Polynomial growth of Bohnenblust--Hille constants on the Hamming cube

We prove that the Bohnenblust–Hille constants for Walsh polynomials on the Hamming cube grow at most polynomially in the degree. More precisely, there is an absolute constant $K$ such that every $f:\{-1,1{\}}^n\to \mathbb{C}$ of degree at most $m$ satisfies $${(\sum _{\mathrm{\mid }S\mathrm{\mid }\le m}\mathrm{\mid }\widehat{f}(S){\mathrm{\mid }}^{2m\mathrm{/}(m+1)})}^{(m+1)\mathrm{/}(2m)}\le K{m}^{27}\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{.}$$ The estimate is uniform in the dimension. The exponent $27$ is not optimized.