A NOTE ON BARKER POLYNOMIALS Academic Article uri icon

abstract

  • We call the polynomial Pn-1(x) = ∑j = 1na-jzn-j a Barker polynomial of degree n-1 if each aj{-1, 1} and Tn(z) = Pn-1(z)P n-1(1/z) = n + ∑k=1n-1ck (z k + z-k, vert ck ≤ 1. Properties of Barker polynomials were studied by Turyn and Storer thoroughly in the early sixties, and by Saffari in the late eighties. In the last few years P. Borwein and his collaborators revived interest in the study of Barker polynomials (Barker codes, Barker sequences). In this paper we give a new proof of the fact that there is no Barker polynomial of even degree greater than 12, and hence Barker sequences of odd length greater than 13 do not exist. This is intimately tied to irreducibility questions and proved as a consequence of the following new result. Theorem. If n 2m + 1 > 13 and Q4m(z) = (2m+1)z 2m + ∑ j=12mbj (z 2m-j} + z2m+j}, where each bj -1, 0, 1} for even values of j, each bj is an integer divisible by 4 for odd values of j, then there is no polynomial P2m in L2m such that Q4m(z) = P2m(z)P2m* (z)$, where P2m (z):= z2mP2m(1/z), and L2m denotes the collection of all polynomials of degree 2m with each of their coefficients in {-1, 1}. A clever usage of Newton's identities plays a central role in our elegant proof. © 2013 World Scientific Publishing Company.

author list (cited authors)

  • BORWEIN, P., & ERDÉLYI, T.

citation count

  • 1

publication date

  • April 2013