Solomon W. Golomb

Backtrack Programming

A widely used method of efftcient search is examined in detail. This examination provides the opportunity to formulate its scope and methods in their full generality. In addL tion to a general exposition of the basic process, some important refinemertts are indicated. Examples are given which illustrate the salient features of this searching process.

Generalized Barker sequences

A generalized Barker sequence is a finite sequence \{a_{r}\} of complex numbers having absolute value 1 , and possessing a correlation function C(\tau) satisfying the constraint |C(\tau)| \leq 1, \tau \neq 0 . Classes of transformations leaving |C(\tau)| invariant are exhibited. Constructions for generalized Barker sequences of various lengths and alphabet sizes are given. Sextic Barker sequences are investigated and examples are given for all lengths through thirteen. No theoretical limit to the length of sextic sequences has been found.

Algebraic constructions for costas arrays

Thus, if ai,,j, = ail,il= ajl,j, = aiJqj4 = 1 in the matrix, we must not have ii2 1 il 3jjL;

Two-dimensional synchronization patterns for minimum ambiguity

) = (i4 4 ,.i, -jd, nor may we have (iz i,,j, -j,)= . ,. . 13 ‘2, 3 Such matri& have been called either Costas Arrays or constellations in Ref. [2], which explores constructions as well as applications for these patterns. It is convenient to represent these arrays on an n X n grid, using dots for the l’s and blanks for the O’s of the matrix. Three examples of 6 X 6 Costas arrays are shown in Fig. 1. Previous constructions [2] for Costas Arrays, for special values of II, have been discovered by L. R. Welch and by A. Lempel. This paper contains the

Binary pseudorandom sequences of period 2/sup n/-1 with ideal autocorrelation

A number of closely related combinatorial problems corresponding to specific assumptions about the type of time-frequency sequence which may be appropriate in a particular application, are formulated in terms of square or rectangular arrays of dots with appropriate constraints on the two-dimensional correlation function. The current state of knowledge concerning each of these problems is summarized. It is hoped that more general constructions may be found, leading to larger families of solutions, as well as better computational algorithms for finding individual solutions which may lie outside of the general families.

Codes with bounded synchronization delay

In this correspondence, we present five new classes of binary sequences of period 2/sup n/-1 with ideal autocorrelation. These sequences, which correspond to new cyclic Hadamard difference sets, were found by extensive computer search. Conjectures on the general construction of these sequences are formulated.

A new construction of 64-QAM golay complementary sequences

In this paper we study “bounded delay codes,≓ which have the property that every message can be uniquely decoded by examining a segment of bounded length, from any starting point. This class is shown to attain the upper bound on codebook size previously encountered (but not always attained) for the familiar subclass of “comma-free codes.≓ The problem of determining the smallest message length s0 which guarantees unique decipherability for such codes is discussed. The bounded delay codes are classified according to the value of s0. An extension is made to the case of variable length codes, in which the upper bound formula of the uniform word-length case is replaced by a system of inequalities.

A new recursive construction for optical orthogonal codes

In this correspondence, we present a new construction for 64-QAM Golay sequences of length n=2/sup m/ for integer m. The peak envelope power (PEP) of 64-QAM Golay sequences is shown to be bounded by 4.66n. The new construction of 64-QAM Golay sequences of length n=2/sup m/ is based on our earlier construction of new offsets of 16-QAM Golay sequences which are also presented here. The total number of offsets of 64-QAM Golay sequences is 496 for m=2,808 for m=3 and 976 for m=4, obtained by computer search. We also computed the PEP distribution for 64-QAM Golay sequences for m=2, m=3, and m=4.

Transform domain analysis of DES

We present a new recursive construction for (n,/spl omega/,/spl lambda//sub a/,/spl lambda//sub c/) optical orthogonal codes. For the case of /spl lambda//sub a/ = /spl lambda//sub c/ = /spl lambda/, this recursive construction enlarges the original family with /spl lambda/ unchanged, and produces a new family of asymptotically optimal codes, if the original family is asymptotically optimal. We call a code asymptotically optimal, following the definition of O. Moreno et al. (see ibid., vol.41, p.448-55, 1995), if, as n, the length of code, goes to infinity, the ratio of the number of codewords to the corresponding Johnson bound approaches unity.

Tiling with sets of polyominoes

The Data Encryption Standard (DES) can be regarded as a nonlinear feedback shift register (NLFSR) with input. From this point of view, the tools for pseudo-random sequence analysis are applied to the S-boxes in DES. The properties of the S-boxes of DES under the Fourier transform, Hadamard transform, extended Hadamard transform, and the Avalanche transform are investigated. Two important results about the S-boxes of DES are found. The first result is that nearly two-thirds of the total 32 functions from GF (2/sup 6/) to GF(2) which are associated with the eight S-boxes of DES have the maximal linear span G3, and the other one-third have linear span greater than or equal to 57. The second result is that for all S-boxes, the distances of the S-boxes approximated by monomial functions has the same distribution as for the S-boxes approximated by linear functions. Some new criteria for the design of permutation functions for use in block cipher algorithms are discussed.