On the Maximal Rate of Non-Square STBCs from Complex Orthogonal Designs

A linear processing complex orthogonal design (LPCOD) is a ptimesn matrix epsiv, (pgesn) in k complex indeterminates x₁,x₂,...,x_k such that (i) the entries of epsiv are complex linear combinations of 0, plusmnx_i, i=1,...,k and their conjugates, (ii) epsiv^Hepsiv=D, where epsiv^H is the Hermitian (conjugate transpose) of epsiv and D is a diagonal matrix with the (i,i)-th diagonal element of the form l₁ ⁽ⁱ⁾|x₁|²+l₂ ⁽ⁱ⁾|x₂|²+...+l_k ⁽ⁱ⁾|x_k|² where l_j ⁽ⁱ⁾,i=1,2,...,n, j=1,2,...,k are strictly positive real numbers and the condition l₁ ⁽ⁱ⁾=l₂ ⁽ⁱ⁾=...=l_k ⁽ⁱ⁾, called the equal- weights condition, holds for all values of i. For square designs it is known that whenever a LPCOD exists without the equal-weights condition satisfied then there exists another LPCOD with identical parameters with l₁ ⁽ⁱ⁾=l₂ ⁽ⁱ⁾=...=l_k ⁽ⁱ⁾=1. This implies that the maximum possible rate for square LPCODs without the equal-weights condition is the same as that of square LPCODs with equal-weights condition. In this paper, this result is extended to a subclass of non-square LPCODs. It is shown that, a set of sufficient conditions is identified such that whenever a non- square (p>n) LPCOD satisfies these sufficient conditions and do not satisfy the equal-weights condition, then there exists another LPCOD with the same parameters n, k and p in the same complex indeterminates with l₁ ⁽ⁱ⁾=l₂ ⁽ⁱ⁾=...=l_k ⁽ⁱ⁾=1.