On a class of polynomials related to Barker sequences

For an odd integer n > 0, we introduce the class LPn 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, i.e., integer sequences having minimal possible autocorrelations. We prove that polynomials P ∈ LPn have large Mahler measures, namely, M(P ) > (n + 1)/2. We conjecture that minimal Mahler measures in the class LPn are attained by the polynomials Rn(z) and Rn(−z), where Rn(z) = (n+ 1) + n ∑ k=−n k odd z is a polynomial with all the coefficients ck = 1. We prove that M(Rn) > n− 2 π logn+O(1). The results of experimental computations on polynomials in the class LPn suggest two conjectures which could shed light on the long-standing Barker problem.