On the Correlation Distribution of Kerdock Sequences - Sequences and Their Applications-SETA 2008 - page 127

1. for τ =2 τ 1 ,

v i ( t + τ )= ϕ ( a i ( t + τ 1 )) ,

i< 2 n− 1
ϕ ( a i ( t + τ 1 )+3) , 2 n− 1

0 ≤

i< 2 n ;

≤

2. for τ =2 τ 1 +1 ,
v i ( t + τ )= ϕ ( a i +2 n − 1 ( t + τ 1 +2 n− 1 )) ,

i< 2 n− 1
ϕ ( a i− 2 n − 1 ( t + τ 1 +2 n− 1 )+1) , 2 n− 1

0 ≤

i< 2 n .
Proof : According to Lemma 5, Remark 1, and the definition of the modified
Graymap,thecaseof τ =2 τ 1 is straightforward. Hereafter we only prove the
result for τ =2 τ 1 +1.
When 0

≤

i< 2 n− 1 , a i +2 n − 1 ( t + τ 1 +2 n− 1 )= Tr 1 ((1 + 2 η i +2 n − 1 ) β t + τ 1 +2 n − 1 ).
Applying Lemma 5, we have
ϕ ( a i +2 n − 1 ( t + τ 1 +2 n− 1 ))
= tr 1 ( ζ i +2 n − 1 α t 1 + τ 1 +2 n − 1 )+ p ( α t 1 + τ 1 +2 n − 1 ) ,t =2 t 1
tr 1 ((1 + ζ i +2 n − 1 ) α t 1 + τ 1 +2 n )+ p ( α t 1 + τ 1 +2 n ) ,t =2 t 1 +1
= tr 1 ((1 + ζ i ) α t 1 + τ 1 +2 n − 1 )+ p ( α t 1 + τ 1 +2 n − 1 ) ,t =2 t 1
tr 1 ( ζ i α t 1 + τ 1 +1 )+ p ( α t 1 + τ 1 +1 ) ,

≤

t =2 t 1 +1

= v i ( t + τ ) ,

i< 2 n− 1 .

wherewemakeuseofthefactthat ζ i +2 n − 1 =1+ ζ i ,0 ≤

i< 2 n ,then
a i− 2 n − 1 ( t + τ 1 +2 n− 1 )+1= Tr 1 ((1 + 2 η i− 2 n − 1 ) β t + τ 1 +2 n − 1 )+1 .
Applying Lemma 5 and (4) to a i− 2 n − 1 +1,wenowhave

When 2 n− 1

≤

ϕ ( a i− 2 n − 1 ( t + τ 1 +2 n− 1 )+1)
= tr 1 ( ζ i− 2 n − 1 α t 1 + τ 1 +2 n − 1 )+ p ( α t 1 + τ 1 +2 n − 1 )+ tr 1 ( α t 1 + τ 1 +2 n − 1 ) ,t =2 t 1

tr 1 ( ζ i− 2 n − 1 α t 1 + τ 1 +2 n )+ p ( α t 1 + τ 1 +2 n )+1 , t =2 t 1 +1

= tr 1 ((1 + ζ i− 2 n − 1 ) α t 1 + τ 1 +2 n − 1 )+ p ( α t 1 + τ 1 +2 n − 1 ) ,t =2 t 1
tr 1 ( ζ i− 2 n − 1 α t 1 + τ 1 +1 )+ p ( α t 1 + τ 1 +1 )+1 , t =2 t 1 +1

= v i ( t + τ ) .

Proof of Theorem 6 : We investigate the following 7 cases for computing the
correlation function:

i = j< 2 n and τ =0.
This is trivial case, R i,i (0) = 2(2 n

Case 1 .0 ≤

− 1).

= j< 2 n− 1 or 2 n− 1 = j< 2 n )and τ =0.

Case 2 .(0 ≤ i ≤ i

In this case,

R i,j ( τ )=2(
x∈ F 2 n

1) tr 1 ( ζ i + ζ j ) x ( − − 1)

= − 2 ,