Daniel J. Katz

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Let p be a prime. We prove various analogues and generalizations of McEliece’s theorem on the p-divisibility of weights of words in cyclic codes over a finite field of characteristic p. Here we consider Abelian codes over various Galois rings. We present four new theorems on p-adic valuations of weights. For simplicity of presentation here, we assume that(More)
The identification of binary sequences with large merit factor (small mean-squared aperiodic autocorrelation) is an old problem of complex analysis and combinatorial optimization, with practical importance in digital communications engineering and condensed matter physics. We establish the asymptotic merit factor of several families of binary sequences and(More)
while ||f ||∞ is the supremum of |f(z)| on the unit circle. The norms L1, L2, L4, and L are of particular interest in analysis. Littlewood was interested in how closely the ratio ||f ||∞/||f ||2 can approach 1 as deg(f) → ∞ for f in the set of polynomials now named after him [26]. Note that if f is a Littlewood polynomial, then ||f ||2 is deg(f)+1. In view(More)
It is shown that the pairs of maximal linear recursive sequences (m-sequences) typically have mean square aperiodic crosscorrelation on par with that of random sequences, but that if one takes a pair of m-sequences where one is the reverse of the other, and shifts them appropriately, one can get significantly lower mean square aperiodic crosscorrelation.(More)
We consider the Rudin-Shapiro-like polynomials, whose L norms on the complex unit circle were studied by Borwein and Mossinghoff. The polynomial f(z) = f0 + f1z + · · ·+ fdz d is identified with the sequence (f0, f1, . . . , fd) of its coefficients. From the L 4 norm of a polynomial, one can easily calculate the autocorrelation merit factor of its(More)
Pairs of binary sequences formed using linear combinations of multiplicative characters of finite fields are exhibited that, when compared with a random sequence pairs, simultaneously achieve significantly lower mean square autocorrelation values (for each sequence in the pair) and significantly lower mean square crosscorrelation values. If we define(More)
We consider Weil sums of binomials of the form WF,d(a) = ∑ x∈F ψ(x d − ax), where F is a finite field, ψ : F → C is the canonical additive character, gcd(d, |F|) = 1, and a ∈ F. If we fix F and d and examine the values of WF,d(a) as a runs through F , we always obtain at least three distinct values unless d is degenerate (a power of the characteristic of F(More)