Generalised class of cubic frequency-hoppingsequences with large family size

D.Y. Peng, T. Peng and P.Z. Fan

Abstract: Based on the cubic polynomials over finite field a generalised class of cubic frequency-hopping sequences with large family size suitable for applications in time/frequency hoppingCDMA systems, multiuser radar and sonar systems is proposed and investigated. It is shown thatthe new time/frequency hopping sequences have at most two hits in their autocorrelation functionsand at most three hits in their crosscorrelation functions except for a special case, and their familysize is much larger than the conventional cubic hopping sequences. To estimate the influence of thespecial case the probability of the occurrence of the special case is calculated. In addition, as animportant performance measure, the average number of hits for the generalised cubic hoppingsequences is also derived and discussed.

1 Introduction

The design of time/frequency hopping sequences suitablefor use in frequency hopping (FH) or time hopping (TH)CDMA systems, multiuser radar and sonar systems remainsof great interest in recent years [1, 2]. Although the natureand physical goals of various TH/FH systems such as ultra-wide-bandwidth (UWB) radio and multiuser radar are quitedifferent, however, the requirements imposed on thehopping signals are almost identical. In FH CDMAsystems the address assignment must be achieved in sucha way that there is no ambiguity about the sender and theinformation it transmits, and the received signal mustinterfere as negligible as possible to the reception of otherusers’ signals [1]. The mutual interference in FH CDMA ismainly controlled by the Hamming crosscorrelation of thehopping sequences [1, 2]. The need for finding FHsequences which have simultaneously good Hammingautocorrelation functions, small Hamming crosscorrelationfunctions and large family size is therefore well motivated.

Recently there has been an increasing interest in UWBTH-CDMA systems for both commercial and militaryapplications [2]. The UWB TH radio communicates withpulses of short duration, thereby spreading the energy of theradio signal very thinly over several gigahertz. For UWBTH-CDMA, due to the large bandwidth occupied, largenumbers of users can be supported. For users in crowdedspaces the most critical parameter of a wireless systemwould be its spatial capacity, measured as kbit/s/m2;typical examples are 0.005kbit/s/m2 (WCDMA/cdma200),1 (IEEE802.11b), 30 (Bluetooth), 83 (IEEE802.11a) and1000kbit/s/m2 (UWB). Compared with traditional wireless

systems, it is therefore, very important for UWB TH-CDMA to have a set of desirable hopping sequences thatcould provide a multiple access means for many users toshare the available bandwidth and, at the same time, havinglow multiple-access interference.

There are a number of hopping sequence families thatpossess both good Hamming auto- and crosscorrelationfunctions, such as one-coincidence sequences [3], linearcongruence sequences [4], quadratic hopping sequences [5],cubic sequences [6], cubic+linear congruence sequences [7],hyperbolic hopping sequences [8], extended hyperbolicsequences [9, 10], etc. In their Hamming crosscorrelationfunctions, one-coincidence sequences have at most one hit,quadratic sequences and hyperbolic sequences have at mosttwo hits, cubic hopping codes have at most three hits, andthe extended hyperbolic sequences having at most four hits.However, the family size of these sequence sets are quitedifferent and small, and the sets with relatively large familysize (approaching L2 for given sequence length L) arequadratic set, cubic + linear set, and extended hyperbolicset, etc. Unfortunately it is very difficult for the existinghopping sequences to satisfy the large capacity requirementin UWB TH-CDMA.

A generalised class of cubic hopping sequences with largefamily size is presented. The new construction is based onthe cubic polynomials over the finite field, and the newhopping sequences are called the generalised cubic hoppingsequences. As an important performance measure theaverage number of hits for the generalised cubic hoppingsequences is derived and discussed in detail.

2 Preliminaries

We first introduce the necessary notation. Let p be aprime number, F¼GF(p) be a finite field with elements{0, 1, y, p–1}. Let C¼ {c(1), c(2), y, c(M)} be a setof M hopping sequences of length L over a given finite fieldF, where

cðkÞ ¼ cðkÞ0 ; cðkÞ1 ; . . . ; cðkÞL�1

� �cðkÞi 2 F ði ¼ 0; 1; . . . ; L� 1Þðk ¼ 1; 2; . . . ; MÞ

D.Y. Peng and P.Z. Fan are with the Institute of Mobile Communications,Southwest Jiaotong University, Chengdu, Sichuan 610031, People’s Republic ofChina

T. Peng is with the Department of Information Sciences, School ofMathematical Sciences, Peking University, Beijing 100871, People’s Republicof China

E-mail: daiyuan_peng@hotmail.com

r IEE, 2005

IEE Proceedings online no. 20045005

doi:10.1049/ip-com:20045005

Paper first received 20th April 2004 and in final revised form 28th January 2005

IEE Proc.-Commun., Vol. 152, No. 6, December 2005 897

For any two such hopping sequences c(r) and c(s) theirperiodic Hamming crosscorrelation function H (r, s; t ) attime delay t ¼ 0, 1, y, L–1 is defined as follows:

H cðrÞ; cðsÞ; t� �

¼XL�1l¼ 0

h cðrÞl ; cðsÞlþt

� �ð1Þ

where

h cðrÞl ; cðsÞlþt

� �¼ 1; if cðrÞl ¼ cðsÞlþt

0; otherwise

�

and the subscript addition l+t is performed modulo L, and(1) becomes a periodic Hamming autocorrelation functionwhen r¼ s. For any given hopping sequence set C themaximum periodic Hamming autocorrelation sidelobeHa(C) and the maximum periodic Hamming crosscorrela-tion Hc(C) of C are, respectively, defined as

HaðCÞ ¼ maxfHðx; x; tÞjx 2 C; t ¼ 1; 2; . . . ; L� 1g

HcðCÞ ¼maxfHðx; y; tÞjx; y 2 C; x 6¼ y;

t ¼ 0; 1; . . . ; L� 1gHopping-sequence design normally involves five para-

meters: the number of frequencies or time slots q to behopped, period L of hopping sequence, the maximumnumber of transmitters M, the maximum Hammingcrosscorrelation value Hc(C) and the maximum Hammingautocorrelation sidelobe Ha(C). It is generally expectedthat both Ha(C) and Hc(C) are as small as possible and M isas large as possible due to the requirements of sequencedesign for FH-CDMA or TH-CDMA systems [1]. Inaddition, an important performance indicator of thehopping sequences is the average number of hits definedas follows.

Definition 1: Let C be a set of M hopping sequences oflength L over a given finite field F, then

SaðCÞ ¼X

x2C; 1�t�L�1Hðx; x; tÞ;

ScðCÞ ¼1

2

Xx; y 2C; x 6¼y; 0�t�L�1

Hðx; y; tÞ

are the overall number of auto-hits and cross-hitsrespectively. We call

AaðCÞ ¼SaðCÞ

MðL� 1Þ ð2Þ

AcðCÞ ¼2ScðCÞ

LMðM � 1Þ ð3Þ

the average number of Hamming autocorrelation and theaverage number of Hamming crosscorrelation of C,respectively. Obviously the average number of hits (orinterference) indicates the average error performance of thehopping systems. It is generally expected that both Aa(C)and Ac(C) are as small as possible [10]. However, generallyspeaking, it is very difficult to derive the average number ofhits for a given hopping sequence.

For simplicity and convenience we denote Sa¼Sa(C),Sc¼Sc(C), Aa¼ Aa(C) and Ac¼Ac(C).

Lemma 1: Let C be a set of M hopping sequences of lengthL over a given finite field F. ThenX

x;y 2C; t

Hðx; y; tÞ ¼ LM þ Sa þ 2Sc ð4Þ

Proof: We haveXx; y 2C; t

Hðx; y; tÞ ¼Xx2C

Hðx; x; 0Þ þX

x2C; t 6¼0Hðx; y; tÞ

þX

x 6¼y2C; t

Hðx; y; tÞ ¼ LMþSaþ2Sc

This completes the proof. G.E.D.Let f ðxÞ ¼ anxn þ an�1xn�1 þ � � � þ a1xþ a0 be a poly-

nomial of degree n over the finite field GF(p), we useN( f)¼N( f (x)¼ 0) to denote the number of solutions of theequation f (x)¼ 0 in GF(p). For a positive integer d, let Ed

be the set of all polynomials of degree smaller than or equalto d in GF(p). The following lemmas come from finite fieldstheory [11].

Lemma 2: Let f (x) be a polynomial of degree n over thefinite field GF(p), then N( f )rn.

Lemma 3: Let d be a positive integer with 0rdrp–1, thenXf 2Ed

Nðf ðxÞ ¼ 0Þ ¼ pdþ1

3 Generalised class of cubic hopping sequences

We now give the definition of the new class of hoppingsequences and derive the relevant properties.

Definition 2: Let f (x)¼ ix3+jx2+kx+l (i, j, k, l2GF(p)) bea polynomial of degree smaller than or equal to three overthe finite field GF(p). A generalised cubic hopping sequencec( f) determined by the cubic polynomial f (x) over the finitefield GF(p) is defined as

cðf Þ ¼ cðf Þ0 ; cðf Þ1 ; . . . ; cðf Þp�1

� �cðf Þn ¼ f ðnÞ ðmodpÞ; ðn ¼ 0; 1; . . . ; p � 1Þ

ð5Þ

C¼ {c( f) 7 f (x)¼ ix3+jx2+kx+l, i, j, k, l2GF(p)} is calledas a generalised cubic hopping sequence set over the finitefield GF(p).

To simplify the notation we write r¼ (i, j, k, l)¼ix3+jx2+kx+l and c(r)¼ c( f). For any two generalisedcubic hopping sequences c(r), c(s)2C, where r¼ (i1, j1 , k1, l1),s¼ (i2, j2, k2, l2)2F4, let H (r, s; t)¼H (i1, j1, k1, l1, i2, j2, k2,

l2; t) be their periodic Hamming correlation function,then

Theorem 1: The generalised cubic hopping sequence set Chave the following properties:

(1) M¼ p4, L¼ q¼ p.

(2) For any two generalised cubic hopping sequences c(r),c(s)2C, where r¼ (i1, j1, k1, l1), s¼ (i2, j2, k2, l2)2F4, wehave

(2.1) If i1a i2, then H (r, s; t )r3.

(2.2) If [i1¼ i2¼ 0 and j1a j2] or [i1¼ i2a0 and ta( j1�j2)(3i1)

–1], then H (r, s; t)r2.

(2.3) If [i1¼ i2¼ j1¼ j2¼ 0 and k1ak2] or [i1¼ i2¼ 0 andj1¼ j2a0 and ta (k1�k2)(2j2)

–1] or [i1¼ i2a0 and t¼( j1–j2)(3i1)

–1 and j21 � j22 þ 3i2ðk2 � k1Þ 6¼ 0], then H (r, s;t)¼ 1.

(2.4) If [i1¼ i2¼ j1¼ j2¼ 0 and k1¼ k2 and k2t+l2�l1a0]or [i1¼ i2¼ 0 and j1¼ j2a0 and t¼ (k1�k2)(2j2)

–1 andk12–k2

2+4j2(l2�l1)a0] or [i1¼ i2a0 and t¼ ( j1�j2)(3i1)–1

and j21 � j22+3i2(k2�k1)¼ 0 and j2( j1�j2)2+3i2(k1+2k2)

( j1�j2) þ27ðl2 � l1Þi22 6¼ 0], then H (r, s; t)¼ 0.

898 IEE Proc.-Commun., Vol. 152, No. 6, December 2005

(2.5) If [i1¼ i2¼ j1¼ j2¼ 0 and k1¼ k2 and k2t+l2�l1¼ 0]or [i1¼ i2¼ 0 and j1¼ j2a0 and t¼ (k1�k2)(2j2)

–1 andk21 � k22+4j2(l2�l1)¼ 0] or [i1¼ i2a0 and t¼ ( j1�j2)

(3i1)–1 and j21 � j22 þ 3i2ðk2 � k1Þ ¼ 0 and j2( j1�j2)

2+

3i2(k1+2k2)( j1�j2)+27(l2�l1) i22 ¼ 0], then H(r, s; t )¼ p.

Proof: The proof of (1) is straightforward. We have bydefinition 2

H cðrÞ; cðsÞ; t� �

¼Xp�1n¼ 0

h cðrÞn ; cðsÞnþt

� �

¼Xp�1n¼ 0

h 0; cðsÞnþt � cðrÞn

� �¼ Nðf ðnÞ ¼ 0Þ

where f (n)¼ a3n3+ a2n

2+a1n+a0 and

a3 ¼ i2 � i1a2 ¼ 3i2tþ j2 � j1a1 ¼ 3i2t2 þ 2j2tþ k2 � k1a0 ¼ i2t3 þ j2t2 þ k2tþ l2 � l1

8>><>>: ðmodpÞ ð6Þ

Case 1: i1a i2. Using lemma 2 we have H (r, s; t)r3because the degree of polynomial f (n) is now three.

Case 2: i1¼ i2¼ 0. Then f (n)¼ a2n2+a1n+a0, where

a2 ¼ j2 � j1a1 ¼ 2j2tþ k2 � k1a0 ¼ j2t2 þ k2tþ l2 � l1

8<: ðmodpÞ

Case 2.1: i1¼ i2¼ 0 and j1a j2. We have H (r, s; t)r2because the degree of polynomial f (n) is now two (quadraticcase).

Case 2.2: i1¼ i2¼ 0 and j1¼ j2. Then f (n)¼ a1n+a0, where

a1 ¼ 2j2tþ k2 � k1a0 ¼ j2t2 þ k2tþ l2 � l1

�ðmodpÞ

Case 2.2.1: i1¼ i2¼ 0 and j1¼ j2¼ 0. Then f (n)¼ a1n+a0,where

a1 ¼ k2 � k1a0 ¼ k2tþ l2 � l1

�ðmodpÞ

Case 2.2.1.1: i1¼ i2 ¼ 0, j1¼ j2¼ 0 and k1ak2. we haveH (r, s; t)¼ 1 because the degree of polynomial f (n) isnow 1.

Case 2.2.1.2: i1¼ i2¼ 0, j1¼ j2¼ 0 and k1¼ k2. We havef (n)¼ k2t+l2–l1. Hence, if

i1 ¼ i2 ¼ j1 ¼ j2 ¼ 0

k1 ¼ k2k2tþ l2 � l1 6¼ 0 ðmodpÞ

then we have that H(r, s; t)¼ 0 because that equationf (n)¼ 0 now has no solution. If

i1 ¼ i2 ¼ j1 ¼ j2 ¼ 0

k1 ¼ k2k2tþ l2 � l1 ¼ 0 ðmodpÞ

we get H(r, s; t)¼ p because that the equation f (n)¼ 0 nowbecomes a equality.

Case 2.2.2: i1¼ i2 ¼ 0 and j1¼ j2a0.

Case 2.2.2.1: i1¼ i2¼ 0 and j1¼ j2a0 and ta (k1–k2) (2j2)–1

(mod p). We have H (r, s; t)¼ 1 because the degree ofpolynomial f (n) is now 1.

Case 2.2.2.2: i1¼ i2¼ 0 and j1¼ j2a0 and t¼ (k1–k2) (2j2)–1

(mod p). Then

f ðnÞ ¼ j2t2 þ k2tþ l2 � l1

¼ ð4j2Þ�1fk21 � k22 þ 4j2ðl2 � l1ÞgThus if

i1 ¼ i2 ¼ 0j1 ¼ j2 6¼ 0t ¼ ðk1 � k2Þð2j2Þ�1k21 � k22 þ 4j2ðl2 � l1Þ 6¼ 0

8>><>>: ðmodpÞ

we have that H(r, s; t)¼ 0 because the equation f (n)¼ 0now has no solution. If

i1 ¼ i2 ¼ 0j1 ¼ j2 6¼ 0t ¼ ðk1 � k2Þð2j2Þ�1k21 � k22 þ 4j2ðl2 � l1Þ ¼ 0

8>><>>: ðmodpÞ

we get H(r, s; t)¼ p because the equation f (n)¼ 0 becomesnow a equality.

Case 3: i1¼ i2a0.

Case 3.1: i1¼ i2a0 and ta ( j1–j2)(3i2)–1 (mod p). Then we

have f (n)¼ a2n2+a1n+a0 (a2a0) by (6), and H (r, s; t)r2

because the degree of polynomial f (n) is now two.

Case 3.2: i1¼ i2a0, t¼ ( j1–j2)(3i2)–1 (mod p) and j21 � j22 þ

3i2ðk2 � k1Þ 6¼ 0 (mod p). Then we have f (n)¼ a1n+a0(a1a0), and H(r, s; t)¼ 1 because the degree of polynomialf (n) is now 1.

Case 3.3: i1¼ i2a0, t¼ ( j1–j2)(3i2)–1 (mod p), j21 � j22þ

3i2ðk2 � k1Þ ¼ 0 ðmodpÞ, and

j2ðj1 � j2Þ2 þ 3i2ðk1 þ 2k2Þðj1 � j2Þþ 27ðl2 � l1Þi22 6¼ 0 ðmodpÞ

Then we have f (n)¼ a0(a0a0), and H(r, s; t)¼ 0 becausethe equation f (n)¼ 0 now has no solution in GF(p).

Case 3.4: i1¼ i2a0, t¼ ( j1�j2)(3i2)–1 (mod p), j21 � j22 þ

3i2ðk2 � k1Þ ¼ 0 ðmodpÞ; and

j2ðj1 � j2Þ2 þ 3i2ðk1 þ 2k2Þðj1 � j2Þþ 27ðl2 � l1Þi22 ¼ 0 ðmod pÞ

Then we get H(r, s; t)¼ p because the equation f (n)¼ 0becomes now a equality.

This completes the proof. Q.E.D.

Table 1 summarises the distribution of H(r, s; t) with respectto i1, i2, j1, j2, k1, k2, l1 and l2.

Early in 1990, S.V. Maric [6] defined a cubic hoppingsequence set C on a finite field GF(p) as following

C ¼ fcðiÞj1 � i � p � 1g

cðiÞ ¼ ðcðiÞ0 ; cðiÞ1 ; . . . ; cðiÞp�1Þ

cðiÞn i� n3 ðmodpÞð0 � n � p � 1ÞThe cubic hopping sequences have the following character-istics,

M ¼ p � 1; L ¼ q ¼ p; Ha ¼ 2; Hc ¼ 3

Later, in 1997, Jovancevic and Titlebaum [7] gavea combined cubic + linear congruence hopping

IEE Proc.-Commun., Vol. 152, No. 6, December 2005 899

sequences, i.e.

C ¼ fcði;jÞj1 � i � p � 1; 0 � j � p � 1g

cði; jÞ ¼ ðcði;jÞ0 ; cði;jÞ1 ; . . . ; cði;jÞp�1Þcði;jÞn ¼ i� n3 þ j� n ðmodpÞð0 � n � p � 1Þ

with the following characteristics:

M ¼ ðp � 1Þp; L ¼ q ¼ p; Ha ¼ 2; Hc ¼ 3

Clearly, for the same sequence length and maximumcorrelation value, the proposed generalised cubic hoppingsequences have a much larger size. For comparison, therelated parameters for several known cubic hoppingsequence sets are listed in Table 2. It can be concludedthat the generalised cubic hopping sequences presentedhave much lager family size than the existing hoppingsequences.

4 Probability of the occurrence of special case

For each pair of different generalised cubic hoppingsequences satisfying the condition (2.5) in theorem 1, thereexists one and only one value of t determined by condition(2.5) in theorem 1 for which the two sequences have exactlyp hits. In this special case ( full collision), the messagesassociated with the two collided sequences cannot bereceived and the two transmitters must retransmit themessage with a different delay t. This must be dealt at thedata link control (DLC) level. To do so, a certain amount ofsignalling must be spent for message recovering, with theconsequence of a higher delay [10].

Let C be a set of M generalised cubic hopping sequencesof length L over a given finite field GF(p), Fc(C) be the totalnumber of possible full collisions of sequence set C. Sincethat LM(M–1)/2 is the total number of possible cross-

correlation values, thus

PcðCÞ ¼2FcðCÞ

LMðM � 1Þ ð7Þ

is the probability of the occurrence of the special case.

Theorem 2: Let C be a generalised cubic hopping sequenceset over a finite field F¼GF(p), we have

PcðCÞ ¼2ðp3 � 1Þ

p4ðp3 þ p2 þ p þ 1Þ ð8Þ

Proof: We use the notations introduced in theorem 1. LetFc(C) be the total number of possible full collisions ofsequence set C. Based on the condition (2.5) in theorem 1we obtain the formula Fc(C)¼F1(C)+F2(C)+F3(C), whereF1(C), F2(C) and F3(C) determined by the followingconditions (2.5.1), (2.5.2) and (2.5.3), respectively:

(2.5.1) i1¼ i2¼ j1¼ j2¼ 0 and k1¼ k2 and k2t+l2–l1¼ 0

(2.5.2) i1¼ i2¼ 0 and j1¼ j2a0 and t¼ (k1–k2)(2j2)–1 and

k12–k2

2+4j2(l2–l1)¼ 0and(2.5.3) i1¼ i2a0 and t¼ ( j1–j2)(3i1)

–1 and j12–j2

2+3i2(k2–k1)¼ 0 and j2( j1–j2)

2+3i2(k1+2k2)( j1–j2)+27(l2–l1)i22 ¼ 0

We derive first the formula of F1(C) which depends oncondition (2.5.1). If k1¼ k2¼ 0, we have l2¼ l1 and c(r)¼ c(s)

by (2.5.1). If k1¼ k2a0 and l1¼ l2, then c(r)¼ c(s). Ifk1¼ k2a0 and l1a l2, we have t¼ (l1–l2)(k2)

–1 by (2.5.1).Because there exist exactly (p–1) pairs of (k1, k2) satisfyingk1¼ k2a0, there exist exactly p(p–1) pairs of (l1, l2)satisfying l1a l2, thus F1(C) equals to p(p–1)2.

We establish now the formula of F2(C) which depends onthe condition (2.5.2). It is obvious that there exists only apair of (i1, i2) satisfying i1¼ i2¼ 0, and there exist exactly

Table 1: Distribution of H (r, s; s) with respect to i1, i2, j1, j2, k1, k2, l1 and l2

Cases H(r, s; t)

i1a i2 r3

i1¼ i2 i1¼ 0 j1a j2 r2

j1¼ j2 j1¼ 0 k1ak2 ¼ 1

k1¼ k2 k2t+ l2 – l1a0 ¼ 0

k2t+ l2 – l1¼0 ¼p

j1a0 ta (k1 –k2) (2 j2)–1 ¼ 1

t¼ (k1 –k2) (2 j2)–1 k1

2 –k22 +4j2(l2 – l1)a0 ¼ 0

k12 –k2

2 +4j2(l2 – l1)¼ 0 ¼p

i1a0 ta ( j1 – j2) (3i1)–1 r2

t ¼ ( j1 – j2) (3i1)–1 j1

2 – j22 +3i2(k2 –k1)a0 ¼ 1

j12 – j2

2 +3i2(k2 –k1)¼0 j2( j1 – j2)2 +3i2(k1 +2k2) ( j1 – j2)+27(l2 – l1)i2

2a0 ¼ 0

j2( j1 – j2)2 +3i2(k1 +2k2) ( j1 – j2)+27(l2 – l1)i2

2¼0 ¼p

Table 2: Family size of various cubic hopping sequences

P 3 5 7 11 13 17 19 23

Cubic hopping sequences 2 4 6 10 12 16 18 22

Cubic+linear hopping sequences 6 20 42 110 156 272 342 506

Generalised cubic hopping sequences 81 625 2401 14641 11661 83521 130321 279841

900 IEE Proc.-Commun., Vol. 152, No. 6, December 2005

(p–1) pairs of ( j1, j2) satisfying j1¼ j2a0. If k1¼ k2, wehave t¼ 0, l2¼ l1 and c(r)¼ c(s) by (2.5.2). If k1ak2 andj2a0 are given, using condition (2.5.2), we have t¼(k1�k2)(2j2)

–1 and l2¼ l1+(k22�k1

2)(4j2)–1, thus that there

exist exactly p pairs of (l1, l2). Because there exist exactlyp(p–1) pairs of (k1, k2) satisfying k1ak2, thus F2(C) equalsto p2(p–1)2.

We derive finally the formula of F3(C) which depends oncondition (2.5.3). There exist exactly (p–1) pairs of (i1, i2)satisfying i1¼ i2a0. Based on condition (2.5.3) we havet¼ 0, k1¼ k2, l2¼ l1 and c(r)¼ c(s) if j1¼ j2, andk2¼ k1+ðj22 � j21Þ(3i2)

–1 and l2¼ l1+[3i2(k1+2k2)( j2–j1)–

j2( j2–j1)2]ð27i22Þ

�1 if j1a j2. Therefore there exist exactly ppairs of (k1, k2) and p pairs of (l1, l2) when j1a j2 and i2a0are given. Because that there exist exactly p(p–1) pairs( j1, j2) satisfying j1a j2, thus F3(C) equals to p3(p–1)2.

Therefore we obtain

FcðCÞ ¼ F1ðCÞ þ F2ðCÞ þ F3ðCÞ¼ pðp � 1Þ2 þ p2ðp � 1Þ2 þ p3ðp � 1Þ2

¼ pðp � 1Þ2ðp2 þ p þ 1Þ

PcðCÞ ¼2FcðCÞ

LMðM � 1Þ ¼2pðp � 1Þ2ðp2 þ p þ 1Þ

p � p4ðp4 � 1Þ

¼ 2ðp3 � 1Þp4ðp3 þ p2 þ p þ 1Þ

and the desired result follows. G.E.D.As an example, when p¼ 13 we have Pc(C)¼

0.0000646118. That is, the probability of the special caseoccurrence is extremely small.

5 Derivation of average number of hits

As for the average number of hits of C, we have thefollowing results.

Theorem 3: Let C¼ {c( f)} be a set of generalised cubichopping sequences over the finite field GF(p). Then theaverage number Aa(C) of Hamming autocorrelation andthe average number Ac(C) of Hamming crosscorrelation ofC are given as follows, respectively:

AaðCÞ ¼ 1

AcðCÞ ¼p4 þ p3 þ p2 þ p � 1

p4 þ p3 þ p2 þ p

ð9Þ

Proof: Given i2, j2, k2, l2, t¼ 0, 1,y, p–1, it is noted that a3,a2, a1 and a0 run independently over all elements in GF(p)as i1, j1, k1 and l1 take independently all elements in GF(p)by (6). Thus we getX

r; s; t

Hðr; s; tÞ ¼X

i2; j2; k2; l2;t

Xi1; j1; k1; l1

Nðf ðnÞ ¼ 0Þ

¼X

i2; j2; k2; l2;t

p4 ¼ p9

by lemma 3, andXr; s; t

Hðr; s; tÞ ¼ p5 þ Sa þ 2Sc

by lemma 1. Therefore

Sa þ 2Sc ¼ p9 � p5 ð10ÞFor any hopping sequence c(r)2C where r¼

(i, j,k, l)2F4, its Hamming autocorrelation function canbe written as H(r, r; t)¼H(i, j,k, l; t)¼N(g(n)¼ 0), where

g(n)¼ b2n2+b1n+b0 and

b2 ¼ 3itb1 ¼ 3it2 þ 2jtb0 ¼ it3 þ jt2 þ kt

8<: ðmodpÞ ð11Þ

For any ta0, it is noted that b2, b1 and b0 runindependently over all elements in GF(p) as i, j and k takeindependently all elements in GF(p) by (11). Therefore bylemma 3 we have

Sa ¼Xr; t 6¼0

Hðr; r; tÞ ¼Xl; t 6¼0

Xi; j; k

Hði; j; k; l; tÞ

¼Xl; t 6¼0

Xi; j;k

NðgðnÞ ¼ 0Þ ¼X

l; t 6¼0p3 ¼ p4ðp � 1Þ

Thus we have Sc¼ p4(p5–2p+1)/2 by (10). Noticing thatM¼ p4 and L¼ p, hence we obtain now by definition 1

Aa ¼p4ðp � 1ÞMðL� 1Þ ¼ 1

Ac ¼p5 � 2p þ 1

LMðM � 1Þ ¼p5 � 2p þ 1

p5 � p

Therefore the desired result follows. G.E.D.It is obvious that the generalised cubic hopping sequence

c(r) becomes a trivial constant hopping sequence which haspoor correlations when r¼ (0, 0, 0, l ). Let D¼ {d(r)} be ahopping sequence set which are obtained by deleting allconstant sequences from the generalised cubic hoppingsequence set C, that is D¼ {c(r)2C 7 ra (0, 0, 0, l)}.

Theorem 4: The hopping sequences set D have the followingproperties:

(i) M¼ p4– p and L¼ q¼ p.

(ii) The average number Aa(D) of Hamming autocorrela-tion and the average number Ac(D) of Hamming cross-correlation of D are given as follows, respectively,

AaðDÞ ¼p2 þ p

p2 þ p þ 1

AcðDÞ ¼p6 þ p5 þ p4 � p3 � 3p2 � 2pp6 þ p5 þ p4 � p3 � 2p2 � 1

ð12Þ

Proof: By using the notations and results in the proofs oftheorems 1 and 2, it is noted that there exist exactly pconstant hopping sequences. Thus the proof of (i) isstraightforward.

For any constant hopping sequence c(r)¼ (l, l, y, l), itsHamming autocorrelation functions H (c(r), c(r); t)¼ p foreach value of t. Therefore we have

SaðDÞ ¼ SaðCÞ � p2ðp � 1Þ ¼ p2ðp � 1Þðp2 � 1Þ

AaðDÞ ¼SaðDÞ

ðp4 � pÞðp � 1Þ ¼p2 þ p

p2 þ p þ 1

For any two different constant hopping sequencesc(r)¼ (l, l,y, l) and c(s)¼ (h,h,y,h), their Hamming cross-correlation functions H(c(r), c(s); t)¼ 0 for each value of t.

Let a constant hopping sequence c(s)¼ (h,h,y,h) begiven. For any generalised cubic hopping sequence c(r)2C,where r¼ (i, j, k, l)2F4, the Hamming correlation functionsof c(r) and c(s) can be written as H(r, s; t)¼N( f (n)¼ 0),where f (n)¼ in3+jn2+kn+(l–h). Thus by lemmas 3 and 1we getX

r; t

Hðr; s; tÞ ¼Xt

Xi; j; k; l

Nðf ðnÞ ¼ 0Þ ¼ p5

IEE Proc.-Commun., Vol. 152, No. 6, December 2005 901

On the other hand, we also have the equation

p9 ¼Xr; s; t

Hðr; s; tÞ ¼X

r¼ð0; 0; 0; l1Þ; s; tHðr; s; tÞ

þX

r; s¼ð0; 0; 0; l2Þ ; tHð r; s; tÞ

�X

r¼ð0; 0; 0; l1Þ ;s¼ð0; 0; 0; l2Þ; tHð r; s; tÞ

þX

r 6¼ð0; 0; 0; l1Þ; s 6¼ð0; 0; 0; l2Þ; tHðr ; s; tÞ

¼ 2p6 �X

r¼ð0; 0; 0; l1Þ ; tHðr; r; tÞ

þ pðp4 � pÞ þ SaðDÞ þ 2ScðDÞ¼ 2p6 � p3 þ pðp4 � pÞ þ SaðDÞ þ 2ScðDÞ

Therefore

SaðDÞ þ 2ScðDÞ ¼ p9 � 2p6 � p5 þ p3 þ p2

¼ p2ðp3 � 1Þðp4 � p � 1Þ

ScðDÞ ¼1

2p3ðp � 1Þðp5 þ p4 þ p3 � p2 � 3p � 2Þ

AcðDÞ ¼2ScðDÞ

LMðM � 1Þ

¼ p6 þ p5 þ p4 � p3 � 3p2 � 2pp6 þ p5 þ p4 � p3 � 2p2 � 1

and the desired result follows. G.E.D.

6 Concluding remarks

A new class of time/frequency hopping sequences has beenproposed based on cubic polynomial theory. It has beenshown that these sequences possess good correlationproperties and have larger family size than the well-knowcubic hopping sequences. Moreover, the occurrence of thespecial case on these sequences are established.

The average number of hits for the hopping sequences isan important performance measure for UWB TH-CDMAsystems. As far as we are aware, the average number of hitshas not hither to been known analytically for Maric’s cubichopping sequences, Jovancevic–Titlebaum’s cubic hopping

sequences, quadratic hopping sequences, hyperbolic hop-ping sequences, extended hyperbolic hopping sequences, etc.In this paper the average number of hits on these newsequences are derived analytically.

7 Acknowledgments

This work was supported by the National ScienceFoundation of China (NSFC) and the Research GrantsCouncil of Hong Kong (RGC) joint research scheme(60218001/N_HKUST617/02), NSFC 60302015/60472089,the Foundation for the Authors of National ExcellentDoctoral Dissertation of PR China (FANEDD)under grant 200341, the National Key Laboratory ofCommunications (UESTC), and the Youth ScienceFoundation of Sichuan Provincial Science and TechnologyDepartment.

8 References

1 Fan, P.Z., and Darnell, M.: ‘Sequence design for communicationsapplications’ (Research Studies Press, Wiley, London, 1996)

2 Win, M.Z., and Scholtz, R.A.: ‘Ultra-wide bandwidth time-hoppingspread-spectrum impulse radio for wireless multiple-access commu-nications’, IEEE Trans. Commun., 2002, 58, (4), pp. 679–691

3 Shaar, A.A., and Davies, P.A.: ‘A survey of one-coincidence sequencesfor frequency-hopped spread-spectrum systems’, IEE Proc. F, 1984,131, (7), pp. 719–724

4 Titlebaum, E.L.: ‘Time–frequency hop signals part I: coding basedupon the theory of linear congruences’, IEEE Trans. Aerosp. Electron.Syst., 1981, 17, (4), pp. 490–493

5 Titlebaum, E.L.: ‘Time–frequency hop signals part II: coding basedupon quadratic congruences’, IEEE Trans. Aerosp. Electron. Syst.,1981, 17, (4), pp. 494–499

6 Maric, S.V.: ‘Frequency-hop multiple-access codes based on thetheory of cubic congruences’, IEEE Trans. Aerosp. Electron. Syst.,1990, 26, (6), pp. 1035–1039

7 Jovancevic, A.V., and Titlebaum, E.L.: ‘A new receiving technique forfrequency-hopping CDMA systems: analysis and application’. Proc.of 47th IEEE Conf. on Vehicular Technology, May 1997, Vol. 3,pp. 2187–2190

8 Maric, S.V., and Titlebaum, E.L.: ‘A class of frequency-hop codeswith nearly ideal characteristics for use in multiple-access spreadspectrum communications and radar and sonar systems’, IEEE Trans.Commun., 1992, 40, (9), pp. 1442–1447

9 Iacobucci, M.S., and Benedetto, M.G.D.: ‘Time-hopping codes inimpulse radio multiple-access communication systems’. Proc. of Int.Symp. on 3G Infrastructure and Services, Athens, Greece, 2–3 July2001

10 Iacobucci, M.S., and Benedetto, M.G.D.: ‘Multiple-access design forimpulse radio communication systems’. Proc. of IEEE Int. Conf. onCommunications, 2002, Vol. 4, pp. 817–820

11 Lidl, R., and Niederreiter, H.: ‘Finite fields’ (Cambridge UniversityPress, 1997)

902 IEE Proc.-Commun., Vol. 152, No. 6, December 2005