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We consider a high-dimensional random constrained optimization problem inwhich a set of binary variables is subjected to a linear system of equations.The cost function is a simple linear cost, measuring the Hamming distance withrespect to a reference configuration. Despite its apparent simplicity, thisproblem exhibits a rich phenomenology. We show that different situations arisedepending on the random ensemble of linear systems. When each variable isinvolved in at most two linear constraints, we show that the problem can bepartially solved analytically, in particular we show that upon convergence, thezero-temperature limit of the cavity equations returns the optimal solution. Wethen study the geometrical properties of more general random ensembles. Inparticular we observe a range in the density of constraints at which thesystems enters a glassy phase where the cost function has many minima.Interestingly, the algorithmic performances are only sensitive to another phasetransition affecting the structure of configurations allowed by the linearconstraints. We also extend our results to variables belonging to$\text{GF}(q)$, the Galois Field of order $q$. We show that increasing thevalue of $q$ allows to achieve a better optimum, which is confirmed by theReplica Symmetric cavity method predictions.

The closest vector problem and the zero-temperature p-spin landscape for lossy compression