Designs, Codes and Cryptography

, Volume 79, Issue 3, pp 407–422

On the extendability of particular classes of constant dimension codes

Article

DOI: 10.1007/s10623-015-0115-1

Cite this article as:
Nakić, A. & Storme, L. Des. Codes Cryptogr. (2016) 79: 407. doi:10.1007/s10623-015-0115-1

Abstract

In classical coding theory, different types of extendability results of codes are known. A classical example is the result stating that every \((4, q^2-1, 3)\)-code over an alphabet of order q is extendable to a \((4, q^2, 3)\)-code. A constant dimension subspace code is a set of \((k-1)\)-spaces in the projective space \(\hbox {PG}(n-1,q)\), which pairwise intersect in subspaces of dimension upper bounded by a specific parameter. The theoretical upper bound on the sizes of these constant dimension subspace codes is given by the Johnson bound. This Johnson bound relies on the upper bound on the size of partial s-spreads, i.e., sets of pairwise disjoint s-spaces, in a projective space \(\hbox {PG}(N,q)\). When \(N+1 \equiv 0 \,\,(\text {mod }s+1)\), it is possible to partition \(\hbox {PG}(N,q)\) into s-spaces, also called s-spreads of \(\hbox {PG}(N,q)\). In the finite geometry research, extendability results on large partial s-spreads to s-spreads in \(\hbox {PG}(N,q)\) are known when \(N+1 \equiv 0\,\,(\text {mod }s+1)\). This motivates the study to determine similar extendability results on constant dimension subspace codes whose size is very close to the Johnson bound. By developing geometrical arguments, avoiding having to rely on extendability results on partial s-spreads, such extendability results for constant dimension subspace codes are presented.

Keywords

Random network coding Extendability of codes Minihypers 

Mathematics Subject Classification

05B25 51E20 

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Faculty of Electrical Engineering and ComputingUniversity of ZagrebZagrebCroatia
  2. 2.Department of MathematicsGhent UniversityGhentBelgium

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