  Discrete orthogonal matrices have several applications in information
technology, such as in coding and cryptography. It is often challenging to
generate discrete orthogonal matrices. A common approach widely in use is to
discretize continuous orthogonal functions that have been discovered. The need
of certain continuous functions is restrictive. To simplify the process while
improving the efficiency and flexibility, we present a general method for
generating orthogonal matrices directly through the construction of certain
even and odd polynomials from a set of distinct positive values, bypassing the
need of continuous orthogonal functions. We provide a constructive proof by
induction that not only asserts the existence of such polynomials, but also
tells how to iteratively construct them. Besides the derivation of the method
as simple as a few nested loops, we discuss two well-known discrete transforms,
the Discrete Cosine Transform and the Discrete Tchebichef Transform. How they
can be achieved using our method with the specific values, and show how to
embed them into the transform module of video coding. By the same token, we
also show some examples of how to generate new orthogonal matrices from
arbitrarily chosen values.
