Ron Ferguson

A complete description of golay pairs for lengths up to 100

In his 1961 paper, Marcel Golay showed how the search for pairs of binary sequences of length n with complementary autocorrelation is at worst a 2 -6 problem. Andres, in his 1977 masters thesis, developed an algorithm which reduced this to a 2 n/2-1 search and investigated lengths up to 58 for existence of pairs. In this paper, we describe refinements to this algorithm, enabling a 2 n/2-5 search at length 82. We find no new pairs at the outstanding lengths 74 and 82. In extending the theory of composition, we are able to obtain a closed formula for the number of pairs of length 2 k n generated by a primitive pair of length n. Combining this with the results of searches at all allowable lengths up to 100, we identify five primitive pairs. All others pairs of lengths less than 100 may be derived using the methods outlined.

Polyphase sequences with low autocorrelation

Low autocorrelation for sequences is usually described in terms of low base energy, i.e., the sum of the sidelobe energies, or the maximum modulus of its autocorrelations, a Barker sequence occurring when this value is /spl les/ 1. We describe first an algorithm applying stochastic methods and calculus to the problem of finding polyphase sequences that are good local minima for the base energy. Starting from these, a second algorithm uses calculus to locate sequences that are local minima for the maximum modulus on autocorrelations. In our tabulation of smallest base energies found at various lengths, statistical evidence suggests we have good candidates for global minima or ground states up to length 45. We extend the list of known polyphase Barker sequences to length 63.

Sign changes in sums of the Liouville function

The Liouville function A(n) is the completely multiplicative function whose value is -1 at each prime. We develop some algorithms for computing the sum T(n) = Σ n k=1 λ(k)/k, and use these methods to determine the smallest positive integer n where T(n) < 0. This answers a question originating in some work of Turan, who linked the behavior of T(n) to questions about the Riemann zeta function. We also study the problem of evaluating Polyas sum L(n) = Σ n k=1 λ(k), and we determine some new local extrema for this function, including some new positive values.

Zeros of Partial Sums of the Riemann Zeta Function

The semiperiodic behavior of the zeta function ζ(s) and its partial sums ζN(s) as a function of the imaginary coordinate has been long established. In fact, the zeros of a ζN(s), when reduced into imaginary periods derived from primes less than or equal to N, establish regular patterns. We show that these zeros can be embedded as a dense set in the period of a surface in ℝk+1, where k is the number of primes in the expansion. This enables us, for example, to establish the lower bound for the real parts of zeros of ζN(s) for prime N and justifies the use of methods of calculus to find expressions for the bounding curves for sets of reduced zeros in ℂ.

Number Theory and Polynomials: The merit factor problem

The merit factor problem is of considerable practical interest to communications engineers and theoretical interest to number theorists. For binary sequences, although it is generally believed that the merit factor is bounded, it still has not been completely established that the number of even length Barker sequences, each with merit factor N , is bounded. In this paper, we present an overview of the problem and results of quite extensive searches we have conducted in lengths up to slightly beyond 200.

Norms of cyclotomic Littlewood polynomials

The main result of this paper gives an explicit computation of the L4 norm of any cyclotomic polynomial of the form

On Littlewood Polynomials with Prescribed Number of Zeros Inside the Unit Disk

We investigate the numbers of complex zeros of Littlewood polynomials p(z) (polynomials with coefficientsf 1; 1g) inside or on the unit circlejzj = 1, denoted by N(p) and U (p), respectively. Two types of Littlewood polynomials are considered: Littlewood polynomials with one sign change in the sequence of coefficients and Littlewood polynomials with one negative coefficient. We obtain explicit formulas for N(p), U (p) for polynomials p(z) of these types. We show that if n + 1 is a prime number, then for each integer k, 0 6 k 6 n 1, there exists a Littlewood polynomial p(z) of degree n with N(p) = k and U (p) = 0. Furthermore, we describe some cases where the ratios N(p)=n and U (p)=n have limits as n!1 and find the corresponding limit values.

Permutations with low discrepancy consecutive k -sums

Consider the permutation π = (π1, . . . .,πn) of 1, 2,..., n as being placed on a circle with indices taken modulo n. For given k ≤ n there are n sums of k consecutive entries. We say the maximum difference of any consecutive k-sum from the average k-sum is the discrepancy of the permutation. We seek a permutation of minimum discrepancy. We find that in general the discrepancy is small, never more than k + 6, independent of n. For g = gcd(n, k) > 1, we show that the discrepancy is ≤7/2. For g = 1 it is more complicated. Our constructions show that the discrepancy never exceeds k/2 by more than 9 for large n, while it is at least k/2 for infinitely many n.We also give an analysis for the easier case of linear permutations where we view the permutation as written on a line. The analogous discrepancy is at most 2 for all n,k.