A New Family of Binary Sequences and Its Correlation Distribution

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A New Family of Binary Sequences and Its Correlation Distribution

XiaYongbo，ShangguanXiaotian

(College of Mathematics and Statistics，South-Central University for Nationalities，Wuhan 430074，China)

KeywordsRothaussequences；sequencefamily；correlationfunction；correlationdistribution；BCHcode

1Introduction

Sequencefamilieswithlowcorrelationhavewideapplicationsinspread-spectrumsystems,code-divisionmultiple-access(CDMA)systems,radarsystemandcryptography[1,2].Combininganm-sequenceanditsdecimatedsequencesisanefficientmethodtoconstructsequencefamilieswithlowcorrelationandlargefamilysize.Duringthepastdecades,numerousfamiliesofbinarysequenceswithgoodpropertieshavebeenconstructedbythismethod,seeRef[3-10],forexample.Amongthem,foroddm,Goldsequences[3,4]constituteoneofthefamilieswithoptimalcorrelationachievingtheSidelnikovbound[11],andforevenm,thesmallsetofKasamisequences[7]givessequenceswithoptimalcorrelationachievingtheWelch′slowerbound[12].

Inacode-divisionmultipleaccess(CDMA)system,maximumcorrelationandfamilysizearekeyparametersofasequencefamily.Largefamilysizeisrequiredtosupportalargenumberofsimultaneoususers,andlowcorrelationisrequiredtominimizeinterferenceamongdifferentusers.DuetoWelchbound[12],Sidelnikovbound[11],andLevenshteinbound[13],thereexistsatradeoffbetweenfamilysizeandmaximumcorrelation.Foraspecificapplication,iflowcorrelationismorecrucialthanlargefamilysizeintheapplication,wemayminimizethemaximumcorrelationfirstatthepriceofthedecreaseoffamilysize.Ontheotherhand,iflargefamilysizeismoreimportant,wemaysacrificethelowcorrelationtoincreasethefamilysize.AgoodexampleforcompromisebetweencorrelationandfamilysizeistheRothaussequencefamily[2,6].

Letmbeodd, m≥5.Rothaussequencefamilywasconstructedfrom

Recently, by making use of

(1)

insteadofEq.(1),RathaussequencefamilywasgeneralizedbyZhouandTanginRef[10],wheregcd(k,m)=1anditscorrelationdistributionisalsodetermined.

anewsequencefamilywillbeproposedanditscorrelationdistributionwillalsobedetermined.ThisnewsequencefamilyisidenticaltoRothaussequencefamilyintermsofmaximumcorrelationandfamilysize.However,notallthedecimationsusedherearequadratic(the2-adicexpansionofthemhaveweight2)sincer2isnotaquadraticdecimation.

Theremainderofthispaperisorganizedasfollows.InSection2,weintroducesomepreliminaries.InSection3,wegivetheconstructionofthesequencefamilyandderiveitscorrelationdistribution.TheconcludingremarksaregiveninSection4.

2Preliminaries

Fortwobinarysequences{u(t)}and{v(t)}oflengthN,theirperiodiccorrelationfunctionisdefinedas:

where 0≤τ≤N-1,t+τis computed by moduloN. Two sequences {u(t)} and {v(t)}are said to be cyclically equivalent if there exists an integerτsuch thatu(t+τ)=v(t) for allt. Otherwise, they are said to be cyclically distinct.

For a sequence familySof periodN, its maximum correlationCmaxis defined by:

where|x|denotesthemodulusvalueofacomplexnumberx.Clearly,Cmaxisthemaximummagnitudeofallnontrivialauto-andcross-correlationsofpairsofsequencesinS.IfScontainsMcycliclydistinctsequences,thenSwillbecalledan(N,M,Cmax)sequencefamily.

wherex∈F2mandlis a divisor ofm. Letf(x) be a function fromF2mtoF2. The trace transform off(x) is defined by:

wherebi,j∈F2. The rank of a quadratic formf(x) is determined by the number ofz∈F2msuch that:

f(x)+f(z)+f(x+z)=0 for allx∈F2m.

(2)

It is well known that the rank off(x) must be even, and if there are 2m-2hz′sinF2msatisfying Eq.(2), then the rank off(x) is equal to 2h[2,15].

LetCbe a linear code overF2with lengthnand

dimensionl. Then, we callCis an [n,l] linear code overF2. The Hamming weight of a vector c∈Cis the number of its nonzero entries and is denoted WH(c). LetAibe the number of codewords inCwith Hamming weighti,0≤i≤n. The sequence {A0,A1,A2,…,An} is called the weight distribution ofC. The dual code of c, denotedC⊥, is defined by:

where x·c denotes the dot product. According to MacWilliams identity[15], if the weight distribution ofCis known, then the weight distribution ofC⊥will be uniquely determined, and vice versa.

Lemma2LetCbethecycliccodedefinedinEq.(3).ItsweightdistributionisgiveninTable1.

Tab.1　Weight distribution of C

3New sequence family with large size

From now on, we will fix the following notations:

(i)mis an odd integer andm≥5;

(iii) αisaprimitiveelementofF2m.

Sequencefamilyconstruction:Set

Δ0={(1,0,0)},Δ1={(x0,1,0)|x0∈F2m},

Δ2={(x0,x1,0)|x0,x1∈F2m},Δ=Δ0∪Δ1∪Δ2.

The new sequence familySis defined by:

(4)

where

(5)

In order to study the correlation properties of the sequence familyS, we need to investigate some exponential sums.

(6)

whereS0(a,b)=2mif and only if (a,b)=(0,0).

From the above analysis, Eq.(6) is obtained.

Lemma 4Let:

(7)

S1(a,b,c)=2m-2WH(c),

(8)

Theorem1LetSbethesequencefamilydefinedinEq.(4).Then, Scontains22m+2m+1cycliclydistinctsequencesanditscorrelationdistributionisgiveninTab.2.

Tab.2　Correlation distribution of S

Proof Let

where (ai,bi,ci)∈Δ,i=1,2 . Then, the correlation function betweens1ands2is given by:

Cs1,s2(τ)=

(9)

where 0≤τ<2m-1. In what follows, we will determine the values taken byCs1,s2(τ) and the times they occur when (a1,b1,c1) and (a2,b2,c2) run throughΔ, respectively, andτruns through {0,1,2,…,2m-2}. We consider the following cases.

Case 1: (a1,b1,c1) and (a2,b2,c2) run throughΔ0, respectively. In this case, Eq.(9) becomes:

Obviously, in this case,-1 occurs 2m-2 times and 2m-1 occurs once.

Case 2: (a1,b1,c1) runs throughΔ0and (a2,b2,c2) runs throughΔ1. Then,

Whena2runs throughF2mandτruns through {0,1,2,…,2m-2}, by the proof of Lemma 3, one has:

Cs1,s2(τ)=

Case 3: (a1,b1,c1) runs throughΔ0and (a2,b2,c2) runs throughΔ2. Then,

Case 4:(a1,b1,c1) runs throughΔ1and (a2,b2,c2) runs throughΔ0. Then,

Whena1runs throughF2mandτruns through {0,1,2,…,2m-2}, by the proof of Lemma 3, we have:

Cs1,s2(τ)=

Case 5:(a1,b1,c1) runs throughΔ1and (a2,b2,c2) runs throughΔ1. Then,

By Lemma 3 and its proof, in this case we have：

Cs1,s2(τ)=

Note that in this caseCs1,s2(τ)=2m-1 if and only if (a1,b1,c1)=(a2,b2,c2) andτ=0.

Case 6:(a1,b1,c1) runs throughΔ1and (a2,b2,c2) runs throughΔ2. Then,

Case 7:(a1,b1,c1) runs throughΔ2and (a2,b2,c2) runs throughΔ0. Then,

Case 8:(a1,b1,c1) runs throughΔ2and (a2,b2,c2) runs throughΔ1. Then,

Case 9:(a1,b1,c1) runs throughΔ2and (a2,b2,c2) runs throughΔ2. Then,

Ifτ=0, by Lemma 3 and its proof, when (a1,b1,c1) runs throughΔ2and (a2,b2,c2) runs throughΔ2, the value distribution ofCs1,s2(τ) is as follows:

Summing the value distributions in Cases 1-9, the desired distribution in Tab.2 is obtained. From the above discussion, it is easily seen thatCs1,s2(τ)=2m-1 if and only if (a1,b1,c1)=(a2,b2,c2) andτ=0. By counting the number of elements inΔ, the family size ofSis obtained.

Tab.3　Known exponent pairs {d1,d2} such that C1,d1,d2 and

4Conclusion

Attheendofthispaper,weprovideanexampletoverifyTheorem1.

Example1Letm=5,r=9,r2=81≡19mod(2m-1)andαbeaprimitiveelementofF25.Then,thesequencefamilySisgivenasfollows:

By computer experiment, the correlation distribution ofSis given by:

The numerical result coincides with Theorem 1.

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一類新的二元序列集及其相關(guān)分布

夏永波, 上官曉天

(中南民族大學(xué) 數(shù)學(xué)與統(tǒng)計學(xué)學(xué)院, 武漢 430074)

摘要對于奇整數(shù)m≥5, 構(gòu)造了一類新的周期為2m-1的二元序列集, 并完全確定了衡量新序列集性能的一個重要指標(biāo)——相關(guān)分布. 新序列集的最大相關(guān)值為+1, 集合容量為2(2m)+2m+1, 和著名的Rothaus序列具有類似的相關(guān)性質(zhì), 能適用于無線通信系統(tǒng)如CDMA通信系統(tǒng).

關(guān)鍵詞Rothaus序列；序列族；相關(guān)函數(shù)；相關(guān)分布；BCH碼

中圖分類號O157

文獻標(biāo)識碼A

文章編號1672-4321(2016)01-0145-06