A NOTE ON BARKER POLYNOMIALS

作者：Borwein Peter^*; Erdelyi Tamas

来源：International Journal of Number Theory, 2013, 9(3): 759-767.

DOI：10.1142/S179304211250159X

• Zeros of Littlewood polynomials
• Barker polynomials
• constrained coefficients
• factorization of polynomials

We call the polynomial Pn-1(x) = Sigma(n)(j=1) a(j)z(n-j) a Barker polynomial of degree n-1 if each a(j) is an element of {-1, 1} and
T-n(z) = Pn-1(z)Pn-1(1/z) = n + Sigma(n-1)(k=1) c(k)(z(k) + z(-k)), vertical bar ck vertical bar <= 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
Q(4m)(z) = (2m + 1)z(2m) + Sigma(2m)(j=1) b(j) (z(2m-j) + z(2m+j)),
where each b(j) is an element of {-1, 0, 1} for even values of j, each b(j) is an integer divisible by 4 for odd values of j, then there is no polynomial P-2m is an element of L-2m such that Q(4m) (z) = P-2m (z) P-2m* (z), where
P-2m*(z) := z(2m)P(2m)(1/z),
and L-2m 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.

• 出版日期2013-5