Abstract
Subspace codes, i.e., sets of subspaces of $\mathbb{F}_q^v$, are applied in random linear network coding. Here we give improved upper bounds for their cardinalities based on the Johnson bound for constant dimension codes.
Highlights
Driven by applications in random linear network coding, the field of subspace coding has sparked a lot of interest recently among both engineers and mathematicians
If b1 = 1 we observe that no q4 + q2 + 1 planes can meet in a common point, cf. the proof of Lemma 15, so that we obtain the stated upper bound for b3 in case (2)
We have generalized the underlying idea of the Johnson bound for constant dimension codes to mixed dimension subspace codes
Summary
Driven by applications in random linear network coding, the field of subspace coding has sparked a lot of interest recently among both engineers and mathematicians. In this article we will investigate the Johnson bound for applicability in the case of general mixed dimension subspace codes. The set of all subspaces of Fvq, ordered by the incidence relation ⊆, is called (v − 1)-dimensional projective geometry over Fq and denoted by PG(v − 1, Fq) or PG(Fvq) It forms a finite modular geometric lattice with meet X ∧ Y = X ∩ Y , join X ∨ Y = X + Y , and rank function X → dim(X).
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