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On the extendability of particular classes of constant dimension codes

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.

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Acknowledgments

The research of the first author was supported by a Short Term Scientific Mission (STSM) of the COST-project IC-1104 Random Network Coding and Designs over GF(q) (http://www.network-coding.eu/).

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Correspondence to Leo Storme.

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This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Finite Geometries”.

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Nakić, A., Storme, L. On the extendability of particular classes of constant dimension codes. Des. Codes Cryptogr. 79, 407–422 (2016). https://doi.org/10.1007/s10623-015-0115-1

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Keywords

  • Random network coding
  • Extendability of codes
  • Minihypers

Mathematics Subject Classification

  • 05B25
  • 51E20