Golay Complementary Sequences Over the QAM Constellation

Golay Complementary Sequences Over the

QAM Constellation

Wenping Ma1_{Chen Yang}2_{and Shaohui Sun}3

1 _{National Key Lab. of ISN, Xidian University , Xi’an 710071, P.R.China}

wp [email protected]

2 _{China Electronic Standardization Institute, Beijing 100007, P.R.China}

[email protected]

3_{Datang Mobile Communications Equipment Co.,Lid,Beijing 100083, P.R.China}

Abstract. In this paper,we present new constructions for M2 _-QAM

and 2M Q-P AM Golay complementary sequences of length 2n_for inte-ger n, whereM = 2m _{for integer m. New decision conditions are} pro-posed to judge whether the sequences with offset pairs propro-posed by Ying Li are Golay complementary, and with the new decision conditions, we prove the conjecture 1 and point out some drawbacks in conjecture 2 proposed by Ying Li. We describe a new offset pairs and construct new 64-QAM Golay sequences based on this new offset pairs. We also study the 128-QAM Golay complementary sequences, and propose a new de-cision condition to judge whether the sequences are 128-QAM Golay complementary.

Index Terms: Golay Complementary Sequences, Quadrature Ampli-tude Modulation(QAM), Orthogonal Frequency Division Multiplexing (OF DM), Quadrature Pulse Amplitude Modulation (Q-P AM).

1

Introduction

Complementary binary sequences were first introduced by Marcel Golay [1] to study problems in infrared multislit spectrometry. Nowadays,Golay sequences have many applications in communications, including peak power control for orthogonal frequency division multiplexing(OF DM) signals, channel estimation, and complementary code-code division multiple access(CC-CDM A).

Multicarrier communications including orthogonal frequency division multi-plexing (OF DM) has been receiving increasing attention [3]. However, a major drawback toOF DMapplications is the large peak to mean envelope power ratio (P M EP R). A large peak to mean power ratio (P M EP R) brings disadvantages such as an increased complexity of the analog-to-digital and digital-to-analog converters , a reduced efficiency of the RF power amplifier, and sometimes for certain applications, like ultra-wide-band communications, the peak transmit power is limited by regulations.

J.A.Davis and J.Jedwab[2] discovered an important relation between Golay se-quences and Reed-Muller codes, their method of generating binary and nonbi-nary Golay sequences is known as the GDJ construction. Davis and Jedwab made major progress in attacking the P M EP Rproblem by coding techniques; they proposed the coding scheme forOF DMtransmission for 2h_{-aryP SK} mod-ulation to reduce the P M EP R. In 2000, V.Tarokh and H.Jafarkhani[6] intro-duced a geometric approach to the offset selection problem forP SKmodulation. In 2000, K.G.Paterson and V.Tarokh[5] found the lower bound on the achiev-able rate of a code of a given length, the minimum Euclidean distance and the maximum peak-to-average power ratio (P AP R). 1n 2001, Cornelai Robing and V.Tarokh[7] made significant progress on the construction of complementary se-quences for both amplitude and phase modulation. 1n 2003, Chan Vee Chong,R. Venkataramani, and V.Tarokh[8] explicitly constructed 16-QAM complemen-tary sequences using cosets of second order Reed-Muller codes by setting up the two coordinates. 1n 2003,B. Tarokh and H.R.Sadjadpour[9] derived the upper bound for the P EP for square M-QAM Golay sequences under the assump-tion that all the symbols are equiprobable. 1n 2006, Heekwan Lee and Solomon W.Golomb extended the constructions of 16-QAMGolay sequences to 64-QAM

constellation using the offsets discovered in [10]. 1n 2008, M. Anand and P. Vijay Kumar[11] studied the low correlation sequences over the QAM constellation. In 2008, Ying Li [15, 16] gave some corrections for the sequence pairing descrip-tions of 16-QAM and 64-QAM, he proposed two conjectures to describe the new offset pairs and enumerate all known first order offset pairs.

In this paper, we extend the constructions of 16-QAM Golay sequences and 64-QAM Golay sequences to M2_{-QAM constellations and 2M Q-P AM}

con-stellations using the offsets discovered in [7, 10]. We study the 64-QAM Golay sequences using the offsets discovered in [16], propose two sufficient conditions to judge whether the sequences are Golay complementary sequences,and as a result, we prove the conjectures in [16]. A new Golay complementary sequence constructed by new offset pairs is described in this paper. We also give a suffi-cient condition to judge whether a sequence over 128-QAM using the offsets in [16] is Golay complementary.

2

The Golay Complementary Sequences over

M

_-

_QAM

constellation

TheM2_{-QAM constellation is the set}

{a+bj| −M+ 1≤a, b≤M−1, a, b odd}.

WhereM = 2m_{,this constellation can alternately be described as}

{p2j( m_X−1

k=0

2kjak_)|a

Where by√2jwe mean the element 1 +j. We now present new constructions of

M2_{-QAM Golay sequences that is similar to that of 16-QAM Golay sequences}

and 64-QAM Golay sequences as derived in [7, 10].

Theorem 1. Let A(x) = 2P_kn₌₁−1xπ(k)xπ(k+1)+

P_n

k=1ckxk+c,

a(k)_{(x) =A(x) +s}(k)_(x),

b(k)_{(x) =A(x) +s}(k)_{(x) +µ(x).}

Whereck∈Z4,k= 0,1,· · · , m−1,c∈Z4,πis a permutation from{1,2,· · · , n}

to{1,2,· · ·, n} ,s(k)_{(x)andµ(x)satify the following cases.}

Case 1:s(k)_=d(k) 0 +d

(k) 1 xπ(1),

µ(x) = 2xπ(n).

Case 2:s(k)_{(x) =d}(k)

0 +d(1k)xπ(n),

µ(x) = 2xπ(1).

Case 3:s(k)_=d(k) 0 +d

(k)

1 xπ(ω)+d(2k)xπ(ω+1),1≤ω≤n−1,

2d(₀k)+d₁(k)+d(₂k)= 0,

µ(x) = 2xπ(1) or2xπ(n).

Whered(₀k), d(₁k), d₂(k)∈Z4 ,k= 0,1,· · ·, m−1.

Then the M2_{-QAMsequences}

c(x) =√2j(P_km₌₀−12k_ja(k)_(x)

),d(x) =√2j(Pm_k=0−12k_jb(k)_(x)

) ,M = 2m

are Golay complementary sequences.

Proof:

Case 1: The aperiodic autocorrelation function of sequencesc(x) , where x= 0,1,· · · ,2n_{−1,at delay shiftτ is}

Cc(τ) =

P2n_−1−τ i=0 cic∗i+τ

= 2P2_i=0n−1−τ[Pm_k=0−12k_(j)a(k)_(i)

][Pm_k=0−12k_(j)−a(k)_(i+τ)

]

= 2{Pm_k=0−122k_C

a(k)(τ) +

P

k,f,k6=f2k+fCa(k)_,a(f)(τ)}

Cd(τ) =

P2n_−1−τ i=0 did∗i+τ

= 2{Pm_k=0−122k_C

b(k)(τ) +

P

k,f,k6=f2k+fCb(k)_,b(f)(τ)}

Forτ >0,

Cc(τ) +Cd(τ) = 2{ P_m−1

k=0 22k[Ca(k)(τ) +C_b(k)(τ)]

+P_k,f,k6=f2k+f_[C

a(k)_,a(f)(τ) +C_b(k)_,b(f)(τ)]}

= 2P_k,f,k6=f2k+f_[C

a(k)_,a(f)(τ) +C_b(k)_,b(f)(τ)]

Ca(k)_,a(f)(τ) +C_a(f)_,a(k)(τ) +C_b(k)_,b(f)(τ) +C_b(f)_,b(k)(τ)

=P_i2₌₀n−1−τ(j)A(i)+d(₀k)+d₁(k)(i)π(1)−A(i+τ)−d(0f)−d (f) 1 (i+τ)π(1)

+P_i2₌₀n−1−τ(j)A(i)+d(₀f)+d₁(f)(i)π(1)−A(i+τ)−d(0k)−d (k) 1 (i+τ)π(1)

+P2_i=0n−1−τ(j)A(i)+d(₀k)+d₁(k)(i)π(1)+2(i)π(n)−A(i+τ)−d(0f)−d (f)

1 (i+τ)π(1)−2(i+τ)π(n)

+P2_i=0n−1−τ(j)A(i)+d(₀f)+d₁(f)(i)π(1)+2(i)π(n)−A(i+τ)−d(0k)−d (k)

1 (i+τ)π(1)−2(i+τ)π(n)

=P2_i=0n−1−τ(j)A(i)−A(i+τ)_[(j)d0(k)+d(1k)(i)π(1)−d0(f)−d(1f)(i+τ)π(1)

+(j)d(₀f)+d₁(f)(i)π(1)−d(0k)−d (k)

1 (i+τ)π(1)]×[1 + (−1)(i)π(n)+(i+τ)π(n)]

If (i)π(n)= (6 i+τ)π(n),then 1 + (−1)(i)π(n)+(i+τ)π(n) = 0.

If (i)π(n)= (i+τ)π(n),Let ν denote the largest index for which(i)π(ν)6= (i+

τ)π(ν),then (i)π(k)= (i+τ)π(k),ν < k≤n. Let i0 andj0 denote indexes whose

binary representations differ from those of i and j only at position π(ν + 1) .

Similar to the Proof in [1], we obtain jA(i)−A(i+τ)_=−jA(i0_)−A(i0_+τ)

.

Obviously,ν+ 16= 1, then

(j)d(₀k)+d₁(k)(i)π(1)−d(0f)−d (f)

1 (i+τ)π(1)+ (j)d(0f)+d (f)

1 (i)π(1)−d(0k)−d (k) 1 (i+τ)π(1)

= (j)d(₀k)+d(₁k)(i0₎

π(1)−d(0f)−d (f)

1 (i0+τ)π(1)+ (j)d(0f)+d (f)

1 (i0)π(1)−d(0k)−d (k)

1 (i0+τ)π(1)

P2n_−1−τ

i=0 (j)A(i)−A(i+τ)[(j)d (k) 0 +d

(k)

1 (i)π(1)−d(0f)−d (f) 1 (i+τ)π(1)

+(j)d(₀f)+d₁(f)(i)π(1)−d(0k)−d (k)

1 (i+τ)π(1)] ×[1 + (−1)(i)π(n)+(i+τ)π(n)]

= 0.

We obtain,

Cc(τ) +Cd(τ) = 0

Case 2:The proof is similar to the proof in the case 1.

Case 3:In the Case 3, we haveπ(i)ν+π(i+τ)π(ν+1)= 1,

(i)π(ν+1)= (i+τ)π(ν+1)= 1−(i0)π(ν+1)= 1−(i0+τ)π(ν+1)

(j)d(₀k)+d₁(k)(i)π(ν)+d(2k)(i)π(ν+1)−d(0f)−d (f)

1 (i+τ)π(ν)−d2(f)(i+τ)π(ν+1)

+(j)d(₀f)+d₁(f)(i)π(ν)+d(2f)(i)π(ν+1)−d(0k)−d (k)

1 (i+τ)π(ν)−d2(k)(i+τ)π(ν+1)

= (j)d(₀f)+d(₁f)(i0₎

π(ν)+d(2f)(i0)π(ν+1)−d(0k)−d (k)

1 (i0+τ)π(ν)−d2(k)(i0+τ)π(ν+1)

+(j)d(₀k)+d(₁k)(i0₎

π(ν)+d(2k)(i0)π(ν+1)−d(0f)−d (f)

1 (i0+τ)π(ν)−d2(f)(i0+τ)π(ν+1)

Then, P₂n_−1−τ

i=0 (j)A(i)−A(i+τ)[1 + (−1)(i)π(n)+(i+τ)π(n)]

×[(j)d(₀k)+d(₁k)(i)π(ν)+d(2k)(i)π(ν+1)−d(0f)−d (f)

1 (i+τ)π(ν)−d2(f)(i+τ)π(ν+1)

+(j)d(₀f)+d₁(f)(i)π(ν)+d(2f)(i)π(ν+1)−d(0k)−d (k)

1 (i+τ)π(ν)−d2(k)(i+τ)π(ν+1)]

= 0

thus, they are also complementary sequences.

3

The Golay Complementary Sequences over

Q

-

P AM

constellation

The class ofQ-P AM constellation considered in this paper is the subset of the

M2_{-QAM constellation of size 2M = 2}m+1 _{having representation}

{√2j(ja0₊Pm−1

These representations suggest that quaternary sequences be used in the con-struction of Golay complementary sequences over these constellations.

Theorem 2. Let A(x) = 2P_kn₌₁−1xπ(k)xπ(k+1)+

Pn

k=1ckxk+c,

a(0)_{(x) =A(x) +s}(0)_(x),

b(0)_{(x) =A(x) +s}(0)_{(x) +µ(x),}

Whereck∈Z4,k= 0,1,· · · , m−1,c∈Z4,πis a permutation from{1,2,· · · , n}

to{1,2,· · ·, n} ,s(k)_{(x)andµ(x)satify the following cases.}

Case 1:s(k)_=d(k)

0 +d(1k)xπ(1),

µ(x) = 2xπ(n).

Case 2:s(k)_{(x) =d}(k) 0 +d

(k) 1 xπ(n),

µ(x) = 2xπ(1).

Case 3:s(k)_=d(k) 0 +d

(k)

1 xπ(ω)+d(2k)xπ(ω+1),1≤ω≤n−1,

2d(0)₀ +d(0)₁ +d(0)₂ = 0, d(₁k)=d(₂k),

µ(x) = 2xπ(1) or2xπ(n).

Whered(0)0 , d(0)1 , d(0)2 ∈Z4 ,d(0k), d1(k), d(2k)∈Z2 ,k= 1,· · · , m−1.

Then the 2M-QP AM sequences

c(x) =√2j(ja0_(x)

+Pm_k=1−12k_(j)a0_(x)+2s(k)_(x)

),

d(x) =√2j(jb0_(x)

+Pm_k=1−12k_(j)b0_(x)+2s(k)_(x)

)

are Golay complementary sequences.

The proof is similar to the proof in the theorem 1, we omit it.

Example 1:The 8-aryQ-P AM constellation is a subset of the 16-QAM con-stellation given by

{√2j(ja0_{+ 2j}a0+2a1_)|a

0∈Z4, a1∈Z2}

Then as discussed above, let a0 = f(x1, x2, x3) = x1x2+x2x3 , a1 = 1 +x3

−√2j,−√2j,−√2j,√2j,3√2j,3√2j,−3√2j,3√2j,

−√2j,√2j,−√2j,−√2j,3√2j,−3√2j,−3√2j,−3√2j.

4

Golay Complementary Sequences over 64-

QAM

Using

Offset Pairs Proposed By Ying Li

Based on offset pairs proposed by Ying Li[16], he proposed two constructions called modified case 4 and modified case 5 respectively, based on the construc-tions, he also proposed two conjectures. Now we study the constructions and prove the conjectures. We rewrite original modified case 4 and modified case 5 in[16] as case 4 and case 5 in this paper.

Case 4:LetA(x) = 2Pn_k=1−1xπ(k)xπ(k+1)+

Pn

k=1ckxk+c,

a1(x) =A(x) +s(1)(x),

a2(x) =A(x) +s(2)(x),

B(x) =A(x) +µ(x),

b1(x) =a1(x) +µ(x),

b2(x) =a2(x) +µ(x),

(s(1)_{(x), s}(2)_{(x)) = (d}

0+d1xπ(ω), d00+d01xπ(ω)), with 2≤ω≤n−1.

µ(x) = 2xπ(1) or 2xπ(n),

c(x) = 4jA(x)_{+ 2j}a1(x)_+ja2(x)_{, d(x) = 4j}B(x)_{+ 2j}b1(x)_+jb2(x)_,

Then,

Cc(τ) =

P₂n_−1−τ i=0 cic∗i+τ

=P2_i=0n−1−τ(8jA(i)−A(i+τ)_{+ 4j}a1(i)−a1(i)−a1(i+τ)+ja2(i)−a2(i+τ))

+P_i2₌₀n−1−τ2[4(jA(i)−a1(i+τ)_+ja1(i)−A(i+τ)₎

+2(jA(i)−a2(i+τ)_+ja2(i)−A(i+τ)_{) + (j}a1(i)−a2(i+τ)_+ja2(i)−a1(i+τ)_)]

Cd(τ) =

=P2_i=0n−1−τ(8jB(i)−B(i+τ)_{+ 4j}b1(i)−b1(i+τ)+jb2(i)−b2(i+τ))

+P2_i=0n−1−τ2[4(jB(i)−b1(i+τ)+jb1(i)−B(i+τ))+2(jB(i)−b2(i+τ)+jb2(i)−B(i+τ))+

(jb2(i)−b1(i+τ)_+jb1(i)−b2(i+τ)_)]

The following three equations can be verified easily. P2n_−1−τ

i=0 (jA(i)−a1(i+τ)+ja1(i)−A(i+τ)+jB(i)−b1(i+τ)+jb1(i)−B(i+τ)) (1)

=P2_i=0n−1−τ[jA(i)−A(i+τ)−d0−d1(i+τ)π(ω)_+jA(i)−A(i+τ)−d0−d1(i+τ)π(ω)+2(i)π(1)−2(i+τ)π(1)_]

+P_i2₌₀n−1−τ[j−A(i+τ)+A(i)+d0+d1(i)π(ω)+j−A(i+τ)+A(i)+d0+d1(i)π(ω)+2(i)π(1)−2(i+τ)π(1)]

=P2_i=0n−1−τjA(i+τ)+A(i)_[jd0+d1(i)π(ω)+j−d0−d1(i+τ)π(ω)][1 + (−1)iπ(1)+(i+τ)π(1)]

P2n_−1−τ

i=0 (jA(i)−a2(i+τ)+ja2(i)−A(i+τ)+jB(i)−b2(i+τ)+jb2(i)−B(i+τ)) (2)

=P2_i=0n−1−τ[jA(i)−A(i+τ)−d0

0−d01(i+τ)π(ω)+jA(i)−A(i+τ)−d00−d01(i+τ)π(ω)+2(i)π(1)−2(i+τ)π(1)]

+P2_i=0n−1−τ[j−A(i+τ)+A(i)+d0

0+d01(i)π(ω)+j−A(i+τ)+A(i)+d00+d01(i)π(ω)+2(i)π(1)−2(i+τ)π(1)]

=P2_i=0n−1−τjA(i+τ)+A(i)_[jd0

0+d01(i)π(ω)+j−d00−d01(i+τ)π(ω)][1 + (−1)iπ(1)+(i+τ)π(1)]

P₂n_−1−τ

i=0 (ja1(i)−a2(i+τ)+ja2(i)−a1(i+τ)+jb2(i)−b1(i+τ)+jb1(i)−b2(i+τ)) (3)

=P2_i=0n−1−τ[jA(i)+d0+d1(i)π(ω)−A(i+τ)−d00+d01(i+τ)π(ω)

+jA(i)+d0+d1(i)π(ω)−A(i+τ)−d00−d01(i+τ)π(ω)+2(i)π(1)−2(i+τ)π(1)]

+P2_i=0n−1−τ[j−A(i+τ)−d0−d1(i+τ)π(ω)+A(i)+d00+d01(i)π(ω)

+j−A(i+τ)−d0+d1(i)π(ω)+A(i)+d00+d01(i)π(ω)+2(i)π(1)−2(i+τ)π(1)]

=P2_i=0n−1−τjA(i+τ)+A(i)_[jd0

0+d01(i)π(ω)−d0−d1(i+τ)π(ω)+j−d00−d01(i+τ)π(ω)+d0+d1(i)π(ω)]

×[1 + (−1)iπ(1)+(i+τ)π(1)]

Thus we obtain the following equation

4×(1) + 2×(2) + (3)

+2(jA(i)−a2(i+τ)_+ja2(i)−A(i+τ)_+jB(i)−b2(i+τ)_+jb2(i)−B(i+τ)₎

+(ja1(i)−a2(i+τ)_+ja2(i)−a1(i+τ)_+jb2(i)−b1(i+τ)_+jb1(i)−b2(i+τ)_)]

=P_i2₌₀n−1−τjA(i+τ)+A(i)_{Q(i)[1 + (−1)}iπ(1)+(i+τ)π(1)]

WhereQ(i) = 4(jd0+d1(i)π(ω)_+j−d0−d1(i+π)π(ω)_)+2(jd00+d01(i)π(ω)_+j−d00−d01(i+π)π(ω)₎

+(jd0

0+d01(i)π(ω)−d0−d1(i+τ)π(ω)+j−d00−d01(i+τ)π(ω)+d0+d1(i)π(ω))

Similar to the proof of theorem 1, we set (i)π(ω) = (i +τ)π(ω) = 0 , then

the above equation is

4(jd0_+j−d0_{) + 2(j}d00+j−d00) + (jd00−d0_+j−d00+d0_),

and set (i)π(ω)= (i+τ)π(ω)= 1 , then the above equation is

4(jd0+d1_+j−d0−d1_{) + 2(j}d00+d01+j−d00−d10) + (jd00+d01−d0−d1_+j−d00−d01+d0+d1_),

thenc(x) andd(x) are Golay complementary pair if

4(jd0_+j−d0_{) + 2(j}d00+j−d00) + (jd00−d0_+j−d00+d0_),

(4) = 4(jd0+d1+j−d0−d1) + 2(jd00+d01+j−d00−d10) + (jd00+d01−d0−d1+j−d00−d01+d0+d1),

By exhaust search, there are 32 sets of (d0, d1, d00, d01) satisfying the above equation(4),

and the four sets{(0123),(0321),(1311),(3133)}proposed by Ying Li in [16] also satisfy the equation.

Case 5:LetA(x) = 2Pn_k=1−1xπ(k)xπ(k+1)+

Pn

k=1ckxk+c,

a1(x) =A(x) +s(1)(x),

a2(x) =A(x) +s(2)(x),

B(x) =A(x) +µ(x),

b1(x) =a1(x) +µ(x),

b2(x) =a2(x) +µ(x),

(s(1)_{(x), s}(2)_{(x)) = (d}

0+d1xπ(ω)+d2xπ(k), d00 +d01xπ(ω)+d02xπ(k)),

µ(x) = 2xπ(1) or 2xπ(n),

c(x) = 4jA(x)_{+ 2j}a1(x)_+ja2(x)_{, d(x) = 4j}B(x)_{+ 2j}b1(x)_+jb2(x)_.

It is easy to check the following three equations hold. P₂n_−1−τ

i=0 (jA(i)−a1(i+τ)+ja1(i)−A(i+τ)+jB(i)−b1(i+τ)+jb1(i)−B(i+τ))

=P_i2₌₀n−1−τjA(i)−A(i+τ)_{[1 + (−1)}iπ(1)−(i+τ)π(1)]

×[jd0+d1(i)π(ω)+d2(i)π(k)+j−d0−d1(i+τ)π(ω)−d2(i+τ)π(k)]

P2n_−1−τ

i=0 (jA(i)−a2(i+τ)+ja2(i)−A(i+τ)+jB(i)−b2(i+τ)+jb2(i)−B(i+τ))

=P_i2₌₀n−1−τjA(i)−A(i+τ)_{[1 + (−1)}iπ(1)−(i+τ)π(1)]

×[jd0

0+d01(i)π(ω)+d02(i)π(k)+j−d00−d01(i+τ)π(ω)−d02(i+τ)π(k)]

P₂n_−1−τ

i=0 (ja1(i)−a2(i+τ)+ja2(i)−a1(i+τ)+jb2(i)−b1(i+τ)+jb1(i)−b2(i+τ))

=P_i2₌₀n−1−τjA(i)−A(i+τ)_{[1 + (−1)}iπ(1)−(i+τ)π(1)]×

[jd0

0+d01(i)π(ω)+d02(i)π(k)−d0−d1(i+τ)π(ω)−d2(i+τ)π(k)

+jd0+d1(i)π(ω)+d2(i)π(k)−d00−d01(i+τ)π(ω)−d02(i+τ)π(k)]

then as in case 4, we obtain c(x) and d(x) are Golay complementary pair if the following conditions hold.

4(jd0+d1+d2y_+j−d0−d1−d2z_{) + 2(j}d00+d01+d02y+j−d00−d01−d02z)

+(jd0

0+d01+d02y−d0−d1−d2z_+jd0+d1+d2y−d00−d01−d02z)

= 4(jd0+d2y+j−d0−d2z)+2(jd00+d02y+j−d00−d20z)+(jd00+d02y−d0−d2z+jd0+d2y−d00−d02z) (5)

4(jd0+d1y+j−d0−d1z) + 2(jd00+d01y+j−d00−d01z)

+(jd0

0+d01y−d0−d1z+jd0+d1y−d00−d01z)

= 4(jd0+d1y+d2_+j−d0−d1z−d2_)+2(jd00+d01y+d02+j−d00−d01z−d02) (6)

+(jd0

0+d01y+d02−d0−d1z−d2_+jd0+d1y+d2−d00−d01z−d02)

By exhaust search, there are 52 sets of (d0, d1, d2, d00, d01, d02) satisfying the above

equation(5)and equation (6), and the four sets{(013231),(031213),(133111),

(311333)}proposed by Ying Li in[16] also satisfy the equations.

In the paper[16], the author asserted that there only exist four sets of coeffi-cients for the case 4 and case 5 respectively, but, based on the proposed judgment conditions above, we find there are 32 sets of coefficients for the case 4 and 52 sets of coefficients for the case 5 using exhaust search. We provide these detail materials in the appendix 1 and appendix 2 respectively.

Due to the discussion above, we have proved the conjecture 1 proposed in[16]by Ying Li. Because the conjecture 2 is based on conjecture 1 and there are some drawbacks existing in counting the set elements, some drawbacks can be found in conjecture 2.

Here, we extend the method proposed by Ying Li and introduce a new case called case 6 as follows:

Case 6:LetA(x) = 2Pn_k=1−1xπ(k)xπ(k+1)+

Pn

k=1ckxk+c,

a1(x) =A(x) +s(1)(x),

a2(x) =A(x) +s(2)(x),

B(x) =A(x) +µ(x),

b1(x) =a1(x) +µ(x),

b2(x) =a2(x) +µ(x),

(s(1)_{(x), s}(2)_{(x)) =}

(d0+d1xπ(ω)+d2xπ(k)+d3xπ(l), d00 +d01xπ(ω)+d02xπ(k)+d03xπ(l)),

with 2≤ω≤n−4 ,ω+ 2≤k≤n−2 ,k+ 2≤l≤n.

µ(x) = 2xπ(1) or 2xπ(n),

c(x) = 4jA(x)_{+ 2j}a1(x)_+ja2(x)_{, d(x) = 4j}B(x)_{+ 2j}b1(x)_+jb2(x)_.

It is easy to check the following three equations hold. P₂n_−1−τ

i=0 (jA(i)−a1(i+τ)+ja1(i)−A(i+τ)+jB(i)−b1(i+τ)+jb1(i)−B(i+τ))

=P_i2₌₀n−1−τjA(i)−A(i+τ)_{[1 + (−1)}iπ(1)−(i+τ)π(1)]×

P2n_−1−τ

i=0 (jA(i)−a2(i+τ)+ja2(i)−A(i+τ)+jB(i)−b2(i+τ)+jb2(i)−B(i+τ))

=P_i2₌₀n−1−τjA(i)−A(i+τ)_{[1 + (−1)}iπ(1)−(i+τ)π(1)]×

[jd0

0+d01(i)π(ω)+d02(i)π(k)+d03(i)π(l)+j−d00−d01(i+τ)π(ω)−d02(i+τ)π(k)−d03(i+τ)π(l)]

P₂n_−1−τ

i=0 (ja1(i)−a2(i+τ)+ja2(i)−a1(i+τ)+jb2(i)−b1(i+τ)+jb1(i)−b2(i+τ))

=P_i2₌₀n−1−τjA(i)−A(i+τ)_{[1 + (−1)}iπ(1)−(i+τ)π(1)]×

[jd0

0+d01(i)π(ω)+d02(i)π(k)+d03(i)π(l)−d0−d1(i+τ)π(ω)−d2(i+τ)π(k)−d3(i+τ)π(l)

+jd0+d1(i)π(ω)+d2(i)π(k)+d3(i)π(l)−d00−d01(i+τ)π(ω)−d02(i+τ)π(k)−d03(i+τ)π(l)]

then as in case 4, we obtain c(x) and d(x) are Golay complementary pair if the following conditions hold.

4(jd0+d2y1+d3y3_+j−d0−d2y2−d3y4_{) + 2(j}d00+d02y1+d03y3_+j−d00−d02y2−d03y4₎

+(jd0

0+d02y1+d03y3−d0−d2y2−d3y4_+jd0+d2y1+d3y3−d00−d02y2−d03y4₎

= 4(jd0+d1+d2y1+d3y3+j−d0−d1−d2y2−d3y4)+2(jd00+d01+d02y1+d03y3+j−d00−d01−d20y2−d03y4) (7)

+(jd0

0+d01+d20y1+d03y3−d0−d1−d2y2−d3y4+jd0+d1+d2y1+d3y3−d00−d01−d02y2−d03y4)

4(jd0+d1y1+d3y3+j−d0−d1y2−d3y4) + 2(jd00+d01y1+d03y3+j−d00−d01y2−d03y4)

+(jd0

0+d01y1+d03y3−d0−d1y2−d3y4_+jd0+d1y1+d3y3−d00−d01y2−d03y4₎

= 4(jd0+d1y1+d2+d3y3_+j−d0−d1y2−d2−d3y4_)+2(jd00+d01y1+d20+d03y3_+j−d00−d01y2−d02−d03y4_{) (8)}

+(jd0

0+d01y1+d02+d03y3−d0−d1y2−d2−d3y4_+jd0+d1y1+d2+d3y3−d00−d01y2−d02−d03y4₎

4(jd0+d1y1+d2y3_+j−d0−d1y2−d2y4_{) + 2(j}d00+d01y1+d02y3_+j−d00−d01y2−d02y4₎

+(jd0

0+d01y1+d02y3−d0−d1y2−d2y4+jd0+d1y1+d2y3−d00−d01y2−d02y4)

= 4(jd0+d1y1+d2y3+d3+j−d0−d1y2−d2y4−d3)+2(jd00+d01y1+d20y3+d03+j−d00−d01y2−d02y4−d03) (9)

+(jd0

Wherey1, y2, y3, y4∈ {0,1}.

By exhaust search, there are 76 sets of (d0, d1, d2, d3, d00, d01, d02, d03) satisfying

the above equation(7),equation (8) and equation (9), please see appendix 3. Note. We can choose

(s(1)_{(x), s}(2)_{(x)) =}

(d0+d1xπ(ω1)+d2xπ(ω2)+· · ·+dlxπ(ωl), d

0

0+d01xπ(ω1)+d

0

2xπ(ω2)+· · ·+d

0

lxπ(ωl)),

there are somewhat difficult to obtain sufficient conditions for judging whether the sequence using above offset pairs is Golay Complementary sequence.

5

Golay Complementary Sequences over 128-

QAM

Constellation

It is easy to construct Golay complementary sequences over 128-QAM constel-lation using the offset pairs in the Case 1, Case 2 and Case 3.We consider the case 5 here.

LetA(x) = 2Pn_k−₌₁1xπ(k)xπ(k+1)+

P_n

k=1ckxk+c,

a1(x) =A(x) +s(1)(x),

a2(x) =A(x) +s(2)(x),

a3(x) =A(x) +s(3)(x),

b1(x) =a1(x) +µ(x),

b2(x) =a2(x) +µ(x),

b3(x) =a3(x) +µ(x),

s(1)_{(x) =d}

0+d1xπ(ω)+d2xπ(k),

s(2)_{(x) =d}0

0+d01xπ(ω)+d02xπ(k),

s(3)_{(x) =d}∗

0+d∗1xπ(ω)+d∗2xπ(k),

With 2≤ω≤n−2 ,ω+ 2≤k≤n.

Then the 128-QAM sequences can be constructed as follows

c(x) = 8jA(x)_{+ 4j}a1(x)_{+ 2j}a2(x)_+ja3(x)_,

d(x) = 8jB(x)_{+ 4j}b1(x)_{+ 2j}b2(x)_+jb3(x)_.

It is easy to check the following five equations hold. P₂n_−1−τ

i=0 (jA(i)−a1(i+τ)+ja1(i)−A(i+τ)+jB(i)−b1(i+τ)+jb1(i)−B(i+τ))

=P_i2₌₀n−1−τjA(i)−A(i+τ)_{[1 + (−1)}iπ(1)−(i+τ)π(1)]

[jd0+d1(i)π(ω)+d2(i)π(k)+j−d0−d1(i+τ)π(ω)−d2(i+τ)π(k)]

P2n_−1−τ

i=0 (jA(i)−a2(i+τ)+ja2(i)−A(i+τ)+jB(i)−b2(i+τ)+jb2(i)−B(i+τ))

=P_i2₌₀n−1−τjA(i)−A(i+τ)_{[1 + (−1)}iπ(1)−(i+τ)π(1)]

[jd0

0+d01(i)π(ω)+d02(i)π(k)+j−d00−d01(i+τ)π(ω)−d02(i+τ)π(k)]

P₂n_−1−τ

i=0 (ja1(i)−a2(i+τ)+ja2(i)−a1(i+τ)+jb2(i)−b1(i+τ)+jb1(i)−b2(i+τ))

+P2_i=0n−1−τ(jA(i)−a3(i+τ)+ja3(i)−A(i+τ)+jB(i)−b3(i+τ)+jb3(i)−B(i+τ))

=P_i2₌₀n−1−τjA(i)−A(i+τ)_{[1 + (−1)}iπ(1)−(i+τ)π(1)]×

[jd0

0+d01(i)π(ω)+d02(i)π(k)−d0−d1(i+τ)π(ω)−d2(i+τ)π(k)

+jd0+d1(i)π(ω)+d2(i)π(k)−d00−d01(i+τ)π(ω)−d02(i+τ)π(k)]

+P_i2₌₀n−1−τjA(i)−A(i+τ)_{[1 + (−1)}iπ(1)−(i+τ)π(1)]×

[jd∗

0+d∗1(i)π(ω)+d∗2(i)π(k)+j−d0∗−d∗1(i+τ)π(ω)−d∗2(i+τ)π(k)]

P₂n_−1−τ

i=0 (ja1(i)−a3(i+τ)+ja3(i)−a1(i+τ)+jb1(i)−b3(i+τ)+jb3(i)−b1(i+τ))

=P_i2₌₀n−1−τjA(i)−A(i+τ)_{[1 + (−1)}iπ(1)−(i+τ)π(1)]×

[jd∗

0+d∗1(i)π(ω)+d∗2(i)π(k)−d0−d1(i+τ)π(ω)−d2(i+τ)π(k)

P2n_−1−τ

i=0 (ja2(i)−a3(i+τ)+ja3(i)−a2(i+τ)+jb2(i)−b3(i+τ)+jb3(i)−b2(i+τ))

=P_i2₌₀n−1−τjA(i)−A(i+τ)_{[1 + (−1)}iπ(1)−(i+τ)π(1)]×

[jd∗

0+d∗1(i)π(ω)+d∗2(i)π(k)−d00−d01(i+τ)π(ω)−d02(i+τ)π(k)

+jd0

0+d01(i)π(ω)+d02(i)π(k)−d∗0−d∗1(i+τ)π(ω)−d∗2(i+τ)π(k)_]

then similar to the case 4, c(x) and d(x) are Golay complementary pair if the following conditions hold.

16(jd0+d2y+j−d0−d2z) + 8(jd00+d02y+j−d00−d02z)

+4(jd0

0+d02y−d0−d2z_+jd0+d2y−d00−d02z) + 4(jd∗0+d∗2y+j−d∗0−d∗2z)

+2(jd∗

0+d∗2y−d0−d2z_+jd0+d2y−d∗0−d∗2z)

+(jd∗

0+d∗2y−d00−d02z+jd00+d02y−d∗0−d∗2z)

= 16(jd0+d1+d2y_+j−d0−d1−d2z_)+8(jd00+d01+d02y+j−d00−d01−d02z) (10)

+4(jd0

0+d01+d02y−d0−d1−d2z_+jd0+d1+d2y−d00−d01−d02z)+4(jd∗0+d∗1+d2∗y+j−d∗0−d∗1−d∗2z)

+2(jd∗

0+d∗1+d∗2y−d0−d1−d2z+jd0+d1+d2y−d0∗−d∗1−d∗2z)

+(jd∗

0+d∗1+d∗2y−d00−d01−d02z+jd00+d01+d02y−d∗0−d∗1−d∗2z)

16(jd0+d1y_+j−d0−d2z_{) + 8(j}d00+d01y+j−d00−d01z)

+4(jd0

0+d01y−d0−d1z_+jd0+d1y−d00−d01z) + 4(jd∗0+d∗1y+j−d∗0−d∗1z)

+2(jd∗

0+d∗1y−d0−d1z_+jd0+d1y−d∗0−d∗1z)

+(jd∗

0+d∗1y−d00−d01z+jd00+d01y−d∗0−d∗1z)

= 16(jd0+d1y+d2_+j−d0−d1z−d2_)+8(jd00+d01y+d02+j−d00−d01z−d02) (11)

+4(jd0

0+d01y+d20−d0−d1z−d2+jd0+d1y+d2−d00−d01z−d02)+4(jd∗0+d∗1y+d∗2+j−d∗0−d∗1z−d∗2)

+2(jd∗

0+d∗1y+d∗2−d0−d1z−d2+jd0+d1y+d2−d0∗−d∗1z−d∗2)

+(jd∗

Wherey, z∈ {0,1}.

By exhaust search, there are 260 sets of (d0, d1, d2, d00, d01, d02) satisfying the above

equation (10) and equation (11) , please see appendix 4.

6

Conlusion

We propose a new method to judge whether the sequences over theQAM con-stellation constructed using new offset pairs are Golay complementary sequences. Based on this method, we prove the conjectures[16] and find some new Golay complementary sequences over 64-QAM constellation. We propose a new offset pairs, based on the pairs, we construct new Golay complementary sequences over 64-QAMconstellation. 260 new Golay complementary sequences over 128-QAM

constellation can also be found. One can find many Golay complementary se-quences overQAM andQ−P AM constellation based on the method proposed in this paper.

7

Acknowledgment

This work was supported by National Science Foundation of China under grant No.60773002, the Project sponsored by SRF for ROCS, SEM, 863 Program (2007AA01Z472), and the 111 Project (B08038). The authors are grateful to Miss Xiao Zhang and Mr. Haolei Qin for help.

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