Asymptotic Bounds for the Sizes of Constant Dimension Codes and an Improved Lower Bound

Abstract

We study asymptotic lower and upper bounds for the sizes of constant dimension codes with respect to the subspace or injection distance, which is used in random linear network coding. In this context we review known upper bounds and show relations between them. A slightly improved version of the so-called linkage construction is presented which is e.g. used to construct constant dimension codes with subspace distance d = 4, dimension k = 3 of the codewords for all field sizes q, and sufficiently large dimensions v of the ambient space, that exceed the MRD bound, for codes containing a lifted MRD code, by Etzion and Silberstein.

New Constructions of Subspace Codes Using Subsets of MRD Codes in Several Blocks

“…The linkage construction in [10] and the generalization in [12] were used to give many presently best known lower bounds for constant dimension subspace codes with small parameters, we refer to [11].…”

“…The important ingredient is the Delsarte Theorem on the rank distribution of a MRD code. The idea using matrices having lower and upper bounded ranks was first appeared in [12].…”

Johnson Type Bounds for Mixed Dimension Subspace Codes

“…Nevertheless parametric improved upper bounds for A q (v, v − 4) and A q (8, 3) have been obtained. We illustrate our results with a small (13,9) 34058 34591 34306 34056 Lemma 4 32514 Here "improved cdc" refers to Lemma 1, "ILP E/V" to the ILP of Etzion and Vardy, see Section 2, "SDP" to results based on semidefinite programming, see [2], "johnson" to the results obtained in this paper, and "bklb" to the currently best known lower bound, see [12]. If the entry of "ILP E/V" is written in italics, then the value for subspace distance d − 1 is taken.…”

Subspace packings: constructions and bounds

“…Rounding in the iterations might decrease the bounds, while the relative difference gets negligible for large values of t, c.f. [43]. Instead of blocks containing a certain point P , we can also consider the collection of blocks that are contained in a certain hyperplane H. Proposition 3.…”

“…The two most successful approaches are the echelon-Ferrers Construction (including their different variants) and the so-called linkage construction [38]. We remark that improvements of the original linkage construction were obtained in [43,63]. In Subsection 4.1 a generalization of the linkage construction for λ > 1 will be presented.…”