An expression is derived for the reliability of an r-on-m fault-tolerant configuration (r spares supporting m identical operating units) when both the hazard and the coverage probability are functions of time. In addition, the coverage probability is allowed to be a function of the number of spares that have to be tested before an operational unit is found.

A new decoding technique is presented for correcting errors due to bit-oriented hardware failures in parallel, random-access memories. It is shown that the resulting decoder compares favorably, both in complexity and in decoding delay, with currently implemented bit-switching techniques used for the same purpose.

A recurring communications systems problem involves the determination, with relatively high precision, of the phase of a periodic, noise-corrupted signal. Perhaps the most commonly proposed signals for such purposes are pseudorandom sequences. Indeed, such signals are known to be optimum, at least when the noise is white and Gaussian, in the sense of minimizing the time needed to find the correct phase with a specified reliability. Equipment limitations, however, often preclude the efficient use of these sequences. In particular, suppose the received signal is known to be of the formy(t) = x(t - \nu \Delta T) + n(t), withx(t)a signal periodic with periodT, andn(t)additive white Gaussian noise. The problem is to determine as reliably and rapidly as possible the "phase"\nuof the received signal. The optimum detector involves the formation of theN = \Delta T/TcorrelationsI_{\nu} = \int y(t)x(t - \nu \Delta T) dtand the selection of the largest of these. Frequently, however, equipment constraints force these correlations to be made serially. In this event, the required search time can be as much asNtimes as great as that needed in the absence of such constraints. Even the most sophisticated sequential search algorithms require search times directly proportional toN, the number of contending phases. In this paper a class of binary sequences is demonstrated which require a search time on the order of only (log2 N)2, a substantial improvement when N is large. These sequences are shown to be essentially optimum under the conditions outlined.

In a recent paper Reed and Scholtz (ibid., vol. IT-12, pp. 388-395, July 1966) analyze the error probabilities encountered in using N-orthogonal codes over a white Gaussian channel. In this correspondence the authors derive simple upper and lower bounds on these probabilities and discuss some of the implications of these bounds.

A method is shown by which it is possible to establish the existence (or nonexistence) of the comma-free properties of any group code from a simple observation of its null-space. Using this technique it is then demonstrated that all(n, k){em cyclic} group error-correcting code dictionaries can be made comma-free (without adding further redundancy or altering their error-correcting properties) if, and only if,k \leq (n - 1)/2. Ifkis larger than(n - 1)/2it is still possible to detect bit gains or losses of up ton-k-1code bits.

Efficient use of block codes to communicate over a telemetry channel is dependent upon the knowledge of the instants, in time, that one code waveform, or code word, ends and the succeeding waveform begins. This word synchronization is the subject of concern in this paper. Two basically different single-channel synchronization techniques are discussed. The first relies upon the identification of periodically transmitted prefixes, which unavoidably increase the redundancy of the code without increasing its synchronous reliability. The second method eliminates the need for prefixes entirely and relies instead upon certain properties of the code itself. Both these techniques are investigated more closely as applied to binary orthogonal codes. Asymptotic results are obtained which show that the number of received words necessary to obtain synchronization with a fixed prefix length increases in proportion to the size of the code when the prefix method is used. In contrast, the number of words which need to be observed for the prefix-free codes remains constant independent of the code length. Finally, some results of the search for prefix-free codes of moderate size are reported.