IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 48, NO. 8, AUGUST 2000 1241

Transactions Letters________________________________________________________________

A New Family of Frequency-Hop CodesOscar Moreno, Fellow, IEEE,and Svetislav V. Maric

Abstract—We give an algebraic construction for a new familyof frequency-hop codes. The construction is based on properties offinite fields; it is shown that for each field GF( ), there exists alarge number of codes of length . The codes are also shown topossess the best possible simultaneous two-dimensional autocorre-lation and cross-correlation properties. Moreover, they include afamily of codes with a code length of a power of 2, which are ide-ally suitable for applications in digital communication systems.

I. INTRODUCTION

CONSTRUCTION of frequency-hop (FH) codes for usein various multiuser communication systems has always

been as much art as science [1]. In fact, the literature on theconstruction of FH codes [2]–[5] utilizes all possible tools offinite field theory, linear algebra, number theory, graph theory,etc.

In this paper, we give an algebraic construction of a new FHcode family. As it will be seen below, the constructed codes pos-sess the best possible simultaneous two-dimensional (2-D) ape-riodic autocorrelation (ACF) and cross-correlation (CCF) func-tions (as defined in [1]–[5]), and also have a large number ofsequences of equal lengths for all Galois fields (GF).

We first review the well-known FH code families in orderto compare them with the newly constructed codes. Of mostpractical value are the families of full FH codes [1].

Definition 1: An full FH code is a code in which eachof the time slots and frequency channels is utilized onceand only once.

In other words, if an FH code of length is represented by asequence of integers of length, then the FH code sequence ofa full FH code is a permutation of . Note that the permutationof size , is represented by a permutationmatrix of size , with a property that columnhas a dotin

An example of two full FH codes of size based on theconstruction in this paper is shown in Fig. 1.

Definition 2: A Costas array is an permutation ma-trix—FH code—with the property that vectors connecting two

Paper approved by G. Caire, the Editor for Multiuser Detection and CDMA ofthe IEEE Communications Society. Manuscript received June 29, 1999; revisedJune 1, 2000.

O. Moreno is with the Department of Mathematics, University of Puerto Rico,San Juan, Puerto Rico.

S. V. Maric was with Cambridge University, Cambridge, U.K. He is now withQualcomm, Inc., San Diego, CA 92121 USA.

Publisher Item Identifier S 0090-6778(00)07103-8.

dots of the matrix are all distinct vectors (that is no two vectorsare equal in both amplitude and slope).

Costas arrays have ideal 2-D autocorrelation functions (ACF)and hence are the optimum choice for single-user radar andsonar systems.

Costas arrays—although not under that name—were intro-duced for application in mobile radio by Einarsson [4]. In thatpaper, Einarsson also shown that in the synchronous case (timeshift is zero) Costas arrays have good 2-D CCF’s. However, ithas been shown [6] that for other shifts Costas arrays do nothave ideal 2-D CCF, in fact it is shown in [7] that Costasarrays can have as many as hits for certain shifts in their2-D CCF. Also, they have at most two hits in the 2-D CCF (nextbest possible case) only in pairs.

Another family of FH codes, called linear congruence (LC)FH codes, has completely different correlation properties fromCostas arrays. Namely, it possesses ideal (at most one hit forany time-frequency shift) 2-D CCF, but rather poor 2-D ACF.In fact, they can have as many as , a prime, hits in the2-D ACF. LC codes are constructed by

mod

(1)

For , the LC code has hits for the time shift of. LC codes for radar and sonar systems were introduced

in [8] and for multiple-access mobile radio in [9].Based on these two extreme examples, it is obvious that a

family of codes that will trade either ideal 2-D autocorrelationor cross-correlation properties (one hit in their 2-D ACF and 2-DCCF, respectively), for nearly ideal simultaneous properties (atmost two hits in both their 2-D ACF and 2-D CCF) is needed.The best example of such codes are the Hyperbolic Congruence(HC) FH codes, which have two hits in both the 2-D ACF andCCF [5]. The HC FH codes exist for primes and are constructedusing

mod

(2)

Useful tools for analyzing FH code properties are the auto-and cross-hit array. The auto- and cross-hit array are a discreterepresentation of the 2-D ACF and CCF, respectively, and areobtained by shifting one FH code array with respect to another inthe left–right and up–down direction and observing the number

0090–6778/00$10.00 © 2000 IEEE

1242 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 48, NO. 8, AUGUST 2000

Fig. 1. Example of two 9� 9 FH codes.

TABLE ICOMPARISONBETWEEN WELL-KNOWN FULL FH CODES AND NEWLY CONSTRUCTEDCODES

of hits for each shift.1 For instance, we see that by shifting thefirst code 2 horizontally to the right and 2 vertically, we havetwo simultaneous hits caused by the elements at position (2, 4)and (4, 5) of the first code coinciding with the elements (4, 6)and (6, 7), respectively, of the second code.

Another tool in establishing the correlation properties of FHcodes is the difference function.

Definition 3: The nonperiodic difference function for twoFH codes and is given by

for

(3)

When the difference function is equal to zero for a certain timeand Doppler shift , a hit between two FH codes occurs.Thanks to this property, and using Lagrange’s theorem—thatthe maximum number of solutions of a polynomial in the finitefield is equal to the power of that polynomial—we can establishthe maximum number of hits in the 2-D ACF and CCF of mostalgebraically constructed FH codes.

The importance of having good 2-D cross-correlation prop-erties comes from the design of a fast FH-CDMA system, orig-inally proposed by Viterbi [10]. In a fast FH-CDMA system,each user has the entire bandwidth available and in order to berecognized by the receiver is assigned an address—its uniqueFH code. The messages are then added to the address, thus cre-ating a frequency (vertical) shift. Note that the message is usu-ally broken into slots of length bits hence it is important to

1Since the 2-D ACF is symmetric, in Fig. 3, we plot just the right-hand side ofthe auto-hit array, and also all the empty spaces indicate that for that particularshift there were no hits.

have FH codes of the same length (so far families of FHcodes oflength and also possessing best possible 2-D ACFand CCF were not known).

The interface effects in this type of system (aside from thenoise) come from the simultaneous use of the same frequencychannel. This event is referred to as a coincidence or a hit, andtherefore the number of hits in the 2-D ACF and 2-D CCF isused as a measure of how good is the FH code. The receiveris not able to distinguish between the hit and the single use ofthe frequency channel; hence the greater the number of hits thegreater the probability of errors [5].

In Table I, we summarize the 2-D autocorrelation and cross-correlation properties of well-known full FH codes.

II. NEW FH CODE FAMILY : CONSTRUCTION ANDPROPERTIES

Let be a finite field with elements, where is a primenumber and is an integer. Let be the projective line over ,in other words, is also a field. If we consider

, where and , thenit is known that substitution of elements ofin producesa permutation of the elements of.

Furthermore, if is another fractional linear transforma-tion, similar to , then there are exactly two values ofinfor which . We now invoke the following result ofBerlekamp–Moreno [11].2

Theorem 1: Whenever is irreducible, and primi-tive in , then the polynomial with as a variablegives a cycle (permutation) of length .

REMARK: This is a crucial theorem for the paper, since itestablishes when the polynomial is going to give a full code.

2The proof is also given in [11].

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 48, NO. 8, AUGUST 2000 1243

We assume now that the cycle given bybegins with 0 and ends with , and since , it goes

.Example 1: gives for the cycle

.Example 2: We can plot the fractional linear transformation

using the vertical and horizontal ordering of Example 1 toobtain the permutation matrix below; notice that “0” maps tosince , then “ ” maps to since ,similarly 1 maps to 1 and to 0.

Note that, in general, we can plot permutation matrices corre-sponding to any using the horizontal and vertical orderinggiven by . Also, when plotted,gives a permutation matrix corresponding to a horizontal peri-odical right shift of with period . Similarly,gives us a vertical, up one unit periodical shift. Since composi-tion of fractional linear transformations gives us also fractionallinear transformations, by repeating the above process we canobtain periodical shifts of more units.

Example 3: By definition, a composition of any functionon is a function defined as . Using Example

2, take the function and compose it with. Then, we have . Clearly,

we have another linear transformation since 0 maps to 0; 1 toto , and to 1.

Notice also that in this paper we consider the arrays in Exam-ples 2 and 3 equivalent (one is the shift of another).

Now, recall that there are exactly two values ofin forwhich for any two fractional linear transforma-tions which are different. From this, we obtain that if we con-sider the set of all fractional linear transformations obtainedfrom by the above shifting process (up and right) and if

is not in this set, then the cross correlation of the arraysgiven by and is two. In fact, we can prove Theorem 2by considering the class of inequivalent fractional linear trans-formation under the shifting equivalence (defined above).

Theorem 2: There are distinct fractional linear trans-formations—and thus distinct FH codes giving doubly periodicarrays of length , with autocorrelation and cross-corre-lation value of two.

Proof: Our proof is based upon the fact that composingtwo linear fractional transformation as in Example 3 gives an-other linear fractional transformation. Also, the CCF of twolinear fractional transformations—acting on the projective lineas in Example 2—given by and

is equal to solving the equa-tion on the projective line. By cross-multiplyingthe last equation, we obtain a quadratic equation over the finite

Fig. 2. A family of FH codes obtained using GF(2 ).

field, which is known by Lagrange’s theorem to have at mosttwo roots hence the CCF is at most 2.

Below Example 4, we show how the code length is decreasedto , as well as how we can increase the size of the family to

.Example 4: Consider the field generated by the

primitive root , .

Step 1 ) One cyclic group is constructed by

(4)

giving

(5)

Step 2 ) If we use the equation for the LC code con-struction above

(6)

where now is the element of the field and for each, goes through the cyclic group, we have the FH

codes shown in Fig. 2. By constructing the auto- andcross-hit arrays, we see that the codes have at most twohits in their 2-D ACF’s, and that first three codes haveat most two hits in their mutual 2-D cross-hit arrays.Also, the sixth code is a shift of the first code, the fifthof the second, and fourth of the third code, so they havemore than two hits in their respective cross-hit arrays.

In Fig. 3, we show the auto- and cross-hit of the codes shownin Fig. 1. Note that because of the form of (4), if we start formingthe cyclic group from 0, the last element of the cyclic group isalways , and hence, it gets assigned integer (in this case9). Thanks to this property, the code sequence is easily reducedfrom length to length by simply removing the lastelement. It remains to increase the size of the FH family from

to . This is done by observing that if wesubstitute (1) and (2) in (3) for and , respectively, we havea second-order polynomial and hence by Lagrange’s theoremstill at most two hits—solutions.

Therefore, taking LC codes and anotherHC codes, and merging them into one group, we now have

a family of FH codes of size , with at most two hits inboth their 2-D ACF’s and CCF’s, and of length .

III. CONCLUSION

We have introduced a construction of a new family of FHcodes. It is shown that the new codes possess not only the best

1244 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 48, NO. 8, AUGUST 2000

Fig. 3. Auto-hit array of the FH code shown on the left-hand side of Fig. 1.

possible simultaneous 2-D autocorrelation and 2-D cross-cor-relation properties, but also exist for a much wider number ofintegers than the so far known families (see Table I). In fact, the

new FH code family is constructed over GF and hence in-cludes an important subclass of FH families of length.

REFERENCES

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[5] S. V. Maric and E. L. Titlebaum, “A class of frequency hop codes withnearly ideal characteristics for use in multiple access spread spectrumcommunications and radar and sonar systems,”IEEE Trans. Commun.,vol. 40, Sept. 1992.

[6] A. Freedman and N. Levanon, “Any twoN � N Costas signals musthave at least one common ambiguity sidelobe ifN > 3—A proof,”Proc. IEEE, vol. 73, pp. 1530–1532, Oct. 1985.

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[8] E. L. Titlebaum, “Time frequency hop signals Part 1: Coding based uponlinear congruences,”IEEE Trans. Aerosp. Electron. Syst., vol. AES-17,pp. 494–501, July 1981.

[9] T. Healy, “Coding and decoding for code division multiple user com-munication,”IEEE Trans. Commun., vol. COM-33, pp. 310–316, Apr.1985.

[10] A. J. Viterbi, “A processing satellite transponder for multiple-access bylow rate mobile users,” inProc. Digital Satellite Communication Conf.,Sept. 1968.

[11] F. J. MacWilliams and N. J. Sloane,The Theory of Error-CorrectingCodes. Amsterdam, The Netherlands: North Holland, 1978.