu Z., Chen W.: Notes on the value function. Des. Codes Cryptogr. 54, 11–19 (2010).
Martin J.: Matroids, Demi-Matroids and Chains of Linear Codes. Master Thesis. http://hdl.handle.net/10037/2957 (2010).
Miller E., Sturmfels B.: Combinatorial Commutative Algebra, GTM 227. Springer, New York (2005).
Oxley J.G.: Matroid Theory. Oxford University Press, Oxford (1992).
Acknowledgments
We thank the anonymous referees for helpful remarks.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by T. Helleseth.
Rights and permissions
About this article
Cite this article
Johnsen, T., Verdure, H. Stanley–Reisner resolution of constant weight linear codes. Des. Codes Cryptogr. 72, 471–481 (2014). https://doi.org/10.1007/s10623-012-9767-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-012-9767-2