  This paper addresses the performance of bit-interleaved coded multiple
beamforming (BICMB) [1], [2] with imperfect knowledge of beamforming vectors.
Most studies for limited-rate channel state information at the transmitter
(CSIT) assume that the precoding matrix has an invariance property under an
arbitrary unitary transform. In BICMB, this property does not hold. On the
other hand, the optimum precoder and detector for BICMB are invariant under a
diagonal unitary transform. In order to design a limited-rate CSIT system for
BICMB, we propose a new distortion measure optimum under this invariance. Based
on this new distortion measure, we introduce a new set of centroids and employ
the generalized Lloyd algorithm for codebook design. We provide simulation
results demonstrating the performance improvement achieved with the proposed
distortion measure and the codebook design for various receivers with linear
detectors. We show that although these receivers have the same performance for
perfect CSIT, their performance varies under imperfect CSIT.
