Walsh codes help improving condition of matrix?

[Question]
A problem originated in engineering: H is a (N*SF+L-1)*(N*SF) complex
matrix and is generated with a Matlab command
H=filter(h,1,[eye(N*SF);zeros(L-1,N*SF)]) where h is a L*1 complex
vector. C is a (N*SF)*(N*K) complex matrix and is generated with a
Matlab command C=kron(eye(N),c) where c is a SF*K matrix whose columns
are Walsh codes. I find out with a Matlab program that C'*H'*H*C
always has a better condition than H'*H. Is the observation general
and may it be proven rigorously?

[Difficulty]
I failed to express the eigenvalues of H'*H and C'*H'*H*C.

[Thoughts]
The phenomenon may be observed with the following code.
cond_s=[];
cond_S=[];
k=0;
while k<10000
    c=[1,1;1,-1;1,1;1,-1];
    h=randn(3,1);
    H=filter(h,1,[eye(8);zeros(2,8)]);
    s=H'*H;
    cond_s=[cond_s,cond(s)];
    C=kron(eye(2),c);
    S=C'*s*C;
    cond_S=[cond_S,cond(S)];
    k=k+1;
end
all(cond_s>cond_S) is always TRUE.