New and Updated Semidefinite Programming Bounds for Subspace Codes

TL;DR

This paper advances bounds on subspace codes using semidefinite programming, providing new upper limits for parameters over various finite fields and extending existing results to broader cases.

Contribution

It introduces new semidefinite programming bounds for subspace codes, generalizing previous results to larger fields and parameters.

Findings

Established new upper bounds for A_q(7,4) for q ≤ 101.

Extended SDP bounds for A_q(n,d) to small q and n.

Provided explicit bounds for specific parameters.

Abstract

We show that $A_2(7,4) \leq 388$ and, more generally, $A_q(7,4) \leq (q^2-q+1)[7]_q + q^4 - 2q^3 + 3q^2 - 4q + 4$ by semidefinite programming for $q \leq 101$. Furthermore, we extend results by Bachoc et al. on SDP bounds for $A_2(n,d)$, where $d$ is odd and $n$ is small, to $A_q(n,d)$ for small $q$ and small $n$.