Explicit merit factor formulae for Fekete and Turyn polynomials

Abstract

We give explicit formulas for the L 4 L_{4} norm (or equivalently for the merit factors) of various sequences of polynomials related to the Fekete polynomials \[ f q ( z ) := ∑ k = 1 q − 1 ( k q ) z k f_{q}(z) := \sum ^{q-1}_{k=1} \left (\frac {k}{q}\right ) z^{k} \] where ( ⋅ q ) \left (\frac {\cdot }{q}\right ) is the Legendre symbol. For example for q q an odd prime, \[ ‖ f q ‖ 4 4 := 5 q 2 3 − 3 q + 4 3 − 12 ( h ( − q ) ) 2 \|f_{q}\|_{4}^{4} : = \frac {5q^{2}}{3}-3q+ \frac {4}{3} - 12 (h(-q))^{2} \] where h ( − q ) h(-q) is the class number of Q ( − q ) \mathbb {Q}(\sqrt {-q}) . Similar explicit formulas are given for various polynomials including an example of Turyn’s that is constructed by cyclically permuting the first quarter of the coefficients of f q f_{q} . This is the sequence that has the largest known asymptotic merit factor. Explicitly, \[ R q ( z ) := ∑ k = 0 q − 1 ( k + [ q / 4 ] q ) z k R_{q}(z) := \sum ^{q-1}_{k=0} \left (\frac {k+[q/4] }{q}\right ) z^{k} \] where [ ⋅ ] [\cdot ] denotes the nearest integer, satisfies \[ ‖ R q ‖ 4 4 = 7 q 2 6 − q − 1 6 − γ q \|R_{q}\|_{4}^{4} = \frac {7q^{2}}{6}- {q} - \frac {1}{6} - \gamma _{q} \] where \[ γ q := { h ( − q ) ( h ( − q ) − 4 ) a m p ; if q ≡ 1 , 5 ( mod 8 ) , 12 ( h ( − q ) ) 2 a m p ; if q ≡ 3 ( mod 8 ) , 0 a m p ; if q ≡ 7 ( mod 8 ) . \gamma _{q}: = \begin {cases} h(-q) (h(-q)-4) & \text {if $q \equiv 1,5 \pmod 8$},\\ 12 (h(-q))^{2} & \text {if $q \equiv 3 \pmod 8$}, \\ 0 & \text {if $q \equiv 7 \pmod 8$}. \end {cases} \] Indeed we derive a closed form for the L 4 L_{4} norm of all shifted Fekete polynomials \[ f q t ( z ) := ∑ k = 0 q − 1 ( k + t q ) z k . f_{q}^{t}(z) := \sum ^{q-1}_{k=0} \left (\frac {k+t}{q}\right ) z^{k}. \] Namely ‖ f q t ‖ 4 4 a m p ; = 1 3 ( 5 q 2 + 3 q + 4 ) + 8 t 2 − 4 q t − 8 t a m p ; − 8 q 2 ( 1 − 1 2 ( − 1 q ) ) | ∑ n = 1 q − 1 n ( n + t q ) | 2 , \begin{align*} \| f_{q}^{t} \|_{4}^{4} &= \frac {1}{3}(5q^{2}+3q+4)+8t^{2}-4qt-8t &\quad -\frac {8}{q^{2}}\left ( 1-\frac {1}{2} \left (\frac {-1}{q}\right ) \right ) \left |{\displaystyle \sum _{n=1}^{q-1}n \left (\frac {n+t}{q}\right )} \right |^{2}, \end{align*} and ‖ f q q − t + 1 ‖ 4 4 = ‖ f q t ‖ 4 4 \| f_{q}^{q-t+1} \|_{4}^{4}= \| f_{q}^{t} \|_{4}^{4} if 1 ≤ t ≤ ( q + 1 ) / 2 1 \le t \le (q+1)/2 .