For fixed neuronal responses variances and tuning curves, we compute coding performance – quantified by

information values – for different values of the pair-wise noise correlations. To be physically realizable, the correlation coefficients must form a positive semi-definite matrix. This constraint defines a spectrahedron, or a swelled tetrahedron, for the

cells used. The colors of the points represent

information values. With different parameters

and

(see values in Section “Details for numerical examples and simulations”), the optimal configuration can appear at different locations, either unique (

**A**) or attained at multiple disjoint places (

**B**), but always on the boundary of the spectrahedron. In both panels, plot titles give the maximum value of

attained over the allowed space of noise correlations, and the value of

that would obtained with the given tuning curves, and perfectly deterministic neural responses. This provides an upper bound on the attainable

(see text Section “Noise cancellation”). Interestingly, in panel (

**A**), the noisy population achieves this upper bound on performance, but this is not the case in (

**B**). Details of parameters used are in Section “Details for numerical examples and simulations”.

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