Error correction via linear programming

Suppose we wish to transmit a vector f ϵ R
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reliably. A frequently discussed approach consists in encoding f with an m by n coding matrix A. Assume now that a fraction of the entries of Af are corrupted in a completely arbitrary fashion by an error e. We do not know which entries are affected nor do we know how they are affected. Is it possible to recover f exactly from the corrupted m-dimensional vector y = Af + e?