Complex Orthogonal Designs with Forbidden $2\times 2$ Submatrices

Abstract

Complex orthogonal designs (CODs) are used to construct space-time block codes. COD ${\mathcal{O}}_z$ with parameter $[p,n,k]$ is a $p\times n$ matrix, where nonzero entries are filled by $\pm z_i$ or $\pm z_i^{\ast }$, $i=1,2,...,k$, such that ${\mathcal{O}}_z^H{\mathcal{O}}_z=(|z_1{|}^2+|z_2{|}^2+...+|z_k{|}^2)I_{n\times n}$. Define ${\mathcal{O}}_z$ a first type COD if and only if ${\mathcal{O}}_z$ does not contain submatrix {\pm z_j & 0; \ 0 & \pm z^*_j}$or${\pm z^*_j & 0; \ 0 & \pm z_j}$.Itisalreadyknownthat,allCODswithmaximalrate,i.e.,maximal$k/p$,areofthefirsttype.Inthispaper,wedetermineallachievableparameters$[p, n, k]$offirsttypeCOD,aswellasalltheirpossiblestructures.Theexistenceofparametersisprovedbyexplicit-formconstructions.NewCODswithparameters$[p,n,k]=[\binom{n}{w-1}+\binom{n}{w+1}, n, \binom{n}{w}], $for$0 \le w \le n$, are constructed, which demonstrate the possibility of sacrificing code rate to reduce decoding delay. It's worth mentioning that all maximal rate, minimal delay CODs are contained in our constructions, and their uniqueness under equivalence operation is proved.