Orthogonal Designs and a Cubic Binary Function

TL;DR

This paper introduces an explicit, non-recursive construction of optimal complex orthogonal designs using a cubic binary function, improving design efficiency for wireless communication coding.

Contribution

It provides a novel explicit formula for optimal CODs based on a cubic binary function, avoiding inductive or algorithmic methods.

Findings

Constructs optimal CODs with direct matrix element calculation

Avoids recurrent procedures in design construction

Utilizes a cubic function related to non-associative algebras

Abstract

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.