Huffman coding finds a prefix-free code that minimizes the expected codeword length for a probability mass function. However, there are many practical situations in which the constraints and goals may be different from the linear penalty of minimizing expected length. For example, we may wish to avoid long codewords in some situations, since these may cause an excessive wait for transmission of unlikely events. Therefore, we may desire a nonlinear penalty in the codeword lengths. Such penalties arise naturally in group testing for diagnostic systems, queueing, remote exploration, and military intelligence. We examine families of coding problems with nonlinear penalties. These include generalizations of Huffman coding suitable for communication and analysis, extending instances of source coding found in the literature. Alternative penalty measures previously considered include exponential average and maximal pointwise redundancy. We find a novel framework encompassing these two penalties, as well as the linear and other natural penalties, resulting in properties and algorithms suitable for all of them. We next turn to generalized quasilinear convex penalties; these are penalties that can be formulated as Σi f ( li, pi), where li denotes the length of the ith codeword, pi denotes the corresponding probability, and f is convex and increasing in li. This class of penalties includes several previously proposed penalties, among them a general convex problem proposed by Campbell. We give an efficient algorithm that, for Campbell's problem, performs the optimization in quadratic time and linear space; we also present penalty bounds for this solution. Finally, we consider coding for infinite alphabets, a problem not fully understood even for the linear penalty.
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