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

DOI10.1109/TIT.2012.2195635zbMATH Open1365.94626arXiv1107.3268OpenAlexW2001958075MaRDI QIDQ5271844

Publication date: 12 July 2017

Published in: IEEE Transactions on Information Theory (Search for Journal in Brave)

Abstract: Complex orthogonal designs (CODs) are used to construct space-time block codes. COD

m a t h c a l O_{z}

with parameter

[p, n, k]

is a

p i m e s n

matrix, where nonzero entries are filled by

p m z_{i}

or

p m z_{i}^{*}

,

i = 1, 2, . . ., k

, such that

m a t h c a l O_{z}^{H} m a t h c a l O_{z} = (| z_{1} |^{2} + | z_{2} |^{2} + . . . + | z_{k} |^{2}) I_{n i m e s n}

. Define

m a t h c a l O_{z}

a first type COD if and only if

m a t h c a l O_{z}

does not contain submatrix {pm z_j & 0; 0 & pm z^*_j}

o r

{pm z^*_j & 0; 0 & pm z_j}

. I t i s a l r e a d y k n o w n t h a t, a l l C O D s w i t h m a x i m a l r a t e, i . e ., m a x i m a l

k/p

, a r e o f t h e f i r s t t y p e . I n t h i s p a p e r, w e d e t e r m i n e a l l a c h i e v a b l e p a r a m e t e r s

[p, n, k]

o f f i r s t t y p e C O D, a s w e l l a s a l l t h e i r p o s s i b l e s t r u c t u r e s . T h e e x i s t e n c e o f p a r a m e t e r s i s p r o v e d b y e x p l i c i t - f o r m c o n s t r u c t i o n s . N e w C O D s w i t h p a r a m e t e r s

[p,n,k]=[�inom{n}{w-1}+�inom{n}{w+1}, n, �inom{n}{w}],

f o r

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.

Full work available at URL: https://arxiv.org/abs/1107.3268

Mathematics Subject Classification ID

Other types of codes (94B60) Orthogonal matrices (15B10)