Complex Orthogonal Designs With Forbidden 2$\,\times\,$2 Submatrices

Complex orthogonal designs (CODs) are used to construct space-time block codes. COD O_z with parameter [p, n, k] is a p × n matrix, where nonzero entries are filled ±z_i by ±z_i*, i = 1,2...,k, or , such that O_z^HO_z = (|z₁|² + |z₂|² + ...+ |z_k|²)I_n×n. Define O_z a first type COD if and only if O_z does not contain submatrix (±zj,0:0,±zj*) or (±zj*,0:0,±zj). It is already known that all CODs with maximal rate, i.e., maximal k/p, are of the first type. In this paper, we will determine all achievable parameters [p, n, k] of first type COD, as well as all their possible structures. The existence of parameters is proved by explicit-form constructions. New CODs with parameters [p, n, k] = [(n:w-1) + (n:w+1),n, (n:w)], for 0 ≤ w ≤ n, are constructed, which demonstrate the possibility of sacrificing code rate to reduce decoding delay. It is worth mentioning that all maximal rate, minimal delay CODs are contained in our constructions, and their uniqueness under equivalence operation is proved.