Quasi-Orthogonal Sequences for Code-Division Multiple-Access Systems Kyeongcheol Yang, Member, IEEE, Young-Ky Kim, and P

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982 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 46, NO. 3, MAY 2000 Quasi-Orthogonal Sequences for Code-Division Multiple-Access Systems Kyeongcheol Yang, Member, IEEE, Young-Ky Kim, and P. Vijay Kumar, Member, IEEE

Abstract—In this paper, the notion of quasi-orthogonal se- correlation between two binary sequences and of quence (QOS) as a means of increasing the number of channels the same length is given by in synchronous code-division multiple-access (CDMA) systems that employ Walsh sequences for spreading information signals and separating channels is introduced. It is shown that a QOS sequence may be regarded as a class of bent (almost bent) functions possessing, in addition, a certain window property. Such sequences while increasing system capacity, minimize interference where is computed modulo for all . It is easily to the existing set of Walsh sequences. The window property gives shown that where denotes the the system the ability to handle variable data rates. A general procedure of constructing QOS's from well-known families of Hamming distance of two vectors and . Two sequences are binary sequences with good correlation, including the Kasami and said to be orthogonal if their correlation is zero. Gold sequence families, as well as from the binary Kerdock code Let be a family of binary is provided. Examples of QOS's are presented for small lengths. sequences of period . The family is said to be orthogonal if Some examples of quaternary QOS's drawn from Family are any two sequences are mutually orthogonal, that is, also included. for any and . For example, the Walsh sequence family of Index Terms—Bent functions, code-division multiple-access sys- length is orthogonal. Where there is no chance of confusion, tems, Gold sequences, Kasami sequences, Kerdock codes, quasi-or- we will abbreviate and write instead of . thogonal sequences, Walsh sequences. Consider a synchronous system without multipath time dis- persion, where a sequence family is employed to I. INTRODUCTION both spread the signal bandwidth as well as distinguish between ODE-division multiple-access (CDMA) systems use different users. We consider the case when binary phase-shift C pseudo-noise binary sequences as signature sequences keying (BPSK) is used to modulate the signal. Let to distinguish between the signals of different users. Several be the binary information signal of the th user at spread-spectrum communication systems also use them as the th information bit time. Then each bit is spread into spreading codes that help achieve a low probability of intercept chips by the signature sequence of the th-user channel by spreading the signal energy over a large bandwidth. De- during the th information bit time as follows: sirable characteristics of pseudo-noise binary sequences used for such applications include long-period, low out-of-phase autocorrelation values, low crosscorrelation values, large linear Ignoring noise added to the signal in the channel, the received span, symbol balance, low nontrivial partial-period correlation signal during the th information bit time is given by values, large family size, and ease of implementation [7], [10], [20], [22]. Let be a sequence of length over . We will sometimes identify the sequence with the binary vector . The The receiver at the output of the th-user channel computes

Manuscript received November 30, 1998; revised October 22, 1999. This work was supported in part by the Korea Research Foundation for the pro- gram year of 1998 and in part by the National Science Foundation under Grant CCR-9612864. The material in this paper was presented in part at the Con- ference on Difference Sets, Sequences and Their Correlation Properties, Bad (1) Windsheim, Germany, Aug. 2–14, 1998. K. Yang is with the Department of Electronic and Electrical Engineering, Pohang University of Science and Technology (POSTECH), Pohang, Kyungbuk 790-784, Korea (e-mail: [email protected]). If is orthogonal, then and can be Y. Kim is with Telecommunications R&D Center, Samsung Electronics Co., easily determined from the sign of . When is not or- Pundang P.O. 32, Sungnam, Kyungki-Do 463-050, Korea (e-mail: ykim@ metro.telecom.samsung.co.kr). thogonal, it is necessary to minimize the second term in (1), P. V. Kumar is with Communication Sciences Institute, Electrical Engi- i.e., the interference from other channels. Since is typi- neering–Systems, University of Southern California, Los Angeles, CA 90089- cally assumed to take on values or with equal likelihood, it 2565 USA (e-mail: [email protected]). Communicated by S. W. Golomb, Associate Editor for Sequences. is necessary to minimize the absolute value of the correlations Publisher Item Identifier S 0018-9448(00)02889-3. between two distinct sequences in .

CDMA systems such as the IS-95 system employ Walsh II. QUASI-ORTHOGONAL SEQUENCES sequences of length in the forward link both as spreading Let be an integer. For a positive integer , let sequences and also to separate the different user channels be the sequence of length ,given [24]. Since the IS-95 system is synchronous in the forward by link, it is the inner product between the vectors associated with different user sequences rather than periodic correlation, that is a measure of interference from other channels. Walsh sequences are perfect in the sense that there is no interference (2) between any pair of sequences, as Walsh sequences form an . orthogonal family. However, the orthogonality limits the size . of an orthogonal family—there are only Walsh sequences of length . For this reason, it is impossible to increase the The sequences are said to be the ca- number of channels without either increasing the sequence nonical sequences of length . We will abbreviate and write length or else losing orthogonality between a pair of user se- instead of when there is no chance of confusion. quences. Any binary sequence of length has a Boolean expres- There are many situations where it is not appropriate to in- sion of the form crease the length of the sequence. A loss in orthogonality is in- evitable in such situations and gives rise to interference from (3) other channels. In [1], Bottomley proposed a set of signature sequences drawn from a Kerdock code for a synchronous direct sequence CDMA system where orthogonal spreading is used. where However, in this scheme, signature and spreading sequences are is a sequence obtained via bitwise multiplica- different and there is no requirement on correlation between sig- tion, and the addition of sequences is carried out bitwise modulo nature sequences over a subblock. [14]. For example, the sequence of length In this paper, the notion of quasi-orthogonal sequence may be expressed in the form . With (3) and (QOS) as a means of increasing the number of channels in (2) in mind, we will interchangeably write synchronous code-division multiple-access (CDMA) systems in place of . that employ Walsh sequences for spreading information signals For any integer , , we have the binary expan- and separating channels is introduced. It is shown that a QOS sion , where . sequence may be regarded as a class of bent (almost bent) Using this expansion, we define to be a linear combina- functions possessing, in addition, a certain window property. tion of 's, given by Such sequences while increasing system capacity, minimize interference to the existing set of Walsh sequences. The (4) window property gives the system the ability to handle variable data rates. A general procedure of constructing QOS's from It is not difficult to show that and are mutually or- well-known families of binary sequences with good correlation, thogonal for any . The family , defined by including the Kasami and Gold sequence families, as well as from the binary Kerdock code is provided. Examples of QOS's (5) are presented for small lengths. Complex or quaternary sequences, specifically sequences is a complete set of orthogonal sequences of length drawn from Family [2], [23], in place of binary sequences, known as the family of Walsh sequences of length . were first considered in [5] in order to expand the set of The Walsh family can be interpreted as a subcode of the binary Walsh sequences for CDMA systems. In the final part first-order Reed–Muller code , where is given by of this paper, the correlation properties in “windows” of the sequences in Family are studied. Computer searches were conducted (under a restriction on the subspaces associated with the windows) and used to provide examples of sets of QOS's for all lengths of the form , . where is the all-one sequence and The paper is organized as follows. In Section II, we give are the canonical sequences of length ,given some preliminaries and introduce the concept of quasi-orthog- in (2) (see [14]). In many applications, it is important to find onal sequences. The properties of a QOS sequence are studied the correlation between a binary sequence of length and the here. In Section III, a general procedure for the construction of code sequences in . This correlation is closely related to the QOS's is provided from well-known families including Kasami determination of a parameter of , known as the covering sequences, Gold sequences, and binary Kerdock codes. Quater- radius given by nary quasi-orthogonal sequences are discussed in Section IV, and some examples drawn from the Family are provided. Fi- (6) nally, concluding remarks and some open problems are given in Section V. where runs through all the binary vectors of length in . 984 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 46, NO. 3, MAY 2000

The covering radius of the first-order Reed–Muller TABLE I code has been intensively studied [4], [9], [15]. It is well THE COVERING RADIUS OF THE FIRST-ORDER REED–MULLER CODE FOR SMALL VALUES OF m known that for any even integer , and

(7) for odd integer . The known values of equal the lower bound in (7) for and . However, Patterson and Wiedemann [18] have shown that is strictly greater than the lower bound in (7) for any odd integer . The determination of the exact value of for odd remains TABLE II an open problem. Known results on are listed in Table I @P A FOR SMALL VALUES OF m for small values of . In the IS-95 CDMA mobile communication system [24], the Walsh sequences of length are used in the forward link both to spread the signal bandwidth and to distinguish between the signals of different users. The usage of Walsh sequences limits the number of channels to the size of the Walsh sequence family which is equal to the length of the sequences in the family. The size of a Walsh sequence family cannot be increased while maintaining orthogonality between pairs of sequences because there can be no greater than pairwise orthogonal sequences of length . Proof: Let be a sequence of length The increasing demand for more service makes it desirable and let be a codeword of with to increase the size of the sequence family. Our goal is to do , where denotes the Hamming distance precisely this while keeping the interference introduced by the between and of length . Then is also a code- additional sequences as small as possible. Note that since the word with , since the all-one CDMA system is assumed to be synchronous, it is the inner vector is a codeword in . Therefore, product between pairs of sequences rather than periodic correlation that is the relevant measure of interference. Consider first, the situation where a single sequence of length is added to the Walsh sequence family .We define to be the maximum correlation between From the definitions in (6) and (8), we have and the sequences in , given by

where takes on or , and runs through . The factor is introduced to account for the effect of data modulation upon the inner product. Let be the minimum achiev- able correlation value, given by The proposition now follows from the well-known results on (8) , including (7). where runs through all sequences of length . Then the Proposition 1 tells us that determining is equiva- maximum interference introduced to the existing Walsh family lent to determining the covering radius of the first-order is at least for any sequence of length Reed–Muller code of length . For small values of , that is used to augment the family . Our next step is to is listed in Table II, in which a denotes an upper determine . bound on . For an even integer , a binary-valued function of length Proposition 1: Let be the covering radius of the is said to be bent if the correlation between and first-order Reed–Muller code of length . Then any sequence in has magnitude [10], [14], [19]. we have . Moreover, we have In the case of an odd integer , there are no bent functions for any even integer , and of length since is not an integer. Instead, for an odd integer , a binary-valued function of length will be said to be almost bent if the correlation between and for any odd integer . any sequence in has magnitude (cf. [3]). YANG et al.: QUASI-ORTHOGONAL SEQUENCES FOR CDMA SYSTEMS 985

We are now in a position to introduce the concept of quasi- It is easily checked that orthogonal sequence. Definition 2: Let be the Walsh family of length , given by (5). A family satisfies the Conditions a), b), and c), and therefore is quasi- of sequences of length is orthogonal. said to be quasi-orthogonal if the following are satisfied: a) contains . Lemma 6: Let be a bent (almost bent) function of length b) for any and . with the window property. Let c) For any ,any , and any integers , , where , , and Then the set is quasi-orthogonal. Proof: For any , is also a bent (almost bent) function of length . It also has the window property. Furthermore, and are orthogonal for Remark 3: When is an odd integer , we use the upper any and , since and are orthogonal. bound instead of the exact value of , since It will be convenient to define an equivalence relation be- is unknown. tween Boolean functions of length in terms of the first-order Condition b) for quasi-orthogonality requires that the corre- Reed–Muller code . lation between any two distinct sequences in should be as Definition 7: Two binary-valued Boolean functions and small as possible. Conditions a) and b) imply that any sequence of the same length are said to be equivalent (with re- in should cause minimal possible interference to the spect to )if ; in other words, is existing Walsh family, i.e., for any a linear combination of the canonical sequences and the all-one and . This requires that any sequence of length . Otherwise, they are said to be inequiv- not belonging to should be either bent or almost bent de- alent (with respect to ). pending on . The following is a direct consequence of Lemma 6 and Defi- Condition c), which we will refer to as the window property, nition 7. requires that when any sequence is divided into consecutive subblocks of length , , the Theorem 8: Let be inequivalent partial correlation between and over every subblock bent (almost bent) functions of length with the be as small as possible for any . Since every sub- window property. If is also bent (almost bent) for block of length in corresponds to a sequence in , the any and , then the set condition requires that every consecutive subblock of corresponds either to a bent or almost bent function of length , depending on . This requirement is motivated by practical applications, where repeating a sequence in twice yields a sequence in , and therefore all sequences in can be is quasi-orthogonal, where . Furthermore, the size of used for the transmission of data at twice the normal data rate. In is . such a situation, Condition c) is necessary to ensure minimum Any quasi-orthogonal sequence (QOS) constructed possible interference to the higher data rate users who will em- from Theorem 8 consists of cosets of . Note that ploy correlation over a window of size rather than . are the coset leaders in with respect Remark 4: In terms of Boolean expressions, the window to . Since any sequence in can be expressed as property in part c) of Definition 2 can be reformulated as for some and , the nonzero follows: If , then functions are often called the masking functions of in practical applications. In order to maximize the size of , it is necessary to maximize . The definition below relates to the maximum possible value of . should be a bent or almost bent function of length for any , where . Definition 9: Let be the ensemble of sets of inequivalent functions of length (with respect to ) such that Example 5: Consider the canonical sequences of length , given by a) any function in is bent (almost bent); b) the sum of any two functions is also bent (almost bent); c) any function in has the window property. The number is defined to be the maximum of Then the Walsh sequence of length is where runs through , that is, 986 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 46, NO. 3, MAY 2000

It is well known [14, Corollary 11, p. 429] that III. QUASI-ORTHOGONAL SEQUENCES FROM KNOWN FAMILIES In Section II, QOS's are defined and their existence is verified for any length in (9). A natural question at this stage is to find is bent, for any even . It is easily checked that this func- an efficient way to construct QOS's. In this section we give a sys- tion has the window property. It follows from this and the the- tematic procedure to construct QOS's from known families with orem below that there exists at least one bent (almost bent) func- good correlation properties, including Gold sequences, Kasami tion of length with the window property, i.e., sequences, and Kerdock codes.

(9) A. Construction of QOS's from Known Sequences with Optimal Correlation for any integer . Consider a family of binary sequences of period , satisfying the following Theorem 10: For any positive integer ,wehave conditions. a) An -sequence of period belongs to (Here, is usually assumed to be an -sequence). Proof: Let be a set of inequivalent functions of length b) Any two sequences are cyclically distinct, that is, for any satisfying the conditions given in Definition 9, which and there does not exist , , such that achieves . Then any function in can be for all . considered as a bent function . Now let c) When is added to the out-of-phase auto- and crosscorre- be a concatenation of and , which is of length lation values, the result has magnitude , i.e., . In the notation of Boolean functions, we have

The theorem follows from the fact that is almost bent, since for all except for and . Here is the largest integer less than or equal to . Our goal is to construct a quasi-orthogonal sequence from . Since the Walsh sequence is obtained by permuting and its shifts properly, is included in by the Condition a). Condition b) allows to be the coset leaders in with respect to , if properly permuted. Condition c) guarantees that is bent (almost bent) for any , and is also bent (almost bent), if properly defined. The remaining for any . work is to check if has the window property.

It is quite interesting to determine exactly, but this Algorithm to Construct a QOS from a Given Family does not appear to be an easy problem. For small values of ,it can be checked that . (A.1) Using an -sequence , we define a mapping In the following section, we know from a computer search that , given by

etc. Example 11: In the case of , consider the set of four inequivalent bent functions of length , given by (A.2) For each we define a sequence of length by if if

where is the inverse mapping of . (A.3) Choose all sequences having the window property, Note that has the window property and are and call them . Define also bent for any and , Hence the set

where . Then the family is a quasi-orthog- is quasi-orthogonal, where . onal sequence of size YANG et al.: QUASI-ORTHOGONAL SEQUENCES FOR CDMA SYSTEMS 987

Let be an integer and let be a finite field of ele- B. Construction of QOS's from Binary Kerdock Codes ments. The trace function is a mapping from to , Let be an even integer . Note that is odd. For sim- given by plicity, let , and let be the quadratic form given by

Every primitive element in is associated with an -sequence of length , via for . (See [13] for more details.) The binary Kerdock code of length can be described For an even integer , let be a sequence decimated from as by , that is,

where and where and is chosen so that is not the all-zero sequence. Note that is an -sequence of period . The small set of Kasami sequences can be defined as It is shown in [8] that the binary Kerdock code can be (10) described as the image of the Gray map of a linear code over a quaternary alphabet. The weight distribution of is well Note that the size of is . (See [10], [20], or [22] for known [14]. details.) Proposition 16: For even , let be the number of Proposition 12: The out-of-phase autocorrelation and cross- codewords of weight in the binary Kerdock code of correlation between sequences of the Kasami sequence length . Then in (10) take on three values: . for or Example 13: For an even integer , the Kasami sequence for in (10) can be used to construct quasi-orthogonal se- for quences of length by Proposition 12. Put for and apply In order to express and in a unified way, we intro- Algorithm described above to . duce a variable and put , where For an odd integer , let be an integer relatively prime to . Let be a sequence decimated by from an -sequence (12) of length , that is, where

The Gold sequence can be defined as is the quadratic part of , and

(11)

Note that is a family of size . (See [10], [20], or is the linear part of . This implies that if we take [22] for details.) in , we get all codewords of the first-order Reed–Muller code by proper permutation of using Proposition 14: The out-of-phase autocorrelation and cross- as in Algorithm . Thus the Kerdock code consists of correlation between sequences of the Gold sequence in cosets of , corresponding to the forms de- (11) take on three values: , , where is odd. pending on . Example 15: For an odd integer , the Gold sequence Let be a basis for over . Then in (11) can be used to construct quasi-orthogonal sequences of every element can be expressed as length by Proposition 12. Put (13)

for where for . Using this expression, we put and apply Algorithm to . 988 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 46, NO. 3, MAY 2000

From the weight distribution of in Proposition 16, it Kerdock codes have been used in the case of even integers . is easily checked that is a bent function of An interesting point in these simulations is that the maximum length and that for any and number of inequivalent bent (almost bent) functions from these families depends on the choice of an -sequence or the choice of primitive elements. is also bent. These facts give a way to construct QOS's from IV. QUATERNARY QUASI-ORTHOGONAL SEQUENCES binary Kerdock codes, if we properly choose bent functions with window property from them. When a CDMA communication system uses QPSK (quadrature phase-shift keying) modulation instead of BPSK (binary Lemma 17: There are inequivalent bent functions from phase-shift keying) modulation, it is natural to use quadrature the Kerdock code , which have the window property of phase sequences as signature sequences rather than binary se- size . quences and this is the approach taken in [5]. Also, as pointed Proof: There are inequivalent bent functions out in [5], when it is desired to expand the set of binary Walsh in . From (13) and linearity of the functions currently used on the forward link of the IS-95 CDMA trace function, we have system, one advantage using quaternary rather than binary sequences to augment the family is that the maximum full-period correlation between a Walsh function and a new sequence belonging to the augmented set, is lower in the quaternary case by a factor of for the case when the full period is of the form odd. For these reasons, we investigate in this paper, the question of whether it is possible to construct a QOS family (i.e., a family that in addition to full-period correlation, also enjoys good correlation properties in every window) that consists of where is a linear function and binary Walsh sequences and quaternary sequences. From the is a quadratic function. Therefore, is a bent theory of quaternary sequences, a natural candidate to use is the function of length for any fixed value sequences set known as Family [2], [23]. These sequences were also the basis of the quaternary sequences studied in [5], where the focus was on the full-period correlation properties.1 The construction of quaternary QOS's from sequences in if and only if it has the term , that is, . Family depends on the choice of certain subspaces of a finite There are exactly such 's in for a fixed . . field. The results of an exhaustive search conducted over a The lemma implies that the number of inequivalent bent func- subset of all possible choices of subspaces are presented in this tions of length with window property, which can be obtained section. from the Kerdock code , is at most . A. Definition of Quaternary Quasi-Orthogonal Sequences C. Simulation Results for QOS's from Known Families Let be the ring of integers modulo . A se- Computer simulations were done to find QOS's of length quence , is called a quaternary sequence from known families. For even , Kasami sequences and Ker- of length if for all . The correlation between dock codes can be used. In general, it is possible to get larger two quaternary sequences and of the same length is QOS's from the Kerdock codes than those from Kasami se- given by quences. For example, there are 4, 8, and 20 inequivalent bent functions with window property from Kerdock codes in the case of , respectively, while there are 2, 3, and 6 such functions from Kasami sequences in each case. It may be natural because there are more candidates for inequivalent bent where denotes complex conjugation, is computed functions with window property in the case of Kerdock codes modulo for each , and is a primitive fourth root of unity, ( candidates) than candidates in Kasami sequences. that is, . Two sequences are said to be orthogonal if For an odd integer , the Gold sequence in (11) has their correlation is zero. candidates for inequivalent almost bent functions with window The following result is well-known from the theory of orthog- property. onal transforms. As a special case, it gives a lower bound on the In Tables III–V, inequivalent bent (almost bent) functions maximum of coefficients of the Walsh transform. with window property (called masking functions) for QOS's are listed for small values of , which are of importance in practical systems. In Tables IV and V, implies that the term 1At the time of initial writing of this paper, the authors were unware of [21], where the QOS properties of Family e are also considered and some methods belongs to the masking function in its Boolean expression. of construction provided. However, the techniques used there and the sequences Gold sequences have been used in the case of odd integers , presented are distinct from ours. YANG et al.: QUASI-ORTHOGONAL SEQUENCES FOR CDMA SYSTEMS 989

TABLE III MASKING FUNCTIONS FOR QOS'SOFLENGTH P

Proposition 18: Let be a complex where vector with and let , where is an unitary matrix, i.e., . Then

Using the relation , for any binary sequence , the correlation becomes Proof: Note that the maximum of is at least the average of . It is easily checked that the average of is equal to . Corollary 19: Let be a quaternary sequence of length and let be the binary Walsh sequence of the same where is computed modulo ,so can be re- length. Then the maximum of absolute correlation values be- garded as a special case of quaternary sequence, which we will tween and any in is at least , that is, denote by . Note that takes on two values: or .For an integer , it is natural to identify the binary Walsh sequence with the set over , given by 990 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 46, NO. 3, MAY 2000

TABLE IV MASKING FUNCTIONS FOR QOS' SOFLENGTH IPV @m aUA

TABLE V MASKING FUNCTIONS FOR QOS' SOFLENGTH PST @m aVA YANG et al.: QUASI-ORTHOGONAL SEQUENCES FOR CDMA SYSTEMS 991

TABLE VI MASKING FUNCTIONS FOR QUATERNARY QOS'S OF LENGTH P

(Continued on the following page)

In the same way as in the binary case, it is natural to use duced by a factor of . Condition c) is also called the window by Corollary 19 in order to define quaternary property in the quaternary case, as in the binary case. quasi-orthogonal sequences in the following. Definition 22: Two quaternary sequences and of Definition 20: A family of the same length are said to be equivalent (with respect to quaternary sequences of length over is said to )if is in , where and be quaternary quasi-orthogonal if the following are satisfied: is a module2 generated by and over . a) contains . In the same way as in the binary case, we get the following b) For any distinct two sequences theorem. Theorem 23: Let be inequivalent sequences of length over with window property. If and satisfy Condition b) in Definition 20 for any and , then the set c) For any ,any , and any integers , where , and

is quasi-orthogonal, where . Furthermore, the size of is . As in the binary case, the nonzero quaternary sequences given in Theorem 23 will be called the Remark 21: Compared with the conditions for binary QOS's, masking sequences of the family . Conditions b) and c) in Definition 20 are stronger when or are odd, in the sense that maximum correlation values are re- 2See [12] for details. 992 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 46, NO. 3, MAY 2000

TABLE VI (Continued)

B. Construction of Quaternary Quasi-Orthogonal Sequences Proposition 24: The out-of-phase autocorrelation and cross- The Galois ring GR is an extension of of correlation between sequences of in (14) take on only the degree . is a local ring having a unique maximal ideal following values: and the quotient ring is isomorphic to a finite for odd field with elements (see [8], [11] for details). As a mul- for even tiplicative group the set of units in has a cyclic subgroup of order . Let be an element of order , where and . and let . Any element can The sequence is an be expressed uniquely as for . Let be -sequence of length multiplied by . Thus the modulo- reduction map. Note that is a primitive can be obtained from and its shifts by a proper per- element in in . The trace map from to is defined by mutation in a similar way as Algorithm in Section III-A. Com- bined with Proposition 24, the family is a good candidate for construction of quaternary QOS's. In Table VI, examples of masking functions for quaternary QOS's are listed for small values of . There have been found four nonzero masking func- Note that it is a -linear map. tions for quaternary QOS's for . Let , , be an ordering of the the elements of . The Family of -sequences of period is defined by V. C ONCLUDING REMARKS

(14) We have introduced quasi-orthogonal sequences which can be used in CDMA systems that employ the Walsh sequence where family for channel separation. A general procedure for construction of QOS's is provided from well-known families of binary sequences with good correlation, including Kasami se- for quences, Gold sequences, and binary Kerdock codes. In partic- ular, examples of masking functions for QOS's are presented for Note that the size of is and is referred to as the small lengths. Some examples of quaternary QOS's drawn from Family , whose correlation distribution is well known in the Family are also included. following proposition [2], [10], [23]. Some open problems relating to QOS's are listed as follows. YANG et al.: QUASI-ORTHOGONAL SEQUENCES FOR CDMA SYSTEMS 993

i) What is for odd ? In other words, what is [9] T. Helleseth, T. Kløve, and J. Mykkeltveit, “On the covering radius of the covering radius of the first-order Reed–Muller binary codes,” IEEE Trans. Inform. Theory, vol. IT-24, pp. 257–258, 1978. code for odd ? [10] T. Helleseth and P. V. Kumar, “Sequences with low correlation,” ii) Find the exact value of for any .In in Handbook of Coding Theory, V. Pless and W. C. Huffman, other words, find a maximal set of masking functions for Eds. Amsterdam, The Netherlands: Elsevier Science, 1998. [11] P. V. Kumar, T. Helleseth, and A. R. Calderbank, “An upper bound for quasi-orthogonal sequences of a given length in the Weil exponential sums over Galois rings and applications,” IEEE Trans. binary and quaternary cases. Inform. Theory, vol. 41, pp. 456–468, Mar. 1995. iii) Find a systematic method to construct QOS's in the bi- [12] S. Lang, Algebra, 2nd ed. Menlo Park, CA: Addison-Wesley, 1984. [13] R. Lidl and H. Niederreiter, Finite Fields of Encyclopedia of Mathe- nary and quaternary cases. matics and Its Applications. Reading, MA: Addison-Wesley, 1983, vol. 20. [14] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. Amsterdam, The Netherlands: North-Holland, 1977. ACKNOWLEDGMENT [15] J. Mykkeltveit, “The covering radius of the @IPVY VA Reed–Muller code is 56,” IEEE Trans. Inform. Theory, vol. IT-24, pp. 359–262, 1980. [16] C.-K. Park and K. Yang, “Generation of quasi-orthogonal sequences The authors wish to thank Prof. J. S. No, Prof. H. Chung, and using bent functions,” in 1997 Fall Conf. Korean Institute of Commu- H. W. Kang for some valuable discussions. They would also like nication Sciences, vol. 16, Seoul, Korea, Nov. 15, 1997, pp. 1187–1190. to thank C. K. Park and H. Y. Lee for their computer assistance. in Korean. [17] C.-K. Park, K. Yang, Y. Kim, and H.-W. Kang, “Generation of quasi-orthogonal sequences for CDMA systems,” in 8th Joint Conf. Communi- cations and Informations (JCCI'98), vol. 8, Chunju, Korea, Apr. 22–24, 1998, pp. 1144–1148. in Korean. REFERENCES [18] N. J. Patterson and D. H. J. Wiedemann, “The covering radius of the @P Y ITA Reed-Muller code is at least ITPUT,” IEEE Trans. Inform. [1] G. E. Bottomley, “Signature sequence selection in a CDMA system with Theory, vol. IT-29, pp. 354–256, May 1983. Correction to “The cov- orthogonal coding,” IEEE Trans. Veh. Technol., vol. 42, pp. 62–68, Feb. ering radius of the @PISY ITA Reed-Muller code is at least ITPUT” IEEE 1993. Trans. Inform. Theory, vol. 36, pp. 443, Mar. 1990. [2] S. Boztas, R. Hammons, and P. V.Kumar, “4-phase sequences with near- [19] O. Rothaus, “On ‘bent’ functions,” J. Comb. Theory, ser. A, vol. 20, pp. optimum correlation properties,” IEEE Trans. Inform. Theory, vol. 40, 300–305, 1976. pp. 1101–1113, May 1992. [20] D. V. Sarwate and M. B. Pursley, “Crosscorrelation properties of pseu- [3] C. Carlet, P. Charpin, and V. Zinoviev, “Codes, bent functions and per- dorandom and related sequences,” Proc. IEEE, vol. 68, pp. 593–619, mutations suitable for DES-like cryptosystems,” Des., Codes Cryptogr., May 1980. vol. 15, no. 2, pp. 125–156, Nov. 1998. [21] A. Shanbhag, J. Holtzman, E. G. Tiedemann, and J. Odenwalder, “Re- [4] G. D. Cohen, M. G. Karpovsky, and H. F. Mattson Jr., “Covering vised Proposal for Improved Quasi-Orthogonal Functions,” Tech. Rep. radius—Survey and recent results,” IEEE Trans. Inform. Theory, vol. TR-45.5/98.10.19.19, Burlington, TX, Oct. 19, 1998. IT-31, pp. 328–343, May 1985. [22] M. K. Simon, J. K. Omura, R. A. Scholtz, and B. K. Levitt, Spread Spec- [5] O. Glauser, J. Holtzman, A. Shanbhag, and E. G. Tiedemann, trum Communications. Rockville, MD: Computer Sci., 1985, vol. 1. “Proposal for improved quasi-orthogonal functions,” Tech. Rep. [23] P. Solé, “A quaternary cyclic code and a family of quadriphase TR-45.5.3.1/98.08.18.13, Portland, OR, Aug. 17, 1998. sequences with low correlation properties,” in Coding Theory and Ap- [6] R. Gold, “Optimal binary sequences for spread spectrum multiplexing,” plications, Lecture Notes in Computer Science. New York: Springer- IEEE Trans. Inform. Theory, vol. IT-13, pp. 619–621, Oct. 1967. Verlag, 1989, vol. 388, pp. 193–201. [7] S. W. Golomb, Shift-Register Sequences. San Francisco, CA: Holden- [24] TIA/EIA Interim Std., Mobile Station-Base Station Compatibility Day, 1967. Laguna Hills, CA: Aegean Park, 1982. Standard for Dual Mode Wideband Spread Spectrum Cellular System, [8] A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane, and Telecommunications Industry Assoc., July 1993. P. Solé, “The -linearity of Kerdock, Preparata, Goethals, and related [25] L. R. Welch, “Lower bounds on the maximum cross-correlation of sig- codes,” IEEE Trans. Inform. Theory, vol. 40, pp. 301–319, Mar. 1994. nals,” IEEE Trans. Inform. Theory, vol. IT-20, pp. 397–399, May 1974.