Previous results on skewsymmetric sequences with low autocorrelation obtained by making exhaustive searches for sequences with optimal merit factor are extended up to length 69. A sieve that required both the symmetric and antisymmetric components of a skewsymmetric sequence to have high merit factor to make limited searches up to length 117 was used, and many sequences with merit factor greater than 9 were found.>

Turyn has shown by numerical computation that the merit factor of long Legendre sequences offset by a quarter of their length has in all likelihood the asymptotic value6. The ergodicity postulate is used in this correspondence to calculate that the merit factorFof a Legendre sequence offset by a fractionfof its length has an asymptotic value given by1/F=(2/3)-4|f|+8f^{2}, |f|\leq 1/2, which givesF=6for|f|=1/4.

The asymptotic "merit factor," i.e, the ratio of central to sidelobe energy of extremely long, optimally Iow autocorrelation sequences, formerly calculated as2e^{2}=14.778 \cdotswith the use of an ergodicity hypothesis and a convenient, but faulty, approximation, is recalculated without that approximation and is established at12.32 \cdots.

A short list is given of two-dimensional line structures which are topologically identifiable one from the other.

The value2e^{2}is calculated by approximations as a conjectured asymptotic limit for the best central to sidelobe energy ratio of very long low autocorrelation binary sequences. The same bound is calculated by approximation for skewsymmetric sequences, hence it is concluded that the requirement of skewsymmetry constitutes a powerful and effective sieve for the search for near-optimal sequences. A second sieve based on the use of complementary sequences is discussed and shown to be quite effective for sequence lengths up to 31, and its investigation for longer lengths by means of computers is suggested. A third and a fourth sieve, based on the selection of restricted classes of complementary sequences, are also discussed.

Skew-symmetric sequences of(2n + 1)terms,a_0,a_1,\cdots,a_{2n}, are described for which the "merit factor" \begin{equation} F_h = \frac{\biggl[\sum_{i=0}^{2n} \mid a_i \mid \biggr] ^2}{ 2 \sum_{k=1}^{2n} \biggl[ \sum_{i=0}^{2n-k} \text{sign} (a_i) \cdot a_{i+k} \biggl] ^2} \end{equation} is unusually high.

A subclass of Huffman sequences is examined and evaluated in terms of an analytically defined figure of merit which would be optimal for sequences of pulses of equal amplitude.

It is shown that when discrete experimental data are smoothed by fitting 2m + 1 consecutive data to a polynomial of 2nth degree, with n≪m, and when n and m are increased indefinitely, the smoothing obtained is equivalent to passing the original data through an ideal low-pass filter.

An algorithm is presented for the generation of finite binary sequences having an odd number of terms of value + 1 or -1, whose autocorrelation values are zero for odd shifts.

The concept of the two-dimensional (2-D) parallel computer with square module arrays was first introduced by Unger. It is the purpose of this paper to discuss the relative merits of square and hexagonal module arrays, to propose an operational symbolism for the various basic hexagonal modular transformations which may be performed by these comupters, to illustrate some logical circuit implementation, and to describe a few elementary applications.

A set of complementary series is defined as a pair of equally long, finite sequences of two kinds of elements which have the property that the number of pairs of like elements with any one given separation in one series is equal to the number of pairs of unlike elements with the same given separation in the other series. (For instance the two series, 1001010001 and 1000000110 have, respectively, three pairs of like and three pairs of unlike adjacent elements, four pairs of like and four pairs of unlike alternate elements, and so forth for all possible separations.) These series, which were originally conceived in connection with the optical problem of multislit spectrometry, also have possible applications in communication engineering, for when the two kinds of elements of these series are taken to be +1 and -1, it follows immediately from their definition that the sum of their two respective autocorrelation series is zero everywhere, except for the center term. Several propositions relative to these series, to their permissible number of elements, and to their synthesis are demonstrated.

The method of construction of lossless symbol coding matrices for one-error correction is illustrated for the case when the prime symbol order is three, and the application of this matrix to the penny-weighing problem is described. This method is then extended to those cases in which the symbol order is2^2, 2^3, 2^4, 2^6, 3^2, 3^3, 3^4, 3^5, 5^2, 5^3, 5^4, 7^2, 7^3, andp^2, wherepis any higher prime. This extension is based on the concept of the master iterating matrix. These matrices are given for the first thirteen cases cited, and their existence is demonstrated forp^2. This paper concludes with a short description of Zaremba's condition, and its application to various problems, and more particularly to the hypothetical one-error correcting close-packed code with the symbol order 6.

The counters without short-time internal memory, conceived by Bigelow, disclosed by Ware, and extended by Brown, are discussed from the standpoint of their respective logic. It is shown that the (self-instructed) bidirectional1 counter of Brown has a more rigorous logic than the unidirectional counter of Ware; the operation of Brown's bidirectional counter being subjected to the only restriction that its speed be compatible with the operating speed of its individual toggles, whereas the operation of Ware's counter is predicated upon the existence of unspecified buffering states in the input of each stage, to prevent run-away conditions. These buffering states, which occur naturally in Brown's bidirectional counter, can be provided explicitly in unidirectional counters by replacing the two transfer circuits of Ware's counter stage, which are controlled only by one toggle of the preceding stage, by two of the four transfer circuits of Brown's bidirectional counter stage, all four of which are controlled by both toggles of the preceding stage. This paper introduces the viewpoint that a bidirectional counter of Brown's type is a counter in which the state of one toggle of each stage determines which toggle of the next stage is master, while the state of the other toggle of each stage determines whether the slave of the next stage shall be like or unlike the master. This viewpoint permits a succinct discussion of the several possible interstage connections, and of the several counting codes obtained for each connection.