Pursley and Sarwate established a lower bound on a combined measure of autocorrelation and crosscorrelation for a pair$(f,g)$of binary sequences (i.e., sequences with terms in {−1, 1}). If$f$is a nonzero sequence, then its autocorrelation demerit factor,$\text {ADF}(f)$, is the sum of the squared magnitudes of the aperiodic autocorrelation values over all nonzero shifts for the sequence obtained by normalizing$f$to have unit Euclidean norm. If$(f,g)$is a pair of nonzero sequences, then their crosscorrelation demerit factor,$\text {CDF}(f,g)$, is the sum of the squared magnitudes of the aperiodic crosscorrelation values over all shifts for the sequences obtained by normalizing both$f$and$g$to have unit Euclidean norm. Pursley and Sarwate showed that for binary sequences, the sum of$\text {CDF}(f,g)$and the geometric mean of$\text {ADF}(f)$and$\text {ADF}{(g)}$must be at least 1. For randomly selected pairs of long binary sequences, this quantity is typically around 2. In this paper, we show that Pursley and Sarwate’s bound is met for binary sequences precisely when$(f,g)$is a Golay complementary pair. We also prove a generalization of this result for sequences whose terms are arbitrary complex numbers. We investigate constructions that produce infinite families of Golay complementary pairs, and compute the asymptotic values of autocorrelation and crosscorrelation demerit factors for such families.

Many networks in the brain are sparsely connected, and the brain eliminates synapses during development and learning. How could the brain decide which synapses to prune? In a recurrent network, determining the importance of a synapse between two neurons is a difficult computational problem, depending on the role that both neurons play and on all possible pathways of information flow between them. Noise is ubiquitous in neural systems, and often considered an irritant to be overcome. Here we suggest that noise could play a functional role in synaptic pruning, allowing the brain to probe network structure and determine which synapses are redundant. We construct a simple, local, unsupervised plasticity rule that either strengthens or prunes synapses using only synaptic weight and the noise-driven covariance of the neighboring neurons. For a subset of linear and rectified-linear networks, we prove that this rule preserves the spectrum of the original matrix and hence preserves network dynamics even when the fraction of pruned synapses asymptotically approaches 1. The plasticity rule is biologically-plausible and may suggest a new role for noise in neural computation.