Rudin-Shapiro-Like Polynomials with Maximum Asymptotic Merit Factor

Borwein and Mossinghoff investigated the Rudin-Shapirolike polynomials, which are infinite families of Littlewood polynomials, that is, polynomials whose coefficients are all in {−1, 1}. Each family of Rudin-Shapiro-like polynomials is obtained from a starting polynomial (which we call the seed) by a recursive construction. These polynomials can be regarded as binary sequences. Borwein and Mossinghoff show that the asymptotic autocorrelation merit factor for any such family is at most 3, and found the seeds of length 40 or less that produce the maximum asymptotic merit factor of 3. The definition of RudinShapiro-like polynomials was generalized by Katz, Lee, and Trunov to include polynomials with arbitrary complex coefficients, with the sole condition that the seed polynomial must have a nonzero constant coefficient. They proved that the maximum asymptotic merit factor is 3 for this larger class. Here we show that a family of such Rudin-Shapiro-like polynomials achieves asymptotic merit factor 3 if and only if the seed is the interleaving of a pair of Golay complementary sequences. In this paper, we assume that all the definitions from the introduction and from Section 2 (up to Lemma 2.2) of the paper of Katz, Lee, and Trunov [3] are in force, so one should read that first. Recall that a sequence (f0, f1, . . . , fl−1) ∈ C l of length l is identified with the polynomial f(z) = f0 + f1z + · · ·+ fl−1z l−1 ∈ C[x]. Recall from the introduction of [3] that if f is a sequence, then ADF(f) is the autocorrelation demerit factor of f , which is the sum of the squared magnitudes of the autocorrelation values of f at all nonzero shifts divided by the squared length of f . The autocorrelation merit factor of f is the reciprocal of its demerit factor, and our goal is to obtain sequences with high merit factor, or equivalently, low demerit factor, which indicates low mean square magnitude autocorrelation. Recall from [3, Theorem 1.2] that if (f0, f1, . . .) is a sequence of RudinShapiro-like polynomials generated from the seed f0 ∈ C[z] with nonzero constant coefficient, then (1) lim n→∞ ADF(fn) = −1 + 2 3 · ‖f0‖ 4 4 + ‖f0f̃0‖ 2 2