  The peak-to-mean envelope power ratio (PMEPR) of a code employed in
orthogonal frequency-division multiplexing (OFDM) systems can be reduced by
permuting its coordinates and by rotating each coordinate by a fixed phase
shift. Motivated by some previous designs of phase shifts using suboptimal
methods, the following question is considered in this paper. For a given binary
code, how much PMEPR reduction can be achieved when the phase shifts are taken
from a 2^h-ary phase-shift keying (2^h-PSK) constellation? A lower bound on the
achievable PMEPR is established, which is related to the covering radius of the
binary code. Generally speaking, the achievable region of the PMEPR shrinks as
the covering radius of the binary code decreases. The bound is then applied to
some well understood codes, including nonredundant BPSK signaling, BCH codes
and their duals, Reed-Muller codes, and convolutional codes. It is demonstrated
that most (presumably not optimal) phase-shift designs from the literature
attain or approach our bound.
