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Let and be matrices over a finite field , with same column size n, having linearly independent rows. The problem is to find an optimal estimate of the "information" from the partial "syndrome" , with the condition , for a transmission of -tuples on a -ary totally symmetric memoryless channel. The best estimate has the form , where is the value of maximizing a so-called decision function . Explicit expressions are obtained for ; they allow computation of the critical probabilities of the channel. The theory is applied to multidimensional orthogonal check set decoding.