Explicit Euclidean embeddings in permutation invariant normed spaces

Let (X,‖⋅‖) be a real normed space of dimension N∈N with a basis (ei)1N such that the norm is invariant under coordinate permutations. Assume for simplicity that the basis constant is at most 2. Consider any n∈N and 0<ε<1/4 such that n≤c(log⁡ε−1)−1log⁡N. We provide an explicit construction of a matrix that generates a (1+ε) embedding of ℓ2n into X.