Pythagorean powers of hypercubes

For $n\in \mathbb{N}$ consider the $n$-dimensional hypercube as equal to the vector space ${\mathbb{F}}_2^n$, where ${\mathbb{F}}_2$ is the field of size two. Endow ${\mathbb{F}}_2^n$ with the Hamming metric, i.e., with the metric induced by the ${\mathrm{\ell }}_1^n$ norm when one identifies ${\mathbb{F}}_2^n$ with $\{0,1{\}}^n\subseteq {\mathbb{R}}^n$. Denote by ${\mathrm{\ell }}_2^n({\mathbb{F}}_2^n)$ the $n$-fold Pythagorean product of ${\mathbb{F}}_2^n$, i.e., the space of all $x=(x_1,\dots ,x_n)\in {\prod }_{j=1}^n{\mathbb{F}}_2^n$, equipped with the metric $$\mathrm{\forall }x,y\in \prod _{j=1}^n{\mathbb{F}}_2^n,d_{{\mathrm{\ell }}_2^n({\mathbb{F}}_2^n)}(x,y)=\sqrt{\mathrm{\parallel }{x}_1-y_1{\mathrm{\parallel }}_1^2+\dots +\mathrm{\parallel }{x}_n-y_n{\mathrm{\parallel }}_1^2}\mathrm{.}$$ It is shown here that the bi-Lipschitz distortion of any embedding of ${\mathrm{\ell }}_2^n({\mathbb{F}}_2^n)$ into $L_1$ is at least a constant multiple of $\sqrt{n}$. This is achieved through the following new bi-Lipschitz invariant, which is a metric version of (a slight variant of) a linear inequality of Kwapie{'n} and Sch"utt (1989). Letting $\{e_{jk}{\}}_{j,k\in \{1,\dots ,n\}}$ denote the standard basis of the space of all $n$ by $n$ matrices $M_n({\mathbb{F}}_2)$, say that a metric space $(X,d_X)$ is a KS space if there exists $C=C(X)>0$ such that for every $n\in 2$\mathbb{N}$$,everymapping$f:M_n($\mathbb{F}$2)\to X$ satisfies \begin{equation*}\label{eq:metric KS abstract} \frac{1}{n}\sum{j=1}^n$\mathbb{E}$\left[d_X\Big(f\Big(x+\sum_{k=1}^ne_{jk}\Big),f(x)\Big)\right]\le C $\mathbb{E}$\left[d_X\Big(f\Big(x+\sum_{j=1}^ne_{jk_j}\Big),f(x)\Big)\right], \end{equation*} where the expectations above are with respect to $x\in M_n($\mathbb{F}$_2)$and$k=(k_1,\ldots,k_n)\in {1,\ldots,n}^n$ chosen uniformly at random. It is shown here that $L_1$ is a KS space (with $C= 2e^2/(e^2-1)$, which is best possible), implying the above nonembeddability statement. Links to the Ribe program are discussed, as well as related open problems.