Approximation in $L_p$ by polynomials in the Walsh system

For $0<q=p<\infty$ and $q=1$, $1\le p<\infty$ we calculate the quantity
$$ \varkappa_{2^n}(L_p,L_q)=\sup_{f\in L_p}\frac{E_{2^n}(f)_q} {\dot\omega\bigl(\frac1{2^n},f\bigr)_p}\,, $$
where $E_{2^n}(f)_q$ is the best $L_q$-approximation of the function $f$ by Walsh polynomials of order $2^n$ and
$$ \dot\omega(\delta,f)_p=\sup_{0<t<\delta}\|f(x\dot+t)-f(x)\|_p $$
is the dyadic modulus of continuity of $f$ in $L_p$ determined by the operation $\dot+$ of addition of numbers from the interval $[0,1]$ in the dyadic system.
Bibliography: 21 titles.