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Extremal Mahler measures and _{} norms of polynomials related to Barker sequences

In the present paper, we study the class LPn which consists of Laurent polynomials P (z) = (n+ 1) + n ∑ k=1 k – odd ck(z k + z−k), with all coefficients ck equal to −1 or 1. Such polynomials arise in the study of Barker sequences of even length — binary sequences with minimal possible autocorrelations. By using an elementary (but not trivial) analytic argument, we prove that polynomials Rn(z) with all coefficients ck = 1 have minimal Mahler measures in the class LPn. In conjunction with an estimate M(Rn) > n − 2/π logn + O(1) proved in an earlier paper, we deduce that polynomials whose coefficients form a Barker sequence would possess unlikely large Mahler measures. A generalization of this result to Ls norms is also given.