On a class of polynomials related to Barker sequences

Abstract

Abstract. For an odd integer n > 0, we introduce the class LP n of Laurent polynomialswith all coefficients c k 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 ∈ LP n have large Mahler measures, namely, M (P ) > (n + 1)/2. We conjecture that minimal Mahler measures in the class LP n are attained by the polynomials R n (z) and R n (−z), whereis a polynomial with all the coefficients c k = 1. We prove thatThe results of experimental computations on polynomials in the class LP n suggest two conjectures which could shed light on the long-standing Barker problem.

Extremal Mahler measures and $L_s$ norms of polynomials related to Barker sequences

“…Such auxiliary polynomials arise in a natural way in connection with Barker polynomials of odd degree. Indeed, if p(z) is a Barker polynomial of degree n, then the product p(z)p(1/z) is a Laurent polynomial which belongs to the class LP n ; see [4]. Mahler measure and L s norms of polynomials P ∈ LP n are defined by the respective integral formulas.…”

“…One should note that, while Barker polynomials of large degree are only hypothetical, the class LP n exists and has some very peculiar extremal properties. As in [4], the notation R n (z) shall be reserved for the polynomials which have all coefficients c k = 1:…”

“…After some computer experimentation, we have conjectured in [4] that the polynomials R n (z) and R n (−z) have smallest possible Mahler measures in LP n . Now we are able to prove this result.…”