Integer DCT approximation constraints

gistfile1.txt

The question: preserving symmetry structure of DCT and orthonormality
(up to scale), how many DoFs are left in integer block transforms?

I wrote a small program that computes the matrix form and resulting
constraints.

S is the overall squared scaling factor. In practice, the S=... constraints
only have to be met approximately (resulting in a transform that's still
orthogonal and close to uniform gain). The off-diagonal (0=) constraints
better be exactly satisfied, though, or we lose orthogonality.

N = 4
c(k) ~proportional to cos(k*pi/8)
   c2    c2    c2    c2
   c1    c3   -c3   -c1
   c2   -c2   -c2    c2
   c3   -c1    c1   -c3

Constraints resulting from orthonormality:
S = 2 * (c1^2 + c3^2)
S = 4 * (c2^2)

N = 8
c(k) ~proportional to cos(k*pi/16)
   c4    c4    c4    c4    c4    c4    c4    c4
   c1    c3    c5    c7   -c7   -c5   -c3   -c1
   c2    c6   -c6   -c2   -c2   -c6    c6    c2
   c3   -c7   -c1   -c5    c5    c1    c7   -c3
   c4   -c4   -c4    c4    c4   -c4   -c4    c4
   c5   -c1    c7    c3   -c3   -c7    c1   -c5
   c6   -c2    c2   -c6   -c6    c2   -c2    c6
   c7   -c5    c3   -c1    c1   -c3    c5   -c7

Constraints resulting from orthonormality:
S = 2 * (c1^2 + c3^2 + c5^2 + c7^2)
S = 8 * (c4^2)
S = 4 * (c2^2 + c6^2)
0 = -c1*c3 + c1*c5 + c3*c7 + c5*c7

N = 16
c(k) ~proportional to cos(k*pi/32)
   c8    c8    c8    c8    c8    c8    c8    c8    c8    c8    c8    c8    c8    c8    c8    c8
   c1    c3    c5    c7    c9    c11   c13   c15  -c15  -c13  -c11  -c9   -c7   -c5   -c3   -c1
   c2    c6    c10   c14  -c14  -c10  -c6   -c2   -c2   -c6   -c10  -c14   c14   c10   c6    c2
   c3    c9    c15  -c11  -c5   -c1   -c7   -c13   c13   c7    c1    c5    c11  -c15  -c9   -c3
   c4    c12  -c12  -c4   -c4   -c12   c12   c4    c4    c12  -c12  -c4   -c4   -c12   c12   c4
   c5    c15  -c7   -c3   -c13   c9    c1    c11  -c11  -c1   -c9    c13   c3    c7   -c15  -c5
   c6   -c14  -c2   -c10   c10   c2    c14  -c6   -c6    c14   c2    c10  -c10  -c2   -c14   c6
   c7   -c11  -c3    c15   c1    c13  -c5   -c9    c9    c5   -c13  -c1   -c15   c3    c11  -c7
   c8   -c8   -c8    c8    c8   -c8   -c8    c8    c8   -c8   -c8    c8    c8   -c8   -c8    c8
   c9   -c5   -c13   c1   -c15  -c3    c11   c7   -c7   -c11   c3    c15  -c1    c13   c5   -c9
   c10  -c2    c14   c6   -c6   -c14   c2   -c10  -c10   c2   -c14  -c6    c6    c14  -c2    c10
   c11  -c1    c9    c13  -c3    c7    c15  -c5    c5   -c15  -c7    c3   -c13  -c9    c1   -c11
   c12  -c4    c4   -c12  -c12   c4   -c4    c12   c12  -c4    c4   -c12  -c12   c4   -c4    c12
   c13  -c7    c1   -c5    c11   c15  -c9    c3   -c3    c9   -c15  -c11   c5   -c1    c7   -c13
   c14  -c10   c6   -c2    c2   -c6    c10  -c14  -c14   c10  -c6    c2   -c2    c6   -c10   c14
   c15  -c13   c11  -c9    c7   -c5    c3   -c1    c1   -c3    c5   -c7    c9   -c11   c13  -c15

Constraints resulting from orthonormality:
S = 8 * (c4^2 + c12^2)
S = 16 * (c8^2)
S = 4 * (c2^2 + c6^2 + c10^2 + c14^2)
S = 2 * (c1^2 + c3^2 + c5^2 + c7^2 + c9^2 + c11^2 + c13^2 + c15^2)
0 = -c1*c5 - c1*c13 + c3*c7 - c3*c15 + c5*c7 - c9*c11 + c9*c13 - c11*c15
0 = -c1*c7 - c1*c9 + c3*c5 + c3*c11 + c5*c13 - c7*c15 + c9*c15 - c11*c13
0 = -c1*c3 + c1*c11 - c3*c9 + c5*c9 - c5*c15 + c7*c11 + c7*c13 + c13*c15
0 = -c2*c6 + c2*c10 + c6*c14 + c10*c14