IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 54, NO. 3, MARCH A DS CDMA system is said to be approximately synchronized if the modulated

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1 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 54, NO. 3, MARCH two ases we get infinite lasses of DPM. The most important result is the onstrution of DPM from ternary vetors of lengths at least 13 to permutations of the same length. Using the DPMs (or the DIMs) an known ternary oes, we get new larger permutation arrays in many ases; a ouple of examples are gives as illustrations. REFERENCES [1] A. E. Brouwer, H. O. Hämäläinen, P. R. J. Östergår, an N. J. A. Sloane, Bouns on mixe binary/ternary oes, IEEE Trans. Inf. Theory, vol. 44, no. 1, pp , Jan [2] J.-C. Chang, Distane-inreasing mappings from binary vetors to permutations, IEEE Trans. Inf. Theory, vol. 51, no. 1, pp , Jan [3] J.-C. Chang, New algorithms of istane-inreasing mappings from binary vetors to permutations by swaps, Designs, Coes Cryptogr., vol. 39, pp , Jan [4] J.-C. Chang, Distane-inreasing mappings from binary vetors to permutations that inrease Hamming istanes by at least two, IEEE Trans. Inf. Theory, vol. 52, no. 4, pp , Apr [5] J.-C. Chang, R.-J. Chen, T. Kløve, an S.-C. Tsai, Distane-preserving mappings from binary vetors to permutations, IEEE Trans. Inf. Theory, vol. 49, no. 4, pp , Apr [6] H. C. Ferreira an A. J. H. Vink, Inferene anellation with permutation trellis arrays, in Pro. IEEE Veh. Tehnol. Conf., 2000, pp [7] H. C. Ferreira, A. J. H. Vink, T. G. Swart, an I. e Beer, Permutation trellis oes, IEEE Trans. Commun., vol. 53, no. 11, pp , Nov [8] H. C. Ferreira, D. Wright, an A. L. Nel, Hamming istane-preserving mappings an trellis oes with onstraine binary symbols, IEEE Trans. Inf. Theory, vol. 35, no. 5, pp , Sep [9] Y.-Y. Huang, S.-C. Tsai, an H.-L. Wu, On the onstrution of permutation arrays via mappings from binary vetors to permutations, Designs, Coes Cryptogr., vol. 40, pp , [10] K. Lee, New istane-preserving maps of o length, IEEE Trans. Inf. Theory, vol. 50, no. 10, pp , Ot [11] K. Lee, Cyli onstrutions of istane-preserving maps, IEEE Trans. Inf. Theory, vol. 51, no. 12, pp , De [12] K. Lee, Distane-inreasing maps of all length by simple mapping algorithms, IEEE Trans. Inf. Theory, vol. 52, no. 7, pp , Jul [13] J.-S. Lin, J.-C. Chang, an R.-J. Chen, New simple onstrutions of istane-inreasing mappings from binary vetors to permutations, Inf. Proess. Lett., vol. 100, no. 2, pp , Ot [14] J.-S. Lin, J.-C. Chang, R.-J. Chen, an T. Kløve, Distane-Preserving Mappings from Ternary Vetors to Permutations arxiv, v1 [s.dm]. [15] T.-T. Lin, S.-C. Tsai, an H.-L. Wu, Distane-preserving mappings from ternary vetors to permutations, Manusript, [16] V. S. Pless an W. C. Huffman, Es., Hanbook of Coing Theory. Amsteram, The Netherlans: Elsevier, [17] T. G. Swart an H. C. Ferreira, A generalize upper boun an a multilevel onstrution for istane-preserving mappings, IEEE Trans. Inf. Theory, vol. 52, no. 8, pp , Aug Complete Mutually Orthogonal Golay Complementary Sets From Ree Muller Coes Appuswamy Rathinakumar, Stuent Member, IEEE, an Ajit Kumar Chaturvei, Senior Member, IEEE Abstrat Reently Golay omplementary sets were shown to exist in the subsets of seon-orer osets of a q-ary generalization of the first-orer Ree Muller (RM) oe. We show that mutually orthogonal Golay omplementary sets an also be iretly onstrute from seon-orer osets of a q-ary generalization of the first-orer RM oe. This ientifiation an be use to onstrut zero orrelation zone (ZCZ) sequenes iretly an it also enables the onstrution of ZCZ sequenes with speial subsets. Inex Terms Complementary sets, generalize Boolean funtion, mutually orthogonal Golay omplementary sets, Ree Muller (RM) oes, zero orrelation zone (ZCZ) sequenes. I. INTRODUCTION Zero orrelation zone (ZCZ) sequenes are a generalization of orthogonal sequenes. Their superior orrelation properties an be utilize to improve the spetral effiieny of an approximately synhronize 1 CDMA system over a similar system that uses onventional orthogonal sequenes [3]. Further, CDMA systems employing ZCZ sequenes have been shown to be performing as well as OFDM systems in fast time-varying multipath hannels at a onsierably lower omputational omplexity [14]. Reently, ZCZ sequenes have foun appliations in ternary iret sequene Ultra Wieban (TS-UWB) systems [13]. It has been shown that the TS-UWB (also known as multioe UWB) systems employing appropriate ZCZ sequenes an support ifferent ata rate requirements at a onstant bit error rate performane level [13]. They are also appliable in broaban satellite IP networks, where sequene sets with small autoorrelation an ross orrelation within a etetion aperture are neee [15], [16]. Mutually orthogonal Golay Complementary Sets (MOGCS) are an integral part in the onstrution of ZCZ sequenes. Traitionally, ZCZ sequenes have been onstrute by iterative methos starting from a pair of MOGCS. In [3], several onstrutions of ZCZ sequenes starting from any set of MOGCS were given. Many reursive onstrutions of MOGCS are known [9], 2 [3], [6], [7], [5], [3]. In [1] an [2], a long staning problem of iretly onstruting Golay Complementary Sets (GCS) [6] was solve by onstruting GCS from Ree Muller (RM) oes. Speifially, GCS were shown to be subsets in seon-orer osets of a q-ary generalization RM q (1;m) of the first-orer RM oe. Size of the set was shown to be iretly relate to a graph assoiate with the oset leaer /$ IEEE Manusript reeive July 28, 2004; revise November 29, A. Rathinakumar was with the Department of Eletrial Engineering, Inian Institute of Tehnology, Kanpur, UP , Inia. He is now with the Department of Eletrial an Computer Engineering, University of California, San Diego, La Jolla, CA USA ( rathnam@us.eu). A. K. Chaturvei is with the Department of Eletrial Engineering, Inian Institute of Tehnology, Kanpur, UP , Inia ( ak@iitk.a.in). Communiate by K. G. Paterson, Assoiate Eitor for Sequenes. Digital Objet Ientifier /TIT A DS CDMA system is sai to be approximately synhronize if the moulate sequenes are synhronize up to a small fration of the sequene length. 2 The onept of zero orrelation sequenes first appears in [9] as semiperfet sequenes.

2 1340 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 54, NO. 3, MARCH 2008 We onsier the problem of iret onstrution of omplete mutually orthogonal GCS (MOGCS) 3 from RM oes an show that they an also be onstrute from seon-orer osets of the same q-ary generalization of the first-orer RM oe. Our approah is to establish a framework in whih MOGCS an be ientifie to be in the seon-orer osets of the q-ary generalization of RM oes given in [2]. The importane of suh an ientifiation is that it enables one to relate ifferent sets of ZCZ sequenes by the orresponing MOGCS use in their onstrution. Our work is motivate in part by the observation that given the number of sequenes in a ZCZ set an its interferene free winow (IFW) length, the sequene set an be onstrute as a union of a large number of ZCZ sets ontaining fewer sequenes of the given length. In turn, the orrelation properties of the smaller sets an be esigne better than that of the larger set whih helps improve the spetral effiieny further [3]. Inee, suh a onstrution of orthogonal sets ombining two ZCZ sets eah having unity IFW length was shown in [4], [3]. Diretly onstruting MOGCS from RM oes an haraterizing the MOGCS that form ZCZ sequenes with given parameters is a promising iretion towar onstrution of suh ZCZ sequenes. Beginning with the GCS ientifie in [2], we ientify a set of permutations of the GCS whih generate mutually orthogonal Golay omplementary sets. It is shown that there are 2 k suh permutations on a omplementary set ontaining 2 k1 sequenes. We then onstrut another omplementary set with the same assoiate graph as the original set. The same set of permutations ientifie earlier are applie on this set to generate further 2 k mutually orthogonal Golay omplementary sets. We then onlue our main result by establishing that these two sets are mutually orthogonal. We emphasize that not every permutation of a given GCS will proue an orthogonal omplementary set an ientifying the permutations that o so is an important ontribution of this work. The rest of the orresponene is organize as follows: Setion II provies the neessary bakgroun an establishes basi notations. Setion III ollets all our main results. After ientifying a number of permutations of a omplementary set to generate MOGCS, generating omplete MOGCS is isusse. Some of the appliations of the iret onstrution are pointe out in Setion IV. We onlue the orresponene in Setion V. A. Correlation Parameters II. BACKGROUND AND NOTATION The aperioi ross-orrelation C(a; b)( ) between two length L omplex-value sequenes a an b is efine as an C(a; b)( ) L001 l a l b 3 l ; if 0 L 0 1 L01 l a l0 b 3 l ; if 0L < <0 0; otherwise (1) [a 0 0a a 0 n01], in whih a 0 i! a.ifa an b are q-value vetors, then we efine the funtion C(a; b)(1) to be the ross-orrelation funtion of the assoiate omplex-value vetors a 0 an b 0. The perioi rossorrelation between the sequenes a an b is efine as ab( )C(a; b)( )C(b; a) 3 (L 0 ): Definition 2.1: A set of M, length L vetors fa i g M is sai to be a Golay omplementary set (GCS) if M A(a i ; a i )() 8 6 0: (2) Notie that a Golay omplementary set is an orere set of sequenes. Definition 2.2: Two Golay omplementary sets fa i mg M an fa i ng M are sai to be mutually orthogonal if M C(a i m; a i n)() 8 : Definition 2.3: The number of Golay omplementary sets that are pairwise mutually orthogonal is at most equal to the number of sequenes in a Golay omplementary set. Suh a olletion of maximum possible number of orthogonal Golay omplementary sets is sai to be omplete. M Definition 2.4: Let fb ngn1 be a set of M sequenes, eah of length L. The zero perioi autoorrelation zone T ACZ an the zero perioi rossorrelation zone T CCZ of this sequene set are efine to be [12] T ACZ maxft j b b ( ); 8n; j j T; 6 0g T CCZ maxft j b b ( ); 8m 6 n; j j T g where, b b ( ) enotes the perioi rossorrelation between the sequenes b m an b n. Definition 2.5: The Interferene Free Winow (IFW) of the sequene set fb n g M n1, enote by T,isefine to be the minimum of the zero autoorrelation zone an the zero rossorrelation zone values, i.e. T minft ACZ ;T CCZ g: The set fb n g M n1 with IFW of T is sai to onstitute a Zero Correlation Zone (ZCZ) sequene set of M sequenes of length L an is enote by ZCZ-(L; M; T ). Two istint sets of sequenes fb 1 mg M m1 an fb 2 ng M n1 are sai to be mutually orthogonal, if b b (0) 0 8 m an n: A(a)() C(a; a)( ) enotes the aperioi autoorrelation of the sequene a. We also efine the orrelation funtions for q-value vetors. This is one by efining! e 2iq an assoiating with eah vetor a [a 0a 1 111a n01], where a i 2 q, a omplex-value vetor a 0 3 A olletion of MOGCS is sai to be omplete if the olletion ontains the maximum number of omplementary sets, whih is known to be equal to the size of the omplementary sets [5]. B. Construting ZCZ Sequenes From MOGCS Here we briefly outline onstruting ZCZ sequenes from MOGCS for motivating the stuy of MOGCS although we o not nee these onstrutions in this paper. Let A be a M 2 MN matrix whose rows form a olletion of MOGCS. Eah omplementary set has M sequenes of length N an the ith row of A onstitutes the ith omplementary set, suh a matrix sai to be a MOGCS matrix. Let it be partitione as A A 1 A 2 111AM, where eah Ai is an M 2 N matrix. We

3 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 54, NO. 3, MARCH will enote the ith row of A by a i, the ith row of A j by ai j an the (i; j)th salar entry of A by a i;j. Let H be a P 2 KQ matrix with omplex value entries having the same magnitue. For onveniene, we will assume that this magnitue is 1.Weefine the matrix operation K as A K H C(h 11;...;h 1K) 111 C(h 1(QK0K1) ;...;h 1(QK) ) C(h 21 ;...;h 2K ) 111 C(h 2(QK0K1) ;...;h 2(QK) ) C(h P 1 ;...;h PK ) 111 C(h P (QK0K1) ;...;h P (QK) ) (viewe as a PM 2 KQMN salar matrix) where C(h ij ;h i(j1) ;...;h i(jk) ) (h ij A 1 ;h i(j1) A 1 ;...;h i(jk) A 1 ) 111 (3) Definition 2.8: For q 2 an 0 r m, RM q(r; m) is efine to be the linear oe over q that is generate by the q -value vetors orresponing to the monomials of egree at most r in x 0;x 1;...;x m01. The rows of a generator matrix for the rth orer generalize Ree Muller oe RM q (r; m) over q an be represente by the olletion of all monomials of egree at most r in x 0;x 1;...;x m01. Let f : f0; 1g m! q be a generalize Boolean funtion in variables x 0 ;x 1 ;...;x m01. Let 0 j 0 <j 1 < 111<j <mbe a list of k inies an write x [x j x j 111x j ]. Let [ ] be a binary wor of length k. Then we efine the vetor fj x to be the omplex-value vetor with omponent i m01 i j j2 j equal to! f(i ;i ;...;i ) if i j for eah 0 <k, an equal to 0, otherwise. Here! is a omplex qth root of unity. In the ase where x an are of length zero, we efine fj x to be the omplex-value vetor assoiate with f. The following simple onsequenes an be easily obtaine from the efinitions. For any x efine as above (h ij A M ;h i(j1) A M ;...;h i(jk) A M ) M2KMN with (h ij A 1 ;h i(j1) A 1 ;...;h i(jk) A 1 ) enoting a sequene of length KN. Let eah element of a olumn vetor be ylially move one row up, upon pre-multipliation by S, a shift operator. We will enote the onjugate transpose of the matrix H by H 3. Consier the binary vetors 11 (1; 0; 1; 0;...; 1; 0) an 21 (1; 1; 0; 1; 1; 0;...; 1; 1; 0). In general, ij is onstrute by a repeate pattern of i ones an j zeros. Define matries D ij iag( ij) an D ij iag( ij), where ij is the binary omplement of ij. Then, the following lemma is an example onstrution of ZCZ sequenes from MOGCS sets [3]. Lemma 2.6: If A is a MOGCS matrix, then the rows of the matrix B A 2H form a ZCZ-(2MNQ;MP;N) if H satisfies the following: 1) HH 3 (MN)I where I is the ientity matrix; 2) HD 11SH 3 HD 11S 3 H 3. Similar onstrutions have been establishe in [3] for K 2. Notations in the following two subsetions are aopte from [2]. We restate them here for ompleteness. C. Generalize Ree Muller Coes For q 2, weefine a length L linear oe over q to be a set of q-value vetors (alle oewors) of length L that is lose uner the operation of aition over the ommutative group q. A set of oewors onstitute a oe C. Byaoset of C, we mean a set of the form a C where a is some fixe vetor over q. The vetor a is alle a oset representative for the oset a C. Definition 2.7: A map f : f0; 1g m! q of f0; 1g-value variables x 0 ;x 1 ;...;x m01 is alle a generalize Boolean funtion. Every suh funtion an be written in algebrai normal form as a sum of monomials of the form x j x j 111x j (in whih j 0 ;j 1 ;...;j r01 are istint). With eah generalize Boolean funtion f we ientify a length 2 m, q-value vetor f [f 0f 1 111f 2 01] in whih f i f(x 0 ;x 1 ;...;x m01 ) where [x 0 x 1 111x m01 ] is the binary expansion of the integer i. A omplex-value vetor f 0 is assoiate with every f, where fi 0! f an! is a omplex qth root of unity. When it is lear from the ontext, we will just use f to refer to all three. f 0 fj x an the Boolean funtion assoiate with the omplex vetor fj x an be written as f 1 x j (1 0 x j ): : 1 : For ompleteness, we repeat the following lemma from [2]. Lemma 2.9: Let f;g : f0; 1g m! q be generalize Boolean funtions in variables x 0;x 1;...;x m01. Let 0 j 0 <j 1 < 111 < j < m be a list of k inies an let [ ] an [ ] be binary-value vetors. Write x [x j x j 111x j ] an suppose 0 i 1 < i 2 < 111 < i l01 < m is a set of inies not in fj 0 ;j 1 ;...;j g. Denote y [x i x i 111x i ], then C(fj x;gj x )( ) ; C(fj xy ;gj xy )(): (4) D. Quarati Forms an Graphs Let Q : f0; 1g m! q be the generalize Boolean funtion efine by Q(x 0 ;x 1 ;...;x m01 ) q ij x i x j 0i<j<m where q ij 2 q, so that Q is a quarati form in m variables over q. We assoiate a labele graph G(Q) on m verties with Q as follows. We label the verties by 0; 1;...;m01 an join verties i an j by an ege labele q ij if q ij 6 0.Iff : f0; 1g m! q is a quarati funtion (i.e., a generalize Boolean funtion orresponing to a oewor of RM q (2;m)), then we efine G(f) to be the graph G(Q) where Q is the quarati part of f. We say that a graph G of the type efine above is a path if either m 1(in whih ase the graph ontains a single vertex an no eges), or; m 2 an G has exatly m 0 1 eges, all labele q2 whih form a Hamiltonian path in G.

4 1342 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 54, NO. 3, MARCH 2008 For m 2, a path on m verties orrespons to a quarati form of the type m01 q 2 1 x (01) x () (5) 1 where is a permutation of f0; 1;...;m0 1g. Let Q : f0; 1g m! q be a quarati form in m variables x 0;x 1;...;x m01. A quarati generalize boolean funtion is of the form [2]: m01 f Q g l x l g 0 l B. MOGCS From Ree Muller Coes In this setion, we prove the main theorem for onstruting mutually orthogonal Golay omplementary sets. Suppose Q : f0; 1g m! q is a quarati form in variables x 0 ;x 1 ;...;x m01.for0 t<2 k, efine the orere set S t (with the natural orer inue by the binary vetor [ ])tobe m01 Q g ix i g 0 q 2 i x j b x j x : ; 2f0; 1g (7) where g 0 ;g l 2 q are arbitrary. Consier the funtion fj x, obtaine by substituting x j in f. It follows that the graph of the funtion fj x is equal to the graph obtaine by eleting vertex j of G(f).By extension, if we have a list of k inies 0 <j 0 < 111<j <man write x [x j x j 111x j ] an [ ] then the graph of the funtion fj x is obtaine by eleting verties j 0;j 1;...;j of G(f). The final graph is inepenent of the hoie of. So for any, the quarati part of the funtion fj x is ompletely esribe by the graph obtaine from G(f) by eleting some verties. III. MUTUALLY ORTHOGONAL GOLAY COMPLEMENTARY SETS The following setion ollets known results on the iret onstrution of GCS from RM oes. We then present one of our main results ientifying permutations of a omplementary set that generate MOGCS. We onlue this setion by onstruting omplete MOGCS by a form of generalization of the reverse an onjugate metho of generating MOGCS introue in [6] an [5]. A. GCS From Ree Muller Coes We repeat [2, Th. 9] to be use later in the proof of our main theorem. Lemma 3.1: If the funtion fj x is a quarati funtion an if G(fj x ) is a path, then the omplex vetor fj x an any vetor of the form (f (q2)x r) j x form a Golay omplementary pair. 4 Where r 2 q is arbitrary an is either the single vertex of G(fj x ) when k m 0 1, or an en vertex of the path in G(fj x ) when 0 k<m01. The following result [2, Th. 12] establishes that the oewors of arbitrary seon-orer osets of RM q (1;m) lie in Golay omplementary sets. Lemma 3.2: Suppose Q : f0; 1g m! q is a quarati form in variables x 0;x 1;...;x m01. Suppose further that G(Q) ontains a set of k istint verties labele j 0 ;j 1 ;...;j with the property that eleting those k verties an all their eges results in a path. Let be the label of either en vertex in this path (or the single vertex of the graph when k m 0 1). Then for any hoie of g 0 ;g i 2 q m01 Q g i x i g 0 q 2 i is a Golay omplementary set of size 2 k1. x j x : ; 2f0; 1g 4 A omplementary pair is a omplementary set with just two sequenes. (6) where t b 2. The following theorem ientifies 2 k mutually orthogonal Golay omplementary sets with 2 k1 sequenes eah. We enote the all-one vetor of appropriate imension by 1 an the moulo 2 aition is enote by 8. Theorem 3.3: Suppose that G(Q) ontains a set of k istint verties labele j 0 ;j 1 ;...;j with the property that eleting those k verties an all their eges results in a path. Let be the label of either en vertex in this path (or the single vertex of the graph when k m 0 1). Then for any hoie of g 0 ;g i 2 q, the set S t is a Golay omplementary set for eah 0 t<2 k. Further, for t 0 6 t, S t an S t are mutually orthogonal omplementary sets. Proof: For any t, the sequenes in the set S t are obtaine by permuting the set of sequenes in (6). It follows iretly from Lemma 3.2 that S t is a Golay omplementary set for every 0 t<2 k. It remains to show that any two istint suh sets are mutually orthogonal. Let x [x j x j 111x j ], [ ] an b [b 0 b 1 111b ]. Let t b 2, t 0 b0 2 an t 6 t 0. To prove that S t an are mutually orthogonal, by Lemma 2.9 we write S t ; where L 1 an L 2 C f q 2 ( b) 1 x x ; ; 6 ; f q 2 ( b0 ) 1 x x L 1 L 2 C f q 2 ( b) 1 x x x f q 2 ( b0 ) 1 x x x C f q 2 ( b) 1 x x x f q 2 ( b0 ) 1 x x x (8) : (9)

5 C f q 2 b 1 1 x x IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 54, NO. 3, MARCH Consier the following sum in L 1 for a fixe ; 1 an 2 whih is zero for all l 6 0by Lemma 3.1. For l C f q 2 ( b) 1 x x x A f j x (0) A f q 2 x x (0) 2 m0k (16) f q 2 ( b0 ) 1 x x x for all 2f0; 1g k. Substituting this bak in (14), we obtain C f q 2 ( b) 1 1 x f q 2 ( b0 ) 1 2 x x x (10) C f q 2 (b b0 ) 1 x x f q 2 x (0) (01) (b8b )1 1 2 m0k1 (17) x (01) 1( 8 ) C f q 2 b11 x x f q 2 b0 1 2 x (11) x sine we are onsiering the ase b 6 b 0,wehaveb 8 b 0 6 (0), an so the linear funtional (b 8 b 0 ) 1 (regar as a funtion of ) is not equal to the zero funtional. Hene it is balane, i.e., takes on the values 0 an 1 equally often as varies. Hene the sum f q 2 b0 1 2 x x 1 (01) 1( 8 ) : (12) (01) (b8b )1 1 2 m0k1 : Sine in the term L 1, the funtion 1 ( ) takes the values 0 an 1 equally often, thus (12) vanishes for all l. Now onsier the following sum in L 2 for a given an : The oewor represente by the quarati Q will be in the seonorer RM oe. In our onstrution, Q is the oset leaer of the seonorer oset use to onstrut MOGCS. C f q 2 ( b) 1 x x x f q 2 ( b0 ) 1 x x x C f q 2 ( b) 1 x x f q 2 ( b0 ) 1 x : (13) x The above sum an be rewritten as C f q 2 (b b0 ) 1 x x C. Complete Mutually Orthogonal Golay Complementary Sets In this setion, we isuss the onstrution of omplete omplementary sets from the RM oes. For any given generalize Boolean funtion f in m variables x 0 ;x 1 ;...;x m01, we enote by ~ f, the funtion f (1 0 x 0 ; 1 0 x 1 ;...; 1 0 x m01 ). Let x enote the vetor 1 0 x.for a omplex sequene a, let ~a enote the sequene obtaine by reversing a an a 3 its omplex onjugate. Note that the quarati forms in the funtions f an ~ f are the same, thus, they have the same assoiate graph. The following lemma follows iretly from the above isussion an Lemma 3.2. Lemma 3.4: Suppose that there exist a set of k istint verties in the graph G(f ) suh that eleting those k verties an all their eges results in a path. Let be the label of either en vertex in this path (or the single vertex of the graph when k m 0 1). Then for eah 0 t<2 k, the orere set S t given by f q 2 x x ~f q 2 xj b xj x : ; 2f0; 1g (18) (01) (b8b )1 1 C f q 2 x x Note that C f q 2 x f q 2 x : (14) x ; x f q 2 x A f j x A f q 2 x x (15) x is a Golay omplementary set of size 2 k1, where t b 2. Consier the set S t (with the natural orer inue by the binary vetor [ ]), the following orollary is evient from Theorem 3.3. Corollary 3.5: The omplementary sets S t an S t orthogonal whenever t 6 t 0. are mutually The following theorem ientifies 2 k1 mutually orthogonal omplementary sets of size 2 k1.bys 3 t, we enote the orere set ontaining the omplex onjugate of the orresponing sequenes in S t.

6 1344 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 54, NO. 3, MARCH 2008 Theorem 3.6: Suppose that the quarati funtion f is as in Lemma 3.4, then the 2 k1 omplementary sets given by The algebra in (21) (23) applies equally to the seon orrelation term in (20) as well. It an be verifie that fs t : 0 t<2 k g[f S 3 t : 0 t<2 k g form omplete mutually orthogonal Golay omplementary sets. Proof: It is enough to show that any omplementary set S t is mutually orthogonal to any other omplementary set S 3 t. Let C f q 2 ( b) 1 x x ; f ~ 3 q 2 ( b0 ) 1 x C f q 2 (b) 1 x ; f ~ 3 q 2 ( b0 ) 1 x x L (19) where Thus C f q 2 ( b) 1 x x ~f 3 q 2 ( b0 ) 1 x x C f j x ; f ~ 3 q 2 x x x(81) 1 2 k 1 (01) (b8b )1 : (24) L ; C f q 2 ( b) 1 x x ~f 3 q 2 ( b0 ) 1 x x C f q 2 ( b) 1 x x x L C f q 2 x ; f ~ 3 j x(81) x C f j x ; f ~ 3 q 2 x 1 2 k x(81) ~f 3 q 2 ( b0 ) 1 x x x : (20) For a given 1 an 2, onsier the following sum of the first orrelation term in (20) C f q 2 ( b) 1 x x ~f 3 q 2 ( b0 ) 1 x x x C f q 2 ( b) 1 x x ~f 3 q 2 ( b0 ) 1 (1 0 x) C f q 2 x 1 x x ; ~ f 3 j x x (21) (01) b1 8b 1 1 (01) (8b )11 1( 8 ) 1 (01) C f q 2 x x 1 (01) b1 8b 1 8b 11 1 ; ~ f 3 j x (01) 1( 8 81) : (22) The sum in (22) is zero whenever ( ) 6 1. So when ( 1 2 ), the first orrelation term in (20) vanishes. Thus, summing (22) over all 1 6 2, we obtain 6 1 C f q 2 x x 1 (01) b1 8b 1 8b 11 2 k C f q 2 x ; ~ f 3 j x x 1 (01) b 11 2 k b18b 1(18) 1 (01) C f q 2 x x ; ~ f 3 j x(81) ; ~ f 3 j x(81) 2 k 1 (01) (b8b )1 : (23) 1 (01) (b8b )1 : (25) By Lemma 2.9 C f q 2 x ; f ~ 3 j x(81) x C f q 2 x ; f ~ 3 j xx xx (81)0 C f q 2 x ; f ~ 3 j xx xx (81)1 C f q 2 x ; f ~ 3 j xx xx 1 (81)0 C f q 2 x ; f ~ 3 j xx xx 1 (81)1 C f j xx ; f ~ 3 j xx (81)0 C f j xx ; f ~ 3 j xx (81)1 0 C f j xx 1; f ~ 3 j xx (81)0 0 C f j xx 1 ; f ~ 3 j xx (81)1 : (26) Similarly C f j x; f ~ 3 q 2 x x(81) C f j xx ; f ~ 3 j xx (81)0 0 C f j xx ; f ~ 3 j xx (81)1 C f j xx 1 ; f ~ 3 j xx (81)0 0 C f j xx 1 ; f ~ 3 j xx (81)1 : (27)

7 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 54, NO. 3, MARCH Substituting (26) an (27) bak in (25) L 2 C f j xx ; ~ f 3 j xx (81)0 0 C f j xx 1; ~ f 3 j xx (81)1 1 2 k 1 (01) (b8b )1 : (28) Sine G(f j x) forms a path, the funtion obtaine by substituting x in f shoul be of the form f j x q 2 m0 x (i01) x (i) m0 0 x (i) g i g i (29) for some permutation, an g i ;g 0 2 q. Let h 1 enote the funtion obtaine from f by substituting x an x for some binary vetor an let h 2 be the orresponing funtion when x an x 1. Further let (m 0 k 0 1) without loss of generality. Then h 1 q 2 m0k02 x (i01) x (i) m0 i: (i)6m0 x (i) g i g 0 (30) h 2 h 1 q 2 x (m0k02) g m0 : (31) The nonzero omponents of the omplex vetors a f j xx an b f j xx 1 are given by the funtions h 1 an h 2 respetively. The nonzero omponents of the vetor ~ f j xx (81)0 are given by the funtion h 3 q 2 m0k02 m0 i: (i)6m0 g m0 (1 0 x (i01) )(1 0 x (i) ) (1 0 x (i) )g i g 0 q 2 (1 0 x (m0k02)) ~ h 2 (32) an similarly the funtion orresponing to the sequene ~ f j xx (81)1 is ~ h 1. Moreover, sine the nonzero omponents in the sequene ~f 3 j xx (81)0 our when xx (1 8 )1, this sequene is exatly, ~ b 3. Also, the sequene represente by the omplex vetor ~f 3 j xx (81)1 is ~a 3. For any two omplex sequenes a an b, reall the ientity Thus C(a; ~ b 3 ) C( ~ b; a 3 )(0l) C(b; ~a 3 ): C f j xx ; ~ f 3 j xx (81)0 proving that L is zero, thus ompleting the proof. C f j xx 1; ~ f 3 j xx (81)1 All the sequenes in the omplementary sets ientifie by the Theorem 3.6 lie in the same oset whose oset leaer is given by the quarati form Q in the funtion f. Totally, 2 k2 sequenes are hosen from this seon-orer oset of the first-orer RM oe an orere to form two mutually orthogonal Golay omplementary sets. All other omplementary sets are obtaine by ertain permutations of the sequenes in eah of these sets. As note, Theorem 3.6 is a form of generalization of [6, Th. 11] where onstrution of a mutually orthogonal Golay omplementary set of a given omplementary pair was isusse 5 (whih an be easily extene to any omplementary set with even number of sequenes). The onstrution there involve reversing the sequenes an multiplying one of the inverte sequenes by 01. For polyphase sequenes, it was note in [5] that an aitional onjugation is neessary. Theorem 3.6 is a generalization of this approah in the sense that it onstruts 2 k omplementary sets whih are mutually orthogonal in aition to being orthogonal to the omplementary sets onstrute in Theorem 3.3. We outline an appliation of our iret onstrution in the next setion. IV. ZCZ SEQUENCES AND REED MULLER CODES The ZCZ sequenes an be onstrute by ertain operations (Kroneker prout followe by a length epenent permutation of olumns) on the matries forme by orering mutually orthogonal Golay omplementary sets as its rows. It was shown in [3] that this operation is equivalent to the onstrution of ertain orthogonal omplementary sets starting from smaller sets. Sine we have assoiate eah of these omplementary set to a oset of the RM oe, we have a iret relation between a ZCZ sequene an the orresponing oset. In orer to iretly onstrut the ZCZ sequenes from RM oes, it remains to ientify the subset of permutations that were use in Theorem 3.3 that woul generate ZCZ sequenes. The onept of orthogonal sets of ZCZ sequenes an large ZCZ sets with smaller subsets of larger interferene free winow than the original set were introue in [3]. Construtions there involves searhing for matries satisfying a number of onitions an beomes omputationally infeasible for large alphabets or large matrix imensions. We believe that the iret onstrution of Golay omplementary sets in this paper an be use to algebraially onstrut suh ZCZ sets. On possible approah is to assoiate a graph with a ZCZ sequene set an relate the IFW property of the ZCZ sequenes to that graph. Thus, ifferent subsets of the ZCZ set an all be assoiate with a subgraph of that graph. Further work is neee to onlusively answer these questions. V. CONCLUSION Motivate by the iret onstrution of Golay omplementary sets, we have presente a iret onstrution of omplete mutually orthogonal Golay omplementary sets from seon-orer osets of a q-ary generalization of the first-orer RM oes. The motivation behin the onstrution is to be able to onstrut ZCZ sequene sets with smaller subsets having superior orrelation properties than the omplete set. REFERENCES [1] J. A. Davis an J. Jewab, Peak-to-mean power ontrol in OFDM, Golay omplementary sequenes, an Ree-Muller oes, IEEE Trans. Inf. Theory, vol. 45, no. 9, pp , Nov [2] K. G. Paterson, Generalize Ree-Muller oes an power ontrol in OFDM moulation, IEEE Trans. Inf. Theory, vol. 46, no. 1, pp , Jan Mutually orthogonal Golay omplementary pairs were referre to as mates in their work.

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On the Intersetion of -Aitive Perfet Coes Josep Rifà, Senior Member, IEEE, Faina Ivanovna Solov eva, an Merè Villanueva Abstrat The intersetion problem for -aitive (extene an nonextene) perfet oes, i.e., whih are the possibilities for the number of oewors in the intersetion of two -aitive oes C an C of the same length, is investigate. Lower an upper bouns for the intersetion number are ompute an, for any value between these bouns, oes whih have this given intersetion value are onstrute. For all these -aitive oes C an C, the abelian group struture of the intersetion oes C \C is haraterize. The parameters of this abelian group struture orresponing to the intersetion oes are ompute an lower an upper bouns for these parameters are establishe. Finally, for all possible parameters between these bouns, onstrutions of oes with these parameters for their intersetions are given. Inex Terms Aitive oes, extene perfet oes, intersetion, perfet oes. I. INTRODUCTION AND BASIC DEFINITIONS Let n be the n-imensional vetor spae of all n-tuples over the finite fiel 2. The Hamming istane (v; s) between two vetors v; s 2 n is the number of oorinates in whih v an s iffer. A binary oe C of length n is a nonempty subset of n. The elements of a oe are alle oewors. The minimum istane of a oe C is the minimum value of (a; b), where a; b 2 C an a 6 b. The error orreting apability of a oe C is the value e b 01 2 an C is alle an e-error orreting oe. Two binary oes C 1 an C 2 of length n are isomorphi if there exists a oorinate permutation suh that C 2 f() j 2 C 1 g. They are equivalent if there exists a vetor a 2 n an a oorinate permutation suh that C 2 fa () j 2 C 1 g. A binary perfet 1-error orreting oe (briefly in this orresponene, binary perfet oe) C of length n is a subset of n, with minimum istane 3, suh that all the vetors in n are within istane one from a oewor. For any t > 1 there exists exatly one binary linear perfet oe of length 2 t 0 1, up to equivalene, whih is the well-known Hamming oe.anextene oe of the oe C is a oe resulting from aing an overall parity hek igit to eah oewor of C. The intersetion problem for binary perfet oes was propose by Etzion an Vary in [8]. They presente a omplete solution to this problem for binary Hamming oes: for eah t 3, there exist two Hamming oes H 1 an H 2 of length n 2 t 0 1 suh that the /$ IEEE Manusript reeive Deember 2, 2006; revise May 28, This work has been partially supporte by the Spanish MEC an the European FEDER Grant MTM an also by the UAB Grant PNL Part of this manusript was proue uring a visit by F. I. Solov eva to the Autonomous University of Barelona uner the Grant VIS The material in this orresponene was presente in part at the International Workshop on Coing an Cryptography (WCC 07), Versailles, Frane, April J. Rifà an M. Villanueva are with the Department of Information an Communiations Engineering, Universitat Autònoma e Barelona, Bellaterra, Spain ( josep.rifa@autonoma.eu; mere.villanueva@ autonoma.eu). F. I. Solov eva is with the Sobolev Institute of Mathematis an Novosibirsk State University, Novosibirsk, Russia ( sol@math.ns.ru). Communiate by T. Etzion, Assoiate Eitor for Coing Theory. Digital Objet Ientifier /TIT