This paper builds on Mittelholzer and Lahtonen's study of group codes for the Gaussian channel based on reflection groups. A careful analysis of the action of a reflection group on its roots leads to the development of improved methods for encoding and decoding. The new algorithm is proved to achieve maximum likelihood decoding. The complexity of decoding is analyzed, and it is shown that a proper choice of the sequence of subgroups used in the algorithm can yield significant gains in the efficiency of decoding.

In this correspondence, the optimality of variant-I permutation codes initially proposed by Slepian [see proc. IEEE, vol. 53, no. 3, p. 228-236, Mar. 1965] is shown in a simple way.

Slepian (1962) observed that the code points in an n-dimensional variant-1 permutation modulation code actually lie in an n-1-dimensional subspace. In this correspondence, a change of coordinates which gives the representation explicitly in n-1 dimensions is derived. Slepian's optimum decoding algorithm can be adapted to the codes in this form using an idea of Biglieri and Elia (1976).

First it is shown that all binary Reed-Muller codes with one digit dropped can be made cyclic by rearranging the digits. Then a natural generalization to the nonbinary case is presented, which also includes the Reed-Muller codes and Reed-Solomon codes as special cases. The generator polynomial is characterized and the minimum weight is established. Finally, some results on weight distribution are given.

A class of cyclic codes is introduced by a polynomial approach that is an extension of the Mattson-Solomon method and of the Muller method. This class of codes contains several important classes of codes as subclasses, namely, BCH codes, Reed-Solomon codes, generalized primitive Reed-Muller codes, and finite geometry codes. Certain fundamental properties of this class of codes are derived. Some subclasses are shown to be majority-logic decodable.

Bose and Ray-Chaudhuri have recently described a class of binary codes which for arbitrarymandtaret-error correcting and have length2^m - 1of which no more thanmtdigits are redundancy. This paper describes a simple error-correction procedure for these codes. Their cyclic structure is demonstrated and methods of exploiting it to implement the coding and correction procedure using shift registers are outlined. Closer bounds on the number of redundancy digits are derived.

This paper describes a search for optimum group codes using the IBM 704 computer. Some theory of the relationship between equivalent codes used in narrowing the search is described, and the method of searching is outlined. The newly found optimum codes are listed, along with a number of counter-examples to typical conjectures on binary group codes.

The problem of signal detectability treated in this paper is the following: Suppose an observer is given a voltage varying with time during a prescribed observation interval and is asked to decide whether its source is noise or is signal plus noise. What method should the observer use to make this decision, and what receiver is a realization of that method? After giving a discussion of theoretical aspects of this problem, the paper presents specific derivations of the optimum receiver for a number of cases of practical interest. The receiver whose output is the value of the likelihood ratio of the input voltage over the observation interval is the answer to the second question no matter which of the various optimum methods current in the literature is employed including the Neyman - Pearson observer, Siegert's ideal observer, and Woodward and Davies' "observer." An optimum observer required to give a yes or no answer simply chooses an operating level and concludes that the receiver input arose from signal plus noise only when this level is exceeded by the output of his likelihood ratio receiver. Associated with each such operating level are conditional probabilities that the answer is a false alarm and the conditional probability of detection. Graphs of these quantities called receiver operating characteristic, or ROC, curves are convenient for evaluating a receiver. If the detection problem is changed by varying, for example, the signal power, then a family of ROC curves is generated. Such things as betting curves can easily be obtained from such a family. The operating level to be used in a particular situation must be chosen by the observer. His choice will depend on such factors as the permissable false alarm rate, a priori probabilities, and relative importance of errors. With these theoretical aspects serving as an introduction, attention is devoted to the derivation of explicit formulas for likelihood ratio, and for probability of detection and probability of false alarm, for a number of particular cases. Stationary, band-limited, white Gaussian noise is assumed. The seven special cases which are presented were chosen from the simplest problems in signal detection which closely represent practical situations. Two of the cases form a basis for the best available approximation to the important problem of finding probability of detection when the starting time of the signal, signal frequency, or both, are unknown. Furthermore, in these two cases uncertainty in the signal can be varied, and a quantitative relationship between uncertainty and ability to detect signals is presented for these two rather general cases. The variety of examples presented should serve to suggest methods for attacking other simple signal detection problems and to give insight into problems too complicated to allow a direct solution.