I am interested in other examples of finite and complete orthogonal basis. I am not confident in my use of the term “complete”, so what I mean specifically is a set of basis vectors that can be used in a transformation from one domain (or vector space) to another with no loss, duplication or distortion in the transformation. (A constant scaling factor is acceptable, hence not restricted to being “orthonormal”.)
Two examples I am familiar with (which related to communication systems is the difference between OFDMA and CDMA):
OFDMA: The set of exponentials in the Discrete Fourier Transform mapping N samples in time to N samples in frequency using $e^{j 2\pi n/N}$ for $n = 0$ to $N-1$, which has the following resulting code set with $W_N^{nk} = (e^{j/N})^{nk}$:
$$W_N = \begin{bmatrix} W_N^0 & W_N^0 & \ldots & W_N^0\\ W_N^0 & W_N^1 & \ldots & W_N^{N-1}\\ \vdots & \vdots & & \vdots \\ W_N^0 & W_N^{N-1} & \ldots & W_N^{(N-1)^2} \\ \end{bmatrix}$$
as a matrix with each row indexed by $k$ for $k=0 \ldots N-1$ and each column indexed by $n$ for $n = 0 \ldots N-1$.
Note that $$W_1 = \begin{bmatrix} 1 & 1\\1 & -1\ \end{bmatrix}$$
CDMA: The set of Walsh-Hadamard Codes mapping N samples in time to N samples in a $W_N$ code space , with a resulting code set built on the pattern:
$$W_N = \begin{bmatrix} W_{N-1} & W_{N-1}\\W_{N-1} & -W_{N-1}\ \end{bmatrix}$$
With $$W_1 = \begin{bmatrix} 1 & 1\\1 & -1\ \end{bmatrix}$$
Interesting to me is the observation that both start of with the same 2 element form, but the DFT expands using the complex plane into possible elements all on the unit circle, while the Walsh-Hadamard codes commonly used for CDMA are limited to two elements (+1 and -1). In particular, I am looking for the possibility of other known finite and complete orthogonal code sets that also use the complex plane, but more samples beyond the unit circle, or alternatively the reason why such codes can't exist.
