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Networks of neurons in the brain encode preferred patterns of neural activity via their synaptic connections. Despite receiving considerable attention, the precise relationship between network connectivity and encoded patterns is still poorly understood. Here we consider this problem for networks of threshold-linear neurons whose computational function is to learn and store a set of binary patterns (e.g., a neural code ) as “permitted sets” of the network. We introduce a simple encoding rule that selectively turns “on” synapses between neurons that coappear in one or more patterns. The rule uses synapses that are binary , in the sense of having only two states (“on” or “off”), but also heterogeneous , with weights drawn from an underlying synaptic strength matrix S . Our main results precisely describe the stored patterns that result from the encoding rule, including unintended “spurious” states, and give an explicit …