Abstract
In random network coding so-called constant dimension codes (CDCs) are used for error correction and detection. Most of the largest known codes contain a lifted maximum rank distance (LMRD) code as a subset. For some special cases, Etzion and Silberstein have demonstrated that one can obtain tighter upper bounds on the maximum possible cardinality of CDCs if we assume that an LMRD code is contained. The range of applicable parameters was partially extended by Heinlein. Here we fully generalize those bounds, which also sheds some light on recent constructions.
Highlights
L ET V ∼= Fvq be a v-dimensional vector space over the finite field Fq with q elements
We speak of a constant dimension code (CDC)
We refine the approach by splitting the counts according by the dimension of the intersection with the special subspace that is disjoint to all codewords of the lifted maximum rank distance (LMRD)
Summary
L ET V ∼= Fvq be a v-dimensional vector space over the finite field Fq with q elements. We refine the approach by splitting the counts according by the dimension of the intersection with the special subspace that is disjoint to all codewords of the LMRD. This gives an integer linear programming problem, see Lemma 6, from which we conclude an explicit upper bound, see Corollary 7. We prove those results for the maximum number Bq(v1, v2, d; k) of k-spaces in Fvq with minimum subspace distance d such that there exists a v2space W which intersects every chosen k-space in dimension at least d/2, which is more general
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