Daniel Heinlein

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

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. It exceeds the MRD bound, for codes containing a lifted MRD code, by Etzion and Silberstein.

Binary subspace codes in small ambient spaces

Codes in finite projective spaces equipped with the subspace distance have been proposed for error control in random linear network coding. Here we collect the present knowledge on lower and upper bounds for binary subspace codes for projective dimensions of at most

Tables of subspace codes

7

An upper bound for binary subspace codes of length 8, constant dimension 4 and minimum distance 6

. We obtain several improvements of the bounds and perform two classifications of optimal subspace codes, which are unknown so far in the literature.