  We study quantized beamforming in wireless amplify-and-forward
relay-interference networks with any number of transmitters, relays, and
receivers. We design the quantizer of the channel state information to minimize
the probability that at least one receiver incorrectly decodes its desired
symbol(s). Correspondingly, we introduce a generalized diversity measure that
encapsulates the conventional one as the first-order diversity. Additionally,
it incorporates the second-order diversity, which is concerned with the
transmitter power dependent logarithmic terms that appear in the error rate
expression. First, we show that, regardless of the quantizer and the amount of
feedback that is used, the relay-interference network suffers a second-order
diversity loss compared to interference-free networks. Then, two different
quantization schemes are studied: First, using a global quantizer, we show that
a simple relay selection scheme can achieve maximal diversity. Then, using the
localization method, we construct both fixed-length and variable-length local
(distributed) quantizers (fLQs and vLQs). Our fLQs achieve maximal first-order
diversity, whereas our vLQs achieve maximal diversity. Moreover, we show that
all the promised diversity and array gains can be obtained with arbitrarily low
feedback rates when the transmitter powers are sufficiently large. Finally, we
confirm our analytical findings through simulations.
