Efficient coding has served as a guiding principle in understanding the neural code. To date, however, it has been explored mainly in the context of peripheral sensory cells with simple tuning curves. By contrast, ‘deeper’ neurons such as grid cells come with more complex tuning properties which imply a different, yet highly efficient, strategy for representing information. I will show that a highly efficient code is not specific to a population of neurons with finely tuned response properties: it emerges robustly in a shallow network with random synapses. Here, the geometry of population responses implies that optimality obtains from a tradeoff between two qualitatively different types of error: ‘local’ errors (common to classical neural population codes) and ‘global’ (or ‘catastrophic’) errors. This tradeoff leads to efficient compression of information from a high-dimensional representation to a low-dimensional one. After describing the theoretical framework, I will use it to re-interpret recordings of motor cortex in behaving monkey. Our framework addresses the encoding of (sensory) information; if time allows, I will comment on ongoing work that focuses on decoding from the perspective of efficient coding.