# Isomorphism for non-degenerate symplectic form

In this paper by Leung et. al (Entanglement can increase asymptotic rates of zero-error classical communication over classical channels) the authors use Haemers' rank bound to bound the independence number of a graph $\Gamma$. Specifically, the independence number $\alpha(\Gamma)$ is shown to be $\leq 2m+1$ by showing that a certain matrix $M$ "fits" the adjacency matrix of $\Gamma$ and that the rank of this matrix is $2m+1$.

The relevant portions from Sec 2.1 of the paper (paraphrased in some cases) are provided here:

On a $2m$-dimensional space, the canonical symplectic form is $$$$\sigma(u,v) := u^T \left( \begin{array}{cc} 0 & \mathbb{I}_m \\ -\mathbb{I}_m & 0 \end{array} \right)v.$$$$ where $\mathbb{I}_m$ is the $m \times m$ identity matrix. Any non-degenerate symplectic space with finite dimensional vector space $V$ is isomorphic to the canonical symplectic space $(V,\sigma)$.

The vertices of the symplectic graph $\Gamma$ are the points of the projective space i.e, the $2^{2m} - 1$ non-zero elements of $\mathbb{F}_2^{2m}$. There is an edge between $u$ and $v$ if $\sigma(u,v)=0$.

Let $$U_m := \{ v \in \mathbb{F}_2^{2m+1} : \langle v,v\rangle = 0 \}$$ be the $2m$-dimensional subspace of $\mathbb{F}_2^{2m+1}$ that consists of vectors which have an even number of entries equal to one. The restriction of the standard inner product $\langle\cdot,\cdot\rangle$ on $\mathbb{F}_2^{2m+1}$ to the subspace $U_m$ is a non-degenerate symplectic form, so there is an isomorphism $T: (\mathbb{F}_2^{2m}, \sigma) \to (U_m, \langle\cdot,\cdot\rangle)$ such that $$\forall u,v \in \mathbb{F}_2^{2m}: \sigma(u,v) = \langle T(u),T(v)\rangle.$$

My question is: What is the explicit form of the map $T$?

I can well believe that this isomorphism exists but what I really want is an explicit recipe e.g. for the case $m=2$, given $(u_1, u_2, u_3, u_4)\in \mathbb{F}_2^4$ what is $T(u_1, u_2, u_3, u_4)\in \mathbb{F}_2^5$?

• This might be suited for MathOverflow since it refers to a research paper and thus ostensibly is research-level mathematics. mathoverflow.net – Chill2Macht Jan 16 '17 at 11:08