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 $k = 2,3$. 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 = zeros(1,2^n);   % verification array

% produce an array with rows all possible +/-1 vectors
temp = dec2bin(0:((2^n)-1),n);
for j = 1:2^n
    for k = 1:n
        s(j,k) = 2*str2num(temp(j,k))-1;
    end
 
    h = s(j,:);
    h2 = [h(1),fliplr(h(2:end))]';
    C = toeplitz(h2,h);

    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
        if det(C) == 0
            'det = 0 for j = ',j
            continue;
        else
           v(j) = c(1);h
 
        end
    end

end

y = v;

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;

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 $k = 2,3$. 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 = zeros(1,2^n);   % verification array

% produce an array with rows all possible +/-1 vectors
temp = dec2bin(0:((2^n)-1),n);
for j = 1:2^n
    for k = 1:n
        s(j,k) = 2*str2num(temp(j,k))-1;
    end

    h = s(j,:);
    h2 = [h(1),fliplr(h(2:end))]';
    C = toeplitz(h2,h);

    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
        if det(C) == 0
            'det = 0 for j = ',j
            continue;
        else
           v(j) = c(1);h

        end
    end

end

y = v;

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 $k = 2,3$. 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 = zeros(1,2^n);   % verification array

% produce an array with rows all possible +/-1 vectors
temp = dec2bin(0:((2^n)-1),n);
for j = 1:2^n
    for k = 1:n
        s(j,k) = 2*str2num(temp(j,k))-1;
    end

    h = s(j,:);
    h2 = [h(1),fliplr(h(2:end))]';
    C = toeplitz(h2,h);

    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
        if det(C) == 0
            'det = 0 for j = ',j
            continue;
        else
           v(j) = c(1);h

        end
    end

end

y = v;