Design of M-Ary Low Correlation Zone Sequence Sets by Interleaving - Sequences and Their Applications-SETA 2008 - page 317

When ( m, n )=(0 , 1), we have

⎧
⎨

s ( t − jd ))

+ s ( t +( i +1) d +1) s ( t + τ 1 +( j +1) d +1) , if τ 0 =0;

id ))( − s ( t + τ 1 −

C ( i, 0) , ( j, 1) ( τ )=

⎩ s ( t − id ) s ( t + τ 1 +( j +1) d +1)

+ s ( t +( i +1) d +1)( − s ( t + τ 1 +1 − jd )) , if τ 0 =1

= −

C s ( τ 1 + ad )+ C s ( τ 1 − ad ) , if τ 0 =0;

C s ( τ 1 + bd +1) − C s ( τ 1 − bd ) , if τ 0 =1 .

When ( m, n )=(1 , 0), we have

⎧
⎨

( − s ( t jd )

+ s ( t +( i +1) d +1) s ( t + τ 1 +( j +1) d +1) , if τ 0 =0;

− id )) s ( t + τ 1 −

C ( i, 1) , ( j, 0) ( τ )=

( − s ( t id )) s ( t + τ 1 +( j +1) d +1)

+ s ( t +( i +1) d +1) s ( t + τ 1 +1

−

⎩

− jd ) , if τ 0 =1

= −

C s ( τ 1 + ad )+ C s ( τ 1 − ad ) , if τ 0 =0;

− C s ( τ 1 + bd +1)+ C s ( τ 1 − bd ) , if τ 0 =1 .

T− 1
2

Case ii) d | : Similar to the Proof of Case i).

Remark : In Case i) of Lemma 1, it is easily checked that ad = − ad mod T if

T− 1
2

0 ≤ i, j ≤ f − 1and( i, j ) =(0 , 0). The condition d implies bd +1 = − bd

mod T for any 0 ≤ i, j ≤ f − 1. Hence, { s ( i,m ) ( t ) } and { s ( j,n ) ( t ) } are cyclically

distinct if ( i, m ) =( j, n ). Cyclic distinctness for Case ii) can also be checked

similarly. Therefore, the set IS in (2) contains 2 f cyclically distinct sequences

of period 2 T .

T − 1

Theorem 2. Let 1 ≤ d ≤ .Theset IS in (2) is a (2 T, 2 f, 2 d, 2 ) M -ary

2

LCZ sequence set, where

⎧
⎨

2 T− 1
4 d

for d

T − 1

2

f =

2 T− 3

4 d for d

⎩

T − 1

| .

2

T− 1
2

Proof. Consider the case that d . Without loss of generality, we can assume

that i ≤ j for C ( i,m ) , ( j,n ) ( τ ). By Lemma 1, we have

| C ( i,m ) , ( j,n ) ( τ ) |≤ 2

except for

( τ 0 ,τ 1 )=(0 ,ad ) , (0 ,T − ad ) , (1 ,bd ) , (1 ,T − bd − 1)

which are equivalent to

τ =2 ad, 2 T − 2 ad, 2 bd +1 , 2 T − 2 bd − 1 ,