  Space-Time Block Codes from square complex orthogonal designs (SCOD) have
been extensively studied and most of the existing SCODs contain large number of
zero. The zeros in the designs result in high peak-to-average power ratio
(PAPR) and also impose a severe constraint on hardware implementation of the
code when turning off some of the transmitting antennas whenever a zero is
transmitted. Recently, rate 1/2 SCODs with no zero entry have been reported for
8 transmit antennas. In this paper, SCODs with no zero entry for $2^a$ transmit
antennas whenever $a+1$ is a power of 2, are constructed which includes the 8
transmit antennas case as a special case. More generally, for arbitrary values
of $a$, explicit construction of $2^a\times 2^a$ rate $\frac{a+1}{2^a}$ SCODs
with the ratio of number of zero entries to the total number of entries equal
to $1-\frac{a+1}{2^a}2^{\lfloor log_2(\frac{2^a}{a+1}) \rfloor}$ is reported,
whereas for standard known constructions, the ratio is $1-\frac{a+1}{2^a}$. The
codes presented do not result in increased signaling complexity. Simulation
results show that the codes constructed in this paper outperform the codes
using the standard construction under peak power constraint while performing
the same under average power constraint.
