  We study the network-wide neighbour discovery problem in wireless networks in
which each node in a network must discovery the network interface addresses
(NIAs) of its neighbours. We work within the rapid on-off division duplex
framework proposed by Guo and Zhang (2010) in which all nodes are assigned
different on-off signatures which allow them listen to the transmissions of
neighbouring nodes during their off slots, leading to a compressed sensing
problem at each node with a collapsed codebook determined by a given node's
transmission signature. We propose sparse Kerdock matrices as codebooks for the
neighbour discovery problem. These matrices share the same row space as certain
Delsarte-Goethals frames based upon Reed Muller codes, whilst at the same time
being extremely sparse. We present numerical experiments using two different
compressed sensing recovery algorithms, One Step Thresholding (OST) and
Normalised Iterative Hard Thresholding (NIHT). For both algorithms, a higher
proportion of neighbours are successfully identified using sparse Kerdock
matrices compared to codebooks based on Reed Muller codes with random erasures
as proposed by Zhang and Guo (2011). We argue that the improvement is due to
the better interference cancellation properties of sparse Kerdock matrices when
collapsed according to a given node's transmission signature. We show by
explicit calculation that the coherence of the collapsed codebooks resulting
from sparse Kerdock matrices remains near-optimal.
