On a class of polynomials related to Barker sequences

Abstract:

For an odd integer $n > 0$, we introduce the class $\mathcal {L}P_n$ of Laurent polynomials \[ P(z) = (n+1) + \sum _{\substack {k = 1 \\ k \text { odd}}}^{n}c_k (z^k+z^{-k}), \] with 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 $P \in \mathcal {L}P_n$ have large Mahler measures, namely, $M(P) > (n+1)/2$. We conjecture that minimal Mahler measures in the class $\mathcal {L}P_n$ are attained by the polynomials $R_n(z)$ and $R_n(-z)$, where \[ R_n(z) = (n+1) + \sum _{\substack {k = -n \\ k \text { odd}}}^{n} z^k \] is a polynomial with all the coefficients $c_k=1$. We prove that \[ M(R_n) > n - \frac {2}{\pi } \log {n} + O(1). \] The results of experimental computations on polynomials in the class $\mathcal {L}P_n$ suggest two conjectures which could shed light on the long-standing Barker problem.