Open access peer-reviewed chapter

# A Direct Construction of Intergroup Complementary Code Set for CDMA

Written By

Palash Sarkar and Sudhan Majhi

Submitted: January 23rd, 2019 Reviewed: May 8th, 2019 Published: July 10th, 2019

DOI: 10.5772/intechopen.86751

From the Edited Volume

## Coding Theory

Edited by Sudhakar Radhakrishnan and Muhammad Sarfraz

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## Abstract

A collection of mutually orthogonal complementary codes (CCs) is said to be complete complementary codes (CCCs) where the number of CCs are equal to the number of constituent sequences in each CC. Intergroup complementary (IGC) code set is a collection of multiple disjoint code groups with the following correlation properties: (1) inside the zero-correlation zone (ZCZ), the aperiodic autocorrelation function (AACF) of any IGC code is zero for all nonzero time shifts; (2) the aperiodic cross-correlation function (ACCF), of two distinct IGC codes, is zero for all time shifts inside the ZCZ when they are taken from the same code groups; and (3) the ACCF, for two IGC codes from two different code groups, is zero everywhere. IGC code set has a larger set size than CCC, and both can be applicable in multicarrier code-division multiple access (CDMA). In this chapter, we present a direct construction of IGC code set by using second-order generalized Boolean functions (GBFs), and our IGC code set can support interference-free code-division multiplexing. We also relate our construction with a graph where the ZCZ width depends on the number of isolated vertices present in a graph after the deletion of some vertices. Here, the construction that we propose can generate IGC code set with more flexible parameters.

### Keywords

• complementary code (CC)
• code-division multiple access (CDMA)
• generalized
• Boolean function (GBF)
• intergroup complementary (IGC) code set
• zero-correlation zone
• (ZCZ) sequences

## 1. Introduction

Code-division multiple access (CDMA) [1] is an important communication technology where sequence signatures with good correlation properties are used to separate multiple users. In CDMA systems, multipath interference (MPI) and multiple access interference (MAI) degrade the performance where MPI and MAI occur due to the multipath propagation, non-ideal synchronization, and non-ideal correlation properties of spreading codes. Spreading code plays a significant role on the overall performance of a CDMA system. The interference-resist capability and system capacity are determined by the correlation properties and available number of spreading codes. Due to ideal auto- and cross-correlation properties, complete complementary codes (CCCs) have been applied to asynchronous multicarrier CDMA (MC-CDMA) [2] communications in order to provide zero interference performance.

Golay proposed a pair of sequences in Golay [3] known as Golay complementary pair (GCP) which is a set of two equal length sequences with the property that the sum of their aperiodic autocorrelation function (AACF) is zero everywhere except at the zero shift. Tseng and Liu [4] extended the idea of GCP to complementary set or complementary code (CC) which contains two or more than two sequences. Davis and Jedwab [5] proposed a direct construction of GCP called Golay-Davis-Jedwab (GDJ) pair by using second-order generalized Boolean functions (GBFs) to reduce peak-to-mean envelope power ratio (PMEPR) for OFDM system. As a generalization of GDJ pair, Paterson introduced a construction of CC Paterson [6] by associating each CC with a graph. Recently, a construction of CC has been reported in Sarkar et al. [7] which is a generalization of Paterson’s CC construction. Later, Rathinakumar and Chaturvedi extended Paterson’s construction to CCC Rathinakumar and Chaturvedi [8] which is a collection of mutually orthogonal CCs. Although CCs have ideal AACF and aperiodic cross-correlation function (ACCF), they are unable to support a maximum number of users as the set size cannot be larger than the flock size [9, 10, 11], where the flock size denotes the number of constituent sequences in each CC. The application of CCC has been extended for the enabling of interference-free MC-CDMA communication by designing a fractional-delay-resilient receiver in Liu et al. [12].

The binary Z-complementary sequences were first introduced by Fan et al. [13] and later extended to quadriphase Z-complementary sequences by Li et al. [14].

Recently, a construction of binary Z-complementary pairs has been reported in Adhikary et al. [15]. A direct construction of polyphase Z-complementary codes has been reported in Sarkar et al. [16], which is an extension of Rathinakumar’s CCC construction. Due to favorable correlation properties of Z-complementary codes, it can be easily utilized for MC-CDMA system as spreading sequences to mitigate MPI and MAI efficiently [17]. The theoretical bound given in Liu et al. [18] shows that the Z-complementary codes have a much larger set size than CCCs.

IGC code set was first proposed by Li et al. [19] based on CCCs. Their code assignment algorithm shows that the CDMA systems employing the IGC codes (IGC-CDMA) outperform traditional CDMA with respect to bit error rate (BER). However, the ZCZ width of IGC codes [19] is fixed to the length of the elementary codes of the original CCCs, which limits the number of IGC codes. Another improved construction method of IGC codes is proposed in Feng et al. [20] based on the CCCs, interleaving operation, and orthogonal matrix which provides a flexible choice of the ZCZ width. However, there is no such construction which can directly produce IGC code set without having operation on CCCs; it motivates us to give a direct construction of IGC code set.

This chapter contains a direct method to construct IGC code set by applying second-order GBFs. This construction is capable of generating IGC code set with more flexible parameters such as ZCZ width and set size. We also relate our construction with a graph, and it has been shown that ZCZ width and set size of the IGC code set obtained by using our method depend on the number of isolated vertices present in a graph which is achieved by deleting some vertices from a graph.

## 2. Preliminary

### 2.1 Correlations of sequences

The ACCF between two sequences a=a0a1aL1and b=b0b1bL1is defined as follows:

Cabτ=Σi=0L1τai+τbi,0τ<L,Σi=0L+τ1ajbiτJL<τ<0,0,otherwise,E1

where τis an integer. The above defined function in Eq. (1) is said to be AACF of a(or, b) if a=b. The AACF of aat τis denoted by Aaτ.

Definition1An ordered seta0a1,aP1containingPsequences of equal lengthLis calledCC lf

i=0P1Aajτ=LP,τ=0,0,otherwise.E2

Definition2LetC0C1,CK1be a set ofKCCs where each of theCC containsPKPconstituent sequences of length L. Theαthconstituent sequence ofCiisCi,α=Ci,α,0Ci,α,1Ci,α,L1whereα=0, 1, ,P1,i=0, 1, ,K1. The ACCF of theCCs is given by

CCiCjτ=α=0P1CCi,αCj,ατ=0,τ,ij.E3

The code set is said to be CCC whenK=P.

Definition3Given anIGCcode set I(K,P,L,Z) (Li et al. [19]), Kdenotes a number of codes, Pdenotes the number of constituent sequences in each code, Ldenotes the length of each constituent sequence, andZdenotesZCZwidth, whereK=PL/Z. TheKcodes can be divided intoPcode groups denoted byIgg=01P1, each group containsK/P=L/Zcodes. The code setIKPLZhas the following properties:

CCiCjτ=PL,i=j,τ=0,0,i=j,0<τ<Z,0,ij,Ci,CjIg,τ<Z,0,CiIg1,CjIg2,g1g2,τ<L,others,otherwise.E4

### 2.2 Generalized Boolean functions

Let f:01mZq(qis average number, not less than 2) be a function of mvariables x0,x1,,xm1. The product of kdistinct variables χi0χi1χik10i0<i1<<ik1m1is called a monomial of degree k. The monomials 1, x0,,xm1,x0x1,,xm2xm1,,x0x1xm1are the list of 2mmonomials over the variables x0,x1,,xm1. A GBF fcan uniquely be presented as a linear combination of these 2mmonomials, where the coefficient of each monomial belongs to Zq. We denote the complex valued sequence corresponding to the GBF fby ψfand define it as

ψf=ωf0ωf1ωf2m1,E5

where fi=fi0i1im1, ω=exp2π1/q, and i0i1im1are the binary representation of the integer ii=i=0m1ij2j. Let Cbe an order set of PBoolean functions given by C=f0f1fP1. Then the complex valued code corresponding to the set of Boolean function Cis denoted by ψC, given by ψC=ψf0ψf1ψfP1. The code can also be viewed as a matrix where ψfi1is the ith row of the matrix.

For any given GBF fof mvariables, the function f1x01x11xm1is denoted by f˜. For a Zqvalued vector e=e0e1eL1, we denote the vector e¯by e¯0e¯1e¯L1where e¯i=q2eii=01L1. Now, we define the following notations a¯and a, where a¯is derived from aby reversing it and ais the complex conjugate of a.

### 2.3 Quadratic forms and graphs

In this context, we introduce some lemmas and new notations which will be used for our proposed construction.

Definition4Letfbe aGBFof variablesx0,x1,,xm1overZq. Consider a list ofk0k<mindices0j0<j1<jk<m, and writex=xj0xj1xjk1. Considerc=c0c1ck1to be a fixed binary vector. Then we defineψfx=cas a complex valued vector withωfi0'i1''im1as aithcomponent ifijα=cαfor each0α<kand equal to zero otherwise. Fork=0, the complex valued vectorψfx=cis nothing, but the vectorψfis defined before.

Let Q:01mZqbe a quadratic form of mvariables x0,x1,,xm1.Aquadratic GBFis of the form Rathinakumar and Chaturvedi [8]

f=Q+i=0m1gixi+g,E6

where g,giZqare arbitrary.

For a quadratic GBF, f,Gfdenotes the graph of f. The Gfis obtained by joining the vertices xiand xjby an edge if there is a term qi,jxixj0i<jm1in the GBF fwith qi,j0qi,jZq. Consider a function fxj=c, derived by fixing xjat cin f. The graph of fxj=cis denoted by Gfxj=cwhich is obtained by deleting the vertex xjand all the edges which are connected to xjfrom Gf. Then Gfx=cis obtained from Gfby deleting xj0,xj1,,xjk1.Gfx=crepresent the same graph for all c01k. Therefore, for all cin 01kfx=chave the same quadratic form. Note that the quadratic forms of fand f˜are the same; thus, they have associated with the same graph.

Lemma1Construction ofCCC [8].

Letf: 01mZqbe aGBF andfits reversal. Assume thatGfx=cis a path for eachc01kand the edges in the path have the same weightq/2. Letb0b1bk1be the binary representation of the integert. Define the order sets ofGBFs Ctto be

f+q2α=0k1dαxjα+α=0k1bαxjα+dxγ:ddα01,E7

and the order set ofGBFs C˜tto be

f˜+q2α=0k1dαx¯jα+α=0k1bαx¯jα+d¯xγ:d¯dα01,E8

wherexγis one of the end vertices in the path. Then

ψCt:0t<2kψC˜t:0t<2kE9

generates a set ofCCC, whereψdenotes the complex conjugate ofψ.

## 3. Construction of IGC code set from GBFs

In this section, we propose a direct construction of IGC code set by using Boolean algebra and graph theory. Before proposing the main theorem of the construction, we define some sets and vectors and present some lemmas. First we define some notations which will be used throughout in our construction:

• x=xj0xj1xjk1Z2k,x=xmp'xmp+1xm1Z2p.

• b=b0b1bk1, bi=bi,0bi,1bi,k1Z2ki=122k.

• d=d0d1dk1Z2k,d=d1d2dpand dj=dj,1dj,2dj,pZ2pj=122p.

• Γ=gmpJgmp+1''gm1Zqp.

• a·bdenotes the dot products of any two vectors aand bwhich are of the same length.

• ABdenotes the Kronecker product of any two matrices of arbitrary size.

• i,j,i=0,1,,M1and j=1,2,,Ndenotes a matrix of order M×N.

Let fbe a GBFof mvariables x0,x1,,xm1over Zq. For bZ2k,dZ2p, we define order sets Sbdand S˜bdcorresponding to the GBF fas follows:

Sbdf+q2α=0kdαxjα+α=01bαxjα+α=1pdαxmp+α1+dxγ:ddα01,E10

or

Sbd=f+q2d+bx+dx+dxγ:dZ2dZ2k,E11

and

S˜bd=f˜+q2d+bx¯+dx¯+d¯xγ:dZ2dZ2kE12

From the above expression, it is clear that each of the order sets Sbdand S˜bdcontains 2k+1GBFs.

Lemma2Letfbe aGBF ofmvariables with the property that eachc01k, G(fx=c)contains a path overmkp0k<mp0vertices andpisolated vertices labeledmp,mp+1,,m1such that0k+pm2m2. Further, assume that there was no edges between deleted vertices(as defined before, the restricted variables in aGBF are considered as the vertices to be deleted in the graph of the Boolean function) and isolated vertices before the deletion. Letxγbe one of the end vertices of the path inGfx=cand the weight of each edge in the path beq/2. Leta1, a2,,a2mpbe binary vector representations of0,1,2mp1of lengthmpandr1,r2,,r2pbe binary vector representations of0, 1, 2p1of lengthp. Also letlbe a positive integer such thatl=Σi=0k1di2j+d2k. Then for any choice ofg,gjZq, the codesψ(Sbd) andψS˜bdcan be expressed as

ψSbd=ψFblωΓ+q2drj,l=0,1,,2k+11,j=1,2,,2p,ψS˜bd=ψFblωΓ+q2dr¯j,l=0,1,,2k+11,j=1,2,,2p,E13

where

ψFbl=ωFbla1ωFbla2ωFbla2mp,ψFbl'=ωFbla1ωFbla2ωFbla2mp,Fbl=f+q2d+bx+dxγ,Fbl=f˜+q2d+bx¯+d¯xγ,f=Q+i=0mp1gixi+g.E14

Proof1Since there are no edges between the deleted and isolated vertices before the deletion ofkverticesxj0,xj1,,xjk1, the quadratic formQpresented inGfcan be expressed as

Q=q2α=0mkp1xπαxπα+1+0μ<vk1bjμ,jvxjμxjv+α=0mkp1σ=0k1cπαjσxπαxjσ,E15

whereπis a permutation over the set01m1j0j1jk1mpmp+1m1,bjμ,jvZqwhich denotes the weight between the verticesxjμandxjvandcπα,jσZqdenotes the weight between the verticesxπαandx. Therefore, fis aGBF ofmpvariablesx0,x1,,xmp1, and theGBF fofmvariables can be expressed as.

f=f+i=mpm1gixiE16

or

f=f+Γx'.E17

Now we define aGBF Flbovermvariables by

Flb=f+q2d+bx+dx+dxγ=Fbl+Γ+q2dx.E18

Leta1,a2,,a2mbe binary vector representations of0,1,2m1of lengthm, given inTable 1 . The truth table given in Table 1 can also be expressed as the truth table given inTable 2 . Table 3 contains a truth table overmpvariables.

 a1 a2 ⋮ a2m

### Table 1.

Truth table over mvariables.

 a1′r1 a2′r1 ⋮ a2m−p′r1 a1′r2 a2′r2 ⋮ a2m−p′r2 ⋮ a1′r2p a2′r2p ⋮ a2m−p′r2p

### Table 2.

Truth table over mvariables.

 a1′ a2′ ⋮ a2m−p′

### Table 3.

Truth table over mpvariables.

FromTables 1 3 , it is observed that the codeψSbdcan be expressed as

ψSbd=ωFlbaj,l=0,1,2k+11,j=1,2,,2mE19

or

ψSbd=ψFblωΓ+q2drj,l=0,1,,2k+11,j=1,2,,2p,E20

where

ψFbl=ωFbla1ωFbla2ωFbla2m0,l=0,1,2k+11.E21

Similarly, we can show that

ψS˜bd=ψFblωΓ+q2dr¯j,l=0,1,2k+11,j=1,2,,2p.E22

Example1Letfbe aGBF offour variables overZ4, given by

fx0x1x2x3=2x1x2+3x0x1+x2+x0+x2+x3+1.E23

From theGf, given inFigure 1 , it is clear that after the deletion of the vertexx0, the resultant graph contains a path over the verticesx1,x2and an isolated vertexx3. For this examplek=1andp=1. Therefore, the vectorsb,d,d,x,Γandxare of length one and belong toZ2.

Hence, b=bj0=b0=b0,d=d0=d0,d=d1=d1, x=xj0=x0=x0, Γ=gmp=g0=1, andx=xmp=x3=x3. The set of Boolean functionsSb0d1andS˜b0d1are given below:

Sb0d1=f+q2d0x0+b0x0+d1x3+dx2:dd0{01},E24

and

s˜b0d1=f˜+q2d0x¯0+b0x¯0+d1x¯3+d¯x2:dd0{01}.E25

TheGBFs Fb0landFb0lare given by

Fb0l=2x1x2+3x0x1+x2+x0+x2+1+q2d0x0+b0x0+dx2E26

and

Fb0l=2x¯1x¯2+3x¯0x¯1+x¯2+x¯0+x¯2+1+q2d0x¯0+b0x¯0+d¯x2,E27

wherel=0,1,2,3.

The codes corresponding to the sets of Boolean functions are listed below:

1)ψS00=ω1ω2ω1ω1ω2ω2ω0ω3ω2ω3ω2ω2ω3ω3ω1ω0ω1ω0ω1ω3ω2ω0ω0ω1ω2ω1ω2ω0ω3ω1ω1ω2ω1ω2ω1ω1ω0ω0ω2ω1ω2ω3ω2ω2ω1ω1ω3ω2ω1ω0ω1ω3ω0ω2ω2ω3ω2ω1ω2ω0ω1ω3ω3ω0=ω0ψC0ω1ψC0E28

where

ψC0=ω1ω2ω1ω1ω2ω2ω0ω3ω1ω0ω1ω3ω2ω0ω0ω1ω1ω2ω1ω1ω0ω0ω2ω1ω1ω0ω1ω3ω0ω2ω2ω3.
2)ψS˜00=ω0ω1ω3ω3ω0ω0ω1ω0ω3ω0ω2ω2ω3ω3ω0ω3ω2ω1ω1ω3ω2ω0ω3ω0ω1ω0ω0ω2ω1ω3ω2ω3ω0ω1ω3ω3ω2ω2ω3ω2ω3ω0ω2ω2ω1ω1ω2ω1ω2ω1ω1ω3ω0ω2ω1ω2ω1ω0ω0ω2ω3ω1ω0ω1=ω1ψC˜0ω0ψC˜0E29

where

ψC˜0=ω3ω0ω2ω2ω3ω3ω0ω3ω1ω0ω0ω2ω1ω3ω2ω3ω3ω0ω2ω2ω1ω1ω2ω1ω1ω0ω0ω2ω3ω1ω0ω1.
3)ψS01=ω1ω2ω1ω1ω2ω2ω0ω3ω0ω1ω0ω0ω1ω1ω3ω2ω1ω0ω1ω3ω2ω0ω0ω1ω0ω3ω0ω2ω1ω3ω3ω0ω1ω2ω1ω1ω0ω0ω2ω1ω0ω1ω0ω0ω3ω3ω1ω0ω1ω0ω1ω3ω0ω2ω2ω3ω0ω3ω0ω2ω3ω1ω1ω2.=ω0ψC0ω3ψC0E30
4)ψS˜01=ω2ω3ω1ω1ω2ω2ω3ω2ω3ω0ω2ω2ω3ω3ω0ω3ω0ω3ω3ω1ω0ω2ω1ω2ω1ω0ω0ω2ω1ω3ω2ω3ω2ω3ω1ω1ω0ω0ω1ω0ω3ω0ω2ω2ω1ω1ω2ω1ω0ω3ω3ω1ω2ω0ω3ω0ω1ω0ω0ω2ω3ω1ω0ω1.=ω3ψC˜0ω0ψC˜0E31
5)ψS10=ω1ω0ω1ω3ω2ω0ω0ω1ω2ω1ω2ω0ω3ω1ω1ω2ω1ω2ω1ω1ω2ω2ω0ω3ω2ω3ω2ω2ω3ω3ω1ω0ω1ω0ω1ω3ω0ω2ω2ω3ω2ω1ω2ω0ω1ω3ω3ω0ω1ω2ω1ω1ω0ω0ω2ω1ω2ω3ω2ω2ω1ω1ω3ω2=ω0ψC1ω1ψC1E32

where

ψC1=ω1ω0ω1ω3ω2ω0ω0ω1ω1ω2ω1ω1ω2ω2ω0ω3ω1ω0ω1ω3ω0ω2ω2ω3ω1ω2ω1ω1ω0ω0ω2ω1.
6)ψS˜10=ω2ω1ω1ω3ω2ω0ω3ω0ω1ω0ω0ω2ω1ω3ω2ω3ω0ω1ω3ω3ω0ω0ω1ω0ω3ω0ω2ω2ω3ω3ω0ω3ω2ω1ω1ω3ω0ω2ω1ω2ω1ω0ω0ω2ω3ω1ω0ω1ω0ω1ω3ω3ω2ω2ω3ω2ω3ω0ω2ω2ω1ω1ω2ω1=ω1ψC˜1ω0ψC˜1E33

where

ψC˜1=ω1ω0ω0ω2ω1ω3ω2ω3ω3ω0ω2ω2ω3ω3ω0ω3ω1ω0ω0ω2ω3ω1ω0ω1ω3ω0ω2ω2ω1ω1ω2ω1.
7)ψS11=ω1ω0ω1ω3ω2ω0ω0ω1ω0ω3ω0ω2ω1ω3ω3ω0ω1ω2ω1ω1ω2ω2ω0ω3ω0ω1ω0ω0ω1ω1ω3ω2ω1ω0ω1ω3ω0ω2ω2ω3ω0ω3ω0ω2ω3ω1ω1ω2ω1ω2ω1ω1ω0ω0ω2ω1ω0ω1ω0ω0ω3ω3ω1ω0.=ω0ψC1ω3ψC1E34
8)ψS˜11=ω0ω3ω3ω1ω0ω2ω1ω2ω1ω0ω0ω2ω1ω3ω2ω3ω2ω3ω1ω1ω2ω2ω3ω2ω3ω0ω2ω2ω3ω3ω0ω3ω0ω3ω3ω1ω2ω0ω3ω0ω1ω0ω0ω2ω3ω1ω0ω1ω2ω3ω1ω1ω0ω0ω1ω0ω3ω0ω2ω2ω1ω1ω2ω1.=ω3ψC˜1ω0ψC˜1E35

Theorem1Letfbe aGBF overmvariables as defined in Lemma1 and Lemma2. SupposeI0,I1,,I2k+11are a list of2k+1code groups defined by

It=Ist:0s<2p=ψSbd:d01pE36

and

I2k+t=Is2k+t:0s<2p=ψS˜bd:d01p,E37

which forms anIGCcode setI2k+p+12k+12m2mp.

Proof2LetψSbd1andψSbd2be any two codes from a code groupIt. Then the ACCF ofψSbd1andψSbd2at the time shiftη2mp+τ(where0η<2p,ηZ,0τ<2mp,τZ) is

CψSbd1ψSbd2η2mp+τ=l=02k+11C(ψFbl,ψFblτi=12pηωΓ+q2d1ri+ηΓ+q2d2ri+l=02k+11C(ψFbl,ψFblτ2mpi=12pη1ωΓ+q2d1ri+η+1Γ+q2d2ri=CψCtψCtτi=12pηωΓ+q2d1ri+ηΓ+q2d2ri+CψCtψCtτ2mpi=12pη1ωΓ+q2d1ri+η+1Γ+q2d2ri=AψCtτi=12pηωr+q2d1ri+ηΓ+q2d2ri+AψCtτ2mpi=12pη1ωΓ+q2d1ri+η+1Γ+q2d2ri.E38

Ford1=d2=d, the ACCF given inEq. (38) reduced toAACFas follows:

AψSbdη2mp+τ=2m+kp+1×Σi=12pηωΓ+q2d1ri+ηΓ+q2d2ri,τ=0,0η<2p,0,0<τ<2mp,0η<2p.E39

Ford1d2, the ACCF given inEq. (38) can be expressed as

CψSbd1ψSbd2η2mp+τ=2m+kp+1×Σi=12pηωΓ+q2d1ri+ηΓ+q2d2ri,τ=0,0<η<2p,0,τ=0,η=0,0,0<τ<2mp,0η<2p.E40

The terms inEqs. (39) and (40) are derived from the autocorrelation properties of theCCψCt. It is also observed that the codes from the same code groupIthave ideal auto- and cross-correlation properties inside theZCZwidth2mp. Similarly, we can show that

AψS˜bdη2mp+τ=2m+kp+1×Σi=12pηωΓ+q2d2r¯iΓ+q2d1r¯i+η,τ=0,0η<2p,0,0<τ<2mp,0η<2p,E41

and ford1d2,

CψS˜bd1'ψS˜bd2η2mp+τ=2m+kp+1×Σi=12pηωΓ+q2d2r¯iΓ+q2d1r¯i+η,τ=0,0<η<2p,0,τ=0,η=0,0,0<τ<2mp,0η<2p.E42

FromEqs. (41) and (42), we get that the codes from the same code groupI2k+thave ideal auto- and cross-correlation properties inside theZCZwidth2mp.

Now we show that the ACCFs between any two codes of any two different code groupsIt1andIt20t1t2<2kare zeros everywhere. LetψSb1d1It1,ψSb2d2It2whereb1,b2are binary vector representations oft1,t2, andd1, d2are any two binary vectors inZ2p. Then

CψSb1d1ψSb2d2η2mp+τ=l=02k+11C(ψFb1l,ψFb2lτi=12pηωΓ+q2d1ri+ηΓ+q2d2ri+l=02k+11C(ψFb1l,ψFb2lτ2mpi=12pη1ωΓ+q2d1ri+η+1Γ+q2d2ri=CψCtψCtτi=12pηωΓ+q2d1ri+ηΓ+q2d2ri+CψCtψCtτ2mpi=12pη1ωΓ+q2d1ri+η+1Γ+q2d2ri=0τ,η.E43

Similarly, we can also show that the ACCFs between any two codes of any two different code groupsI2k+t1andI2k+t20t1t2<2kare zeros everywhere, i.e.,

CψS˜b1d1ψS˜b2d2η2mp+τ=l=02k+11C(ψFb1l,ψFb2lτi=12pηωΓ+q2d2riΓ+q2d1ri+η+l=02k+11C(ψFb1l,ψFb2lτ2mpi=12pη1ωΓ+q2d2riΓ+q2d1ri+η+1=CψC˜t1ψC˜t2τ2mpi=12pη1ωΓ+q2d2riΓ+q2d1ri+η+CψC˜t1ψC˜t2τ2mpi=12pη1ωΓ+q2d2riΓ+q2d1ri+η+1=0τ,η.E44

The results inEqs. (43) and (44) are obtained by using the ideal cross-correlation properties of CCCs. To complete the proof, now we only need to show that the ACCFs of any code fromItuandI2k+tvuvZ1uv2kare zeros everywhere. In this case, tuandtvare any two integers in02kand may or may not be equal. Letψ(Sbud1'Itu,ψS˜bvd2'I2k+tvwherebu,bvare binary vector representations oftu,tv, respectively. Then

CψSbud1ψS˜bvd2η2mp+τ=l=02k+11C(ψFbul,ψFbvlτi=12pηωΓ+q2d1ri+ηΓ+q2d2ri+l=02k+11C(ψFbul,ψFbvlτi=12pηωΓ+q2d1ri+ηΓ+q2d2ri=CψCtuψCtvτ2mpi=12pη1ωΓ+q2d1ri+ηΓ+q2d2ri+CψCtuψC˜tvτ2mpi=12pη1ωΓ+q2d1ri+η+1Γ+q2d2ri=0τ,η.E45

The above result is also obtained by using the ideal cross-correlation properties of CCCs. FromEqs. (39)-(45), we observed that theAACFsand ACCFs of the codes of the same group are zeros inside theZCZwidth2mpand the ACCFs of the codes from different code groups are zeros everywhere. Hence, wecan conclude thatI0,I1,,I2k+11form anIGCcode setI2k+p+12k+12m2mp.

Example2Letfbe aGBFoffour variables as given in Example1. Then the obtainedIGCcode setI8,4,16,8corresponding to theGBF fis given below:

Code group1:

I00=ψS00=ω1ω2ω1ω1ω2ω2ω0ω3ω2ω3ω2ω2ω3ω3ω1ω0ω1ω0ω1ω3ω2ω0ω0ω1ω2ω1ω2ω0ω3ω1ω1ω2ω1ω2ω1ω1ω0ω0ω2ω1ω2ω3ω2ω2ω1ω1ω3ω2ω1ω0ω1ω3ω0ω2ω2ω3ω2ω1ω2ω0ω1ω3ω3ω0I10=ψS01=ω1ω2ω1ω1ω2ω2ω0ω3ω0ω1ω0ω0ω1ω1ω3ω2ω1ω0ω1ω3ω2ω0ω0ω1ω0ω3ω0ω2ω1ω3ω3ω0ω1ω2ω1ω1ω0ω0ω2ω1ω0ω1ω0ω0ω3ω3ω1ω0ω1ω0ω1ω3ω0ω2ω2ω3ω0ω3ω0ω2ω3ω1ω1ω2.E46

Code group2:

I01=ψS10=ω1ω0ω1ω3ω2ω0ω0ω1ω2ω1ω2ω0ω3ω1ω1ω2ω1ω2ω1ω1ω2ω2ω0ω3ω2ω3ω2ω2ω3ω3ω1ω0ω1ω0ω1ω3ω0ω2ω2ω3ω2ω1ω2ω0ω1ω3ω3ω0ω1ω2ω1ω1ω0ω0ω2ω1ω2ω3ω2ω2ω1ω1ω3ω2I11=ψS11=ω1ω0ω1ω3ω2ω0ω0ω1ω0ω3ω0ω2ω1ω3ω3ω0ω1ω2ω1ω1ω2ω2ω0ω3ω0ω1ω0ω0ω1ω1ω3ω2ω1ω0ω1ω3ω0ω2ω2ω3ω0ω3ω0ω2ω3ω1ω1ω2ω1ω2ω1ω1ω0ω0ω2ω1ω0ω1ω0ω0ω3ω3ω1ω0.E47

Code group3:

I02=ψS˜00=ω0ω3ω1ω1ω0ω0ω3ω0ω1ω0ω2ω2ω1ω1ω0ω1ω2ω3ω3ω1ω2ω0ω1ω0ω3ω0ω0ω2ω3ω1ω2ω1ω0ω3ω1ω1ω2ω2ω1ω2ω1ω0ω2ω2ω3ω3ω2ω3ω2ω3ω3ω1ω0ω2ω3ω2ω3ω0ω0ω2ω1ω3ω0ω3I12=ψS˜01=ω2ω1ω3ω3ω2ω2ω1ω2ω1ω0ω2ω2ω1ω1ω0ω1ω0ω1ω1ω3ω0ω2ω3ω2ω3ω0ω0ω2ω3ω1ω2ω1ω2ω1ω3ω3ω0ω0ω3ω0ω1ω0ω2ω2ω3ω3ω2ω3ω0ω1ω1ω3ω2ω0ω1ω0ω3ω0ω0ω2ω1ω3ω0ω3.E48

Code group 4:

I03=ψS˜10=ω2ω3ω3ω1ω2ω0ω1ω0ω3ω0ω0ω2ω3ω1ω2ω1ω0ω3ω1ω1ω0ω0ω3ω0ω1ω0ω2ω2ω1ω1ω0ω1ω2ω3ω3ω1ω0ω2ω3ω2ω3ω0ω0ω2ω1ω3ω0ω3ω0ω3ω1ω1ω2ω2ω1ω2ω1ω0ω2ω2ω3ω3ω2ω3I13=ψS˜11=ω0ω1ω1ω3ω0ω2ω3ω2ω3ω0ω0ω2ω3ω1ω2ω1ω2ω1ω3ω3ω2ω2ω1ω2ω1ω0ω2ω2ω1ω1ω0ω1ω0ω1ω1ω3ω2ω0ω1ω0ω3ω0ω0ω2ω1ω3ω0ω3ω2ω1ω3ω3ω0ω0ω3ω0ω1ω0ω2ω2ω3ω3ω2ω3.E49

The correlation properties of I(8, 4, 16, 8) are described in Figure 2 where Figure 2a presents the absolute value of AACF sum of each code in I8,4,16,8, Figure 2b shows absolute value of ACCF sum between any two distinct codes from the same code group, and Figure 2c presents the absolute value of ACCF sum between any two distinct codes from different code groups.

## 4. Summary

In this chapter, we have presented a direct construction of IGC code set by using second-order GBFs. The AACF sidelobes of the codes of constructed IGC code set are zeros within ZCZ width, and the ACCFs of any two different codes of the same code group are zeros inside the ZCZ width, whereas the ACCFs of any two different codes from two different code groups are zeros everywhere. We have shown that there is a relation between our proposed construction and graph. The ZCZ width of the proposed IGC code set depends on the number of isolated vertices present in a graph after the deletion of some vertices. We also have shown that the ZCZ width of the proposed IGC code set by our construction is flexible and it can extend their applications. It is observed that most of the constructions given in literature are based on CCCs, whereas our construction can produce IGC code set directly.

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Written By

Palash Sarkar and Sudhan Majhi

Submitted: January 23rd, 2019 Reviewed: May 8th, 2019 Published: July 10th, 2019