1 c○1997 American Mathematical Society

k=0 n
p(z ) ± z n+1 − z 2n+2 p(−1/z )
ak z k ,
with each ak ∈ {+1, −1} (thus a Littlewood polynomial), is a polynomial where each a cyclic autocorrelation coefficient satisfies |cj | ≤ K, j = 1, 2, . . . , n.
= (n + 1)2 + n
,
while if p(z ) is a Barker polynomial of odd degree n then p
4
= (n + 1)2 + n + 1
1/4
.
If p(z ) is a Barker polynomial of even degree n then p
s s
t−r ( p t ) t−r . ≤( p r r) t t−s s−r
So with r := 1, s := 2 and t := 4 we deduce, with the above, the following. p
2
≤ p ≥
1/3 1
p
2/3 4
and if p(z ) is a Barker polynomial of even degree n then p
4 4
j = 1, 2, . . . , n.
n − k odd n − k even.
n
=
k=−n
c2 k
we have that if p(z ) is a Barker polynomial of even degree n then p
4
= (n + 1)2 + n
1/4
,
while if p(z ) is a Barker polynomial of odd degree n then p
2 1
(n + 1)3 (n + 1)2 + n
and if p(z ) is a Barker polynomial of odd degree n then p
2 1
≥
(n + 1)3 . (n + 1)2 + n + 1
For any Littlewood polynomial of degree n 1 pn 4 4 −1= 2 (n + 1) 2π
MATHEMATICS OF COMPUTATION Volume 00, Number 0, Pages 000–000 S 0025-5718(XX)0000-0
K-BARKER POLYNOMIALS AND SEQUENCES
P. B. BORWEIN
Abstract. A Barker sequence is a binary sequence with all non–trivial acyclic autocorrelation coefficients of size at most 1. It is widely believed (and confirmed up to size 1022 ) that none exist of length greater than 13. A Barker polynomial is a polynomial whose coefficients form a Barker sequence. This problem has been around for at least 40 years. A K-Barker sequence is a binary sequence with all non–trivial acyclic autocorrelation coefficients of size at most K. It is widely believed that for eack K there are only finitely many such sequences (though this appears very hard and has been open for decades even for k:=1). A K-Barker polynomial is a polynomial whose coefficients form a Barker sequence.
4
P. B. BORWEIN
Proof. A Barker polynomial
n
p(z ) :=
k=0
ak z k ,
with each ak ∈ {+1, −1} (thus a Littlewood polynomial), is a polynomial where each acyclic autocorrelation coefficient satisfies |cj | ≤ 1, Thus, c0 = n + 1, and by parity ck = 0, and Since |ck | = 1, p(z )
K-BARKER POLYNOMIALS
3
are both skewsymmetric. While, if n is even and p is of degree n, then are both skewsymmetric. All skewsymmetric Littlewood polynomials are of the above form. 2. Basic Theorems on Barker Polynomials A K-Barker polynomial p(z ) :=
A real polynomial is skewsymmetric if p(z ) = ±z d p(−1/z ), where d is the degree of p. If n is odd and p is of degree n, then p(z ) ± z n+1 + z 2n+2 p(−1/z )
4
= (n + 1)2 + n + 1
1/4
.
Thus, when a Barker polynomial of degree n exists, it minimizes the L4 norm (and maximizes the so-called merit factor) of polynomials from the class L n . Note that the sequence of coefficients of a Barker polynomial of degree n forms a Barker sequence of length n +1. We prefer to deal with polynomials and care must be taken in the conversion from length to degree when going from sequences to polynomials (the count is out by 1). Also, if 0 < r < s < t then, the norms satisfy p
2π 0
2 log(|pn (eiθ )|) − log(n + 1)
2
dθ ≤
c−2 (n + 1)
2π √ 1 d log(|pn (eiθ )|) dθ = log n + 1 + √ 2π 0 n+1 where |d| ≤ c−1 /2 ≤ 3/2. And the Mahler measure, M of a sequence of Barker polynomials satisfies √ M (pn ) = (1 + n ) n + 1 where √ 0 ≤ n ≤ c2 / n + 1 and c2 ≤ 2.
2π 1 α dθ p eiθ . 2π 0 and for a polynomial p(z ) := an z n + · · · + a1 z + a0 , the L2 norm on D is also given by p 2 = |an |2 + · · · + |a1 |2 + |a0 |2 . 1/α
p
α
:=
In the limiting case
α→0
lim p
α
= exp
1 2π
2π
log p eiθ
0
dθ
=: p 0 .
This is called the Mahler measure and is denoted by M (p). For a polynomial Jensen’s theorem gives pn (z ) := a(z − α1 )(z − α2 ) · · · (z − αn ) M (pn ) = |a| Let Ln :=
c 1997 American Mathematical Society
பைடு நூலகம்
1
2
P. B. BORWEIN
1. Introduction For any polynomial with real coefficients
n
p(z ) :=
k=0
ak z k ,
the k th acyclic autocorrelation coefficient is defined, for −n ≤ k ≤ n, by
n−k j =0
(1.1) So