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We study the network-wide neighbour discovery problem in wireless networks inwhich each node in a network must discovery the network interface addresses(NIAs) of its neighbours. We work within the rapid on-off division duplexframework proposed by Guo and Zhang (2010) in which all nodes are assigneddifferent on-off signatures which allow them listen to the transmissions ofneighbouring nodes during their off slots, leading to a compressed sensingproblem at each node with a collapsed codebook determined by a given node'stransmission signature. We propose sparse Kerdock matrices as codebooks for theneighbour discovery problem. These matrices share the same row space as certainDelsarte-Goethals frames based upon Reed Muller codes, whilst at the same timebeing extremely sparse. We present numerical experiments using two differentcompressed sensing recovery algorithms, One Step Thresholding (OST) andNormalised Iterative Hard Thresholding (NIHT). For both algorithms, a higherproportion of neighbours are successfully identified using sparse Kerdockmatrices compared to codebooks based on Reed Muller codes with random erasuresas proposed by Zhang and Guo (2011). We argue that the improvement is due tothe better interference cancellation properties of sparse Kerdock matrices whencollapsed according to a given node's transmission signature. We show byexplicit calculation that the coherence of the collapsed codebooks resultingfrom sparse Kerdock matrices remains near-optimal.

Compressed Neighbour Discovery using Sparse Kerdock Matrices