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.