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Summary This paper gives a method for constructing minimum-redundancy prefix codes for the general discrete noiseless channel without constraints. The costs of code letters need not be equal, and the symbols encoded are not assumed to be equally probable. A solution had previously been given by Huffman in 1952 for the special case in which all code letters are of equal cost. The present development is algebraic. First, structure functions are defined, in terms of which necessary and sufficient conditions for the existence of prefix codes may be stated. From these conditions, linear inequalities are derived which may be used to characterize prefix codes. Gomory’s integer programming algorithm is then used to construct optimum codes subject to these inequalities; an elegant combinatorial approach to obtain a strictly computational experience is presented to demonstrate the practicability of the method. Finally, some additional coding problems are discussed and a problem of classification is treated.