  Orthogonal designs are fundamental mathematical notions used in the
construction of space time block codes for wireless transmissions. Designs have
two important parameters, the rate and the decoding delay; the main problem of
the theory is to construct designs maximizing the rate and minimizing the
decoding delay. All known constructions of CODs are inductive or algorithmic.
In this paper, we present an explicit construction of optimal CODs. We do not
apply recurrent procedures and do calculate the matrix elements directly. Our
formula is based on a cubic function in two binary n-vectors. In our previous
work (Comm. Math. Phys., 2010, and J. Pure and Appl. Algebra, 2011), we used
this function to define a series of non-associative algebras generalizing the
classical algebra of octonions and to obtain sum of squares identities of
Hurwitz-Radon type.
