  This is an attempt to build Banach space valued theory for certain singular
integrals on Hamming cube. Of course all estimates below are dimension
independent, and we tried to find ultimate sharp assumptions on the Banach
space for a corresponding operators to be bounded. In certain cases we
succeeded, although there are still many open questions, some of them are
listed in the last Section. Using the approach of \cite{IVHV} and also quantum
random variables approach of \cite{ELP} we generalize several theorems of
Pisier \cite{P} and Hyt\"onen-Naor \cite{HN}. We also improve the constant in
$L^1$-Poincar\'e inequality on Hamming cube, the previous results are due to
Talagrand and Ben Efraim--Lust-Piquard.
