Advertisement

Designs, Codes and Cryptography

, Volume 86, Issue 4, pp 875–892 | Cite as

Codes with a pomset metric and constructions

  • Irrinki Gnana Sudha
  • R. S. SelvarajEmail author
Article
  • 218 Downloads

Abstract

Brualdi’s introduction to the concept of poset metric on codes over \(\mathbb {F}_{q}\) paved a way for studying various metrics on \(\mathbb {F}_{q}^{n}\). As the support of vector x in \(\mathbb {F}_{q}^{n}\) is a set and hence induces order ideals and metrics on \(\mathbb {F}_{q}^{n}\), the poset metric codes could not accommodate Lee metric structure due to the fact that the support of a vector with respect to Lee weight is not a set but rather a multiset. This leads the authors to generalize the poset metric structure on to a pomset (partially ordered multiset) metric structure. This paper introduces pomset metric and initializes the study of codes equipped with pomset metric. The concept of order ideals is enhanced and pomset metric is defined. Construction of pomset codes are obtained and their metric properties like minimum distance and covering radius are determined.

Keywords

Multiset Pomset Lee weight Poset codes Covering radius 

Mathematics Subject Classification

06A06 94B05 94B75 

References

  1. 1.
    Barg A., Felix L.V., Firer M., Spreafico M.V.P.: Linear codes on posets with extension property. Discret. Math. 317, 1–13 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Brualdi R.A., Graves J.S., Lawrence K.M.: Codes with a poset metric. Discret. Math. 147, 57–72 (1995).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chakrabarty K., Biswas R., Nanda S.: On Yagers theory of bags and fuzzy bags. Comput. Artif. Intell. 18, 1–17 (1999).MathSciNetzbMATHGoogle Scholar
  4. 4.
    D’Oliveira R.G.L., Firer M.: The packing radius of a code and partitioning problems: the case for poset metrics on finite vector spaces. Discret. Math. 338, 2143–2167 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Girish K.P., John S.J.: General relations between partially ordered multisets and their chains and antichains. Math. Commun. 14, 193–205 (2009).MathSciNetzbMATHGoogle Scholar
  6. 6.
    Girish K.P., John S.J.: Multiset topologies induced by multiset relations. Inf. Sci. 188, 298–313 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Hyun J.Y., Kim H.K.: Maximum distance separable poset codes. Des. Codes Cryptogr. 3, 247–261 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Panek L., Firer M., Kim H.K., Hyun J.Y.: Groups of linear isometries on poset structures. Discret. Math. 308, 4116–4123 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Stanley R.P.: Enumerative Combinatorics, vol. 1, 2nd edn. Cambridge University Press, Cambridge (2012).zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of MathematicsNational Institute of Technology WarangalTelanganaIndia

Personalised recommendations