Constructions and Bounds for Mixed-Dimension Subspace Codes

Abstract

Codes in finite projective spaces equipped with the subspace distance have been proposed for error control in random linear network coding. The resulting so-called \emph{Main Problem of Subspace Coding} is to determine the maximum size $A_q(v,d)$ of a code in $\mathrm{PG}(v-1,{\mathbb{F}}_q)$ with minimum subspace distance $d$. Here we completely resolve this problem for $d\ge v-1$. For $d=v-2$ we present some improved bounds and determine $A_q(5,3)=2q^3+2$ (all $q$), $A_2(7,5)=34$. We also provide an exposition of the known determination of $A_q(v,2)$, and a table with exact results and bounds for the numbers $A_2(v,d)$, $v\le 7$.

35 pages, 2 tables, typo corrected

References in corpus (3)

• Tables of subspace codes
• On putative q-Analogues of the Fano Plane and Related Combinatorial Structures
• A New Approach to the Main Problem of Subspace Coding

Cited by in corpus (1)

• The Expurgation-Augmentation Method for Constructing Good Plane Subspace Codes