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On the extendability of particular classes of constant dimension codes

(2016) DESIGNS CODES AND CRYPTOGRAPHY. 79(3). p.407-422
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Abstract
In classical coding theory, different types of extendability results of codes are known. Aclassical 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 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 PG(N, q). When N + 1 equivalent to 0 (mod s + 1), it is possible to partition PG(N, q) into s-spaces, also called s-spreads of PG(N, q). In the finite geometry research, extendability results on large partial s-spreads to s-spreads in PG(N, q) are known when N + 1 equivalent to 0 (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
Extendability of codes, Random network coding, Minihypers, BAER SUBPLANES, FINITE-FIELDS, BLOCKING SETS, DESIGNS

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Chicago
Nakić, Anamari, and Leo Storme. 2016. “On the Extendability of Particular Classes of Constant Dimension Codes.” Ed. Dina Ghinelli, Dieter Jungnickel, Michel Lavrauw, and Alexander Pott. Designs Codes and Cryptography 79 (3): 407–422.
APA
Nakić, A., & Storme, L. (2016). On the extendability of particular classes of constant dimension codes. (D. Ghinelli, D. Jungnickel, M. Lavrauw, & A. Pott, Eds.)DESIGNS CODES AND CRYPTOGRAPHY, 79(3), 407–422.
Vancouver
1.
Nakić A, Storme L. On the extendability of particular classes of constant dimension codes. Ghinelli D, Jungnickel D, Lavrauw M, Pott A, editors. DESIGNS CODES AND CRYPTOGRAPHY. 2016;79(3):407–22.
MLA
Nakić, Anamari, and Leo Storme. “On the Extendability of Particular Classes of Constant Dimension Codes.” Ed. Dina Ghinelli et al. DESIGNS CODES AND CRYPTOGRAPHY 79.3 (2016): 407–422. Print.
@article{8028443,
  abstract     = {In classical coding theory, different types of extendability results of codes are known. Aclassical 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 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 PG(N, q). When N + 1 equivalent to 0 (mod s + 1), it is possible to partition PG(N, q) into s-spaces, also called s-spreads of PG(N, q). In the finite geometry research, extendability results on large partial s-spreads to s-spreads in PG(N, q) are known when N + 1 equivalent to 0 (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.},
  author       = {Nakić, Anamari and Storme, Leo},
  editor       = {Ghinelli, Dina and Jungnickel, Dieter and Lavrauw, Michel and Pott, Alexander},
  issn         = {0925-1022},
  journal      = {DESIGNS CODES AND CRYPTOGRAPHY},
  keywords     = {Extendability of codes,Random network coding,Minihypers,BAER SUBPLANES,FINITE-FIELDS,BLOCKING SETS,DESIGNS},
  language     = {eng},
  number       = {3},
  pages        = {407--422},
  title        = {On the extendability of particular classes of constant dimension codes},
  url          = {http://dx.doi.org/10.1007/s10623-015-0115-1},
  volume       = {79},
  year         = {2016},
}

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