A random coding bound for the multiple-access channel (MAC) is given, and it is shown to be obtainable universally for all MAC's with common input and output alphabets. For channels for which the random coding bound may be achieved by codes consisting of codewords of a single type for each sender, this bound is also achievable by codes produced by permutations.

In the classification problem for chromosomes there areNchromosomes which must be classified intokpopulationsA_(1}, \cdots ,A_{k}having known probability distributions. It is further known that theseNchromosomes haveN_{i}in classA_{i}, i = 1,2, \cdots ,k. This is a compound decision problem whose optimal solution gives a classification algorithm which is not currently useful in practice because of its long computation time. Two other classification methods are considered, and the results are compared. One is the method often used for the classification of the 46 human chromosomes, where the knowledge about the exact number of chromosome types is disregarded, and only the a priori probability that a chromosome originates from populationA_{i}is used. The other method permits only classifications with the correct number of objects in each class and selects from all the possible classifications that one which has the maximum likelihood function. This last method has some advantages for a small number of objects, particularly if the numbers of objects in the classes are equal.