Let $B^d_2$ denote the unit ball of $\ell_2^d$ and let $T$ be an invertible linear map. 

Consider the function 
$$
H(T) := \log M(TB_2^d, B_2^d), 
$$
which is the packing entropy for $TB_2^d$ by $B_2^d$.

**Question:** 
Is a general, explicit characterization of $H(T)$ available? 

For instance, the Sudakov minoration yields 
$$
H(T) \lesssim \bigg(\mathbb{E}_g \sup_{x \in B_2^d} \langle g, Tx\rangle\bigg)^2 \asymp \mathrm{tr}(T^\ast T).
$$
(Here $g$ is distributed standard Gaussian in $\mathbf{R}^d$, and $\lesssim$ indicates inequality up to absolute constants and $\asymp$ denotes equality up to constants, independent of everything.) 
Of course, this cannot be sharp as $T = I_d$ already shows.