Large sets of cyclically permutable (or cyclically inequivalent) codewords can be created by interleaving the cyclic shifts of several shorter words and selecting a cyclically inequivalent subset from the resulting set, This correspondence presents an efficient algorithm for generating the cyclically inequivalent subset directly.

LetRbe the asymptotic error probability of the nearest neighbor rule (NNR). A particular estimate ofRfromn-classified samples, known to converge toRin probability asnincreases to infinity, is shown to actually converge with probability 1 toRwhenever the NNR itself converges with probability 1.

A systematic procedure for constructing one member of each cyclic equivalence class of an(n, k + 1)Reed-Solomon code is presented. It is shown that if the procedure is modified to exclude those codewords that do not have maximum period, the resulting set of codewords constitutes a synchronizable code having comma freedom of degreen - 2k.

The two category classification problem is treated. No a priori knowledge of the statistics of the classes is assumed. A sequence of labeled samples from the two classes is used to construct a sequence of approximations of a discriminant function that is optimum in the sense of minimizing the probability of misclassification but which requires knowledge of all the statistics of the classes. Depending on the assumptions made about the probability densities corresponding to the two classes, the integrated square error of the approximations converges to0in probability or with probability1. The approximations are nonparametric and recursive for each fixed point of the domain. Rates of convergence are given. The approximations are used to define a decision procedure for classifying unlabeled samples. It is shown that as the number of labeled samples used to construct the approximations increases, the resulting sequence of discriminant functions is asymptotically optimal in the sense that the probability of misclassification when using the approximations in the decision procedure converges in probability or with probability1, depending on the assumptions made, to the probability of misclassification of the optimum discriminant function. The results can be easily extended to the multicategory problem and to the case of arbitrary loss functions, that is, where the costs of misclassification are not necessarily equal to1.

Let {Yn} be a stationary random process where each Y value takes values in a finite set B. Then define Hnand enso that H is the entropy of {Yn} and e is the probability of error in predicting Y0from the infinite past. The purpose of this correspondence is to give an example which shows that {Hn- H} and {en- e} are not, in general, asymptotically identical for a stationary process {Yn}.