Rudin-Shapiro-Like Polynomials with Maximum Asymptotic Merit Factor.

Borwein and Mossinghoff investigated the Rudin-Shapiro-like 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 showed 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 Rudin-Shapiro-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 also $3$ 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.