A quick bit of MATLABing gives 4 total solutions for $k = 1$, namely $h = (-,-,+,-)$, $(-,+,+,+)$, $(+,-,-,-)$, and $(+,+,-,+)$: the corresponding values of $c$ are $2,-2,-2,2$. 

My code returns no solutions for $k = 2,3,4,5$. For larger values this is prohibitive to run. To play with it (especially in case I made any stupid mistakes, as is entirely possible given the time at which I'm writing this), here it is:

    function y = moq51069(K);

    % for MO question 51069

    n = 4*K;
    w = exp(2*pi*i/n);

    v = sparse(1,2^n);   % verification array

    for j = 1:2^n
        temp = dec2bin(j-1,n);
        h = zeros(1,n);
        for k = 1:n
            h(k) = 2*str2num(temp(k))-1;
        end

        Rv = zeros(1,n);
        for m = 1:n
            t = w^(m-1);
            tt = t.^(0:(n-1))';
            Rv(m) = h*tt;
        end

        % test for constant integrality
        c = Rv./h;
        mc = max(max(abs(c - mean(c))));
        if mc < 10^-6
            h2 = [h(1),fliplr(h(2:end))]';
            C = toeplitz(h2,h);
            if det(C) == 0
                'det = 0 for j = ',j
                continue;
            else
               v(j) = c(1);h
            end
        end

    end

    y = v;