A Family of Binary Sequences from Interleaved Construction and their Cryptographic Properties

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1 Contemorary Mathematics A Family of Binary Sequences from Interleaed Construction and their Crytograhic Proerties Jing Jane He, Daniel Panario, and Qiang Wang Abstract. Families of seudorandom sequences with low cross correlation hae imortant alications in communications and crytograhy. Among seeral known constructions of sequences with low cross correlations, interleaed constructions roosed by Gong uses two sequences of the same eriod with two-leel autocorrelation. In this aer, we study the balance roerty and the cross correlation of interleaed sequences such that the base sequences may not hae the same eriod, or they may not hae two-leel autocorrelation. In articular, we study the interleaed sequences of two Legendre sequences of eriods and q, resectiely, where and q are odd rime numbers. 1. Introduction Pseudorandom sequences are widely used in comuter science and engineering including alications to sread sectrum communication systems, radar systems, signal synchronization, simulation and crytograhy [3]. The seudorandom sequences in a good family should be easy to generate (ossibly with hardware or software), hae good distribution roerties which make them aear statistically to be random, hae low cross correlation alues so that each sequence may be searated from the others in the family, and arise from some underlying algebraic structure so they can be analyzed using standard mathematical tools. In this aer we continue the study began in [4] of constructed families of sequences from interleaed structure which enjoy many nice roerties. We seek families of binary sequences with low cross correlation, good randomness, and large linear comlexity. Such kind of families of sequences hae imortant alications in code-diision multile-access (CDMA) communications and crytograhy [3]. Correlation is a measure of the similarity, or relatedness, between two henomena. In signal rocessing, cross-correlation is a measure of similarity of two waeforms as a function of a time-lag alied to one of them. So the sequences with low cross correlation emloyed in CDMA communications can successfully combat interference from the other users who share a common channel Mathematics Subject Classification. Primary 94A55; Secondary 11T06. Key words and hrases. interleaed sequences, cross correlation. Research of Daniel Panario and Qiang Wang is artially suort by NSERC of Canada. Version: December 14, c 0000 (coyright holder)

2 JING JANE HE, DANIEL PANARIO, AND QIANG WANG Let l be a rime number and F l be the finite field of l elements. We consider l-ary sequences which are eriodic sequences with entries in F l. In articular, if l =, then the sequences are binary sequences. Let S be a family of k l-ary sequences with the same eriod, and let C max be the maximum magnitude of the nontriial autocorrelation and cross correlation alues of the sequences in S. It is known that C max which is called the Welch bound (see [7] or [9]). k 1 k 1 Known sequences can be emloyed to construct new families of sequences using the interleaed structure algorithm. For examle, Gong [4] used short two-leel autocorrelation l-ary sequences of eriod to obtain a (,, + 3) signal set which is otimal with resect to the Welch bound. Each sequence in this family of sequences is balanced and has large linear san. In fact, this construction can generate + 1 sequences so that we hae a (, + 1, + 3) signal set (see age 364 in [3]). For l =, Wang and Qi [8] emloyed Legendre sequences with twin rime eriods 3 ( mod 4) and + to construct a (( + ), + 3, 3 + 4) signal set, and the balance roerty of such family was also studied. In this aer, we extend Gong s construction of interleaed structures where two sequences of equal length with two-leel autocorrelation are used, to a general interleaed construction which uses two sequences of different length, and one sequence may not hae two-leel autocorrelation. In articular, we study interleaed Legendre sequences with any two odd rime eriods and q. Without loss of generality, ( we assume ) that q. In the case 3 (mod 4), we obtain a (q, q + 1, q + 1 ( + 1) + q) signal set which generalizes Wang and Qi s result. ( In the case 1 ( mod 4), we obtain a (q, q + 1, q + 1 ( + 1) + 3q ) signal set.. Preliminaries Some notations and reliminaries for sequence construction which are used throughout the aer are gien next. We introduce the terminologies about sequences in the general case, that is, most of the definitions are based on l-ary sequences. Again l is a rime and eery element in the sequence is oer F l. Later we focus on binary sequences. Let be a ositie integer and let a = (a 0,..., a 1 ) be an l-ary sequence of eriod. For any integer i 0, let the left shift oerator act on a by L i (a) = (a i, a i+1,..., a i+ 1 ). In articular, define L (a) = (0,..., 0). Definition.1. Two sequences a = (a 0,..., a 1 ) and b = (b 0,..., b 1 ) of the same eriod are called (cyclically) shift equialent if there exists an integer k such that a i = b i+k, for all i 0. In this case, we write a = L k (b), or simly a b. Otherwise, they are called cyclically shift distinct. Definition.. The cross correlation function C a,b (τ) of two l-ary sequences a and b of eriod is defined in [] as 1 C a,b (τ) = ω ai b (i+τ)( mod ), τ = 0, 1,..., i=0 where ω is a rimitie l-th root of unity. If b = a, then denote C a (τ) = C a,b (τ) as the autocorrelation of a.

3 INTERLEAVED CONSTRUCTION 3 Definition.3. Let s j = (s j,0, s j,1,..., s j, 1 ), 0 j < r, be r shift-distinct l-ary sequences of eriod. Let S = {s 0,..., s r 1 }, and let δ = max C si,s j (τ) for any 0 τ <, 0 i, j < r, where τ 0 if i = j. The set S is said to be a (, r, δ) signal set and δ is called the maximal correlation of S. We say that the set S has low correlation if δ c where c is a constant. 3. Algorithm for Constructing Sequences of Period s t The matrix form of a sequence is an imortant tool to study interleaed structures. Definition 3.1. [4] [5] Fix two ositie integers s and t where both s and t are not equal to 1. Gien an l-ary sequence a = (a 0,..., a s 1 ) of eriod s (a is called base sequence) and a sequence e = (e 0,..., e t 1 ), for each 0 i t such that e i Z s and e t 1 = (e is called shift sequence), let u = (u 0,..., u st 1 ) be an l-ary sequence of eriod s t. We arrange the elements of the sequence u into an s t matrix as follows: A u = u 0 u t 1... u (s 1) u (s 1)t+t 1 satisfying that each column of A u is a shift of a. Let A j be the j th column. Then A = (A 0,..., A t 1 ) and A j = L ej (a) and L (a) = (0,..., 0). The matrix A u is called the matrix form of sequence u, and u is called an interleaed sequence from the base sequence a and the shift sequence e. Gien a base sequence a and a shift sequence e, an interleaed sequence u is uniquely determined. So we also say u is an (s, t)-interleaed sequence associated with (a, e). Moreoer, using another sequence b = (b 0,..., b s 1 ) of the same eriod s, Gong [5] constructed a family of interleaed (s, s)-sequences with the desired roerties. Here we consider the case where s is not neccessarily equal to t. Algorithm 3.. Let s and t be two ositie integers. Suose that a = (a 0,..., a s 1 ) and b = (b 0,..., b t 1 ) are two l-ary sequences of eriods s and t, resectiely. (1) Choose e = (e 0,..., e t 1 ) as the shift sequence for which the first t 1 elements are oer Z s and e t 1 =. Moreoer, if we let d i 1 = e i e i 1, then we choose e such that d 0, d 1..., d 1 is in an arithmetic rogression with common distance d 0. () Construct an interleaed sequence u = (u 0,..., u st 1 ), whose j th column in the matrix form is gien by L ej (a). (3) For 0 i < st 1, 0 j t, define s j = (s j,0,..., s j,st 1 ) as follow: { ui + b s j,i = j+i, 0 j t 1, u i, j = t. (4) Define the family of sequences S = S(a, b, e) as S = {s j j = 0, 1,..., t}, where a is the first base sequence, e is the shift sequence, and b is the second base sequence.

4 4 JING JANE HE, DANIEL PANARIO, AND QIANG WANG Because it is desirable to hae more sequences in a family, without loss of generality, we can assume that s t. We also concentrate on binary sequences (l = ) for the rest of aer. In articular, we are interested in Legendre sequences of rime eriods since they are a nice class of sequences with the randomness roerties. Definition 3.3. Let be an odd rime. The Legendre sequence s = {s i i 0} of eriod is defined as [3] 1, if i 0 (mod ); s i = 0, if i is a quadratic residue modulo ; 1, if i is a quadratic non-residue modulo. The following is the known result for the autocorrelation of Legendre sequences. Proosition 3.4. [1]. Let s be a Legendre sequence of rime eriod as aboe. Then, if 3 (mod 4), C s (τ) = { 1, }, and if 1 (mod 4), C s (τ) = {1, 3, }. We use any two odd rimes and q to denote the eriods of two Legendre sequences a, b. For similar reasons as aboe, we assume without loss of generality that q. In the following we study the family of interleaed (, q)-sequences constructed as in Algorithm 3. by using Legendre sequences a, b of eriods and q, resectiely. Here we gie an examle of interleaed (3, 5)-sequence by using two Legendre sequences of eriod 3 and 5 resectiely. Examle 3.5. (1) Let the first and second base sequences be Legendre sequences a = (1, 0, 1) and b = (1, 0, 1, 1, 0), resectiely. We ick u a shift sequence e = (1,, 1, 1, ). () We construct the interleaed sequence u associated with a and e. The matrix form of u is The interleaed sequence is (3) We hae (4) The family is u = (0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0). s 0 = u + b = ( ); s 1 = u + L 1 (b) = ( ); s = u + L (b) = ( ); s 3 = u + L 3 (b) = ( ); s 4 = u + L 4 (b) = ( ). S = {s 0, s 1, s, s 3, s 4, s 5 = u}.

5 INTERLEAVED CONSTRUCTION 5 4. Balance Proerty For any sequence s, let N 0 (s) denote the number of zeros of the sequence s. Theorem 4.1. Let us choose a as the first base sequence with eriod and b as the second base sequence with eriod w. Then using Algorithm 3. we construct a family S(a, b, e) = {s j j = 0, 1,..., w} with the roerty that the number N 0 (s j ) of zeros in one eriod of each sequence s j is: (w 1) N 0 (a) +, j = w; N 0 (a) (N 0 (b) 1) + ( N 0 (a)) (w N 0 (b)) +, b j+ = 0, j w 1; N 0 (a) N 0 (b) + ( N 0 (a)) (w N 0 (b) 1), b j+ = 1, j w 1. Proof. Case 1. j = w. In this case s w = u. Then we arrange the elements of u into the w matrix A u = [L e0 (a),..., L ew (a), 0]. Since each of the first w 1 columns is just a shift of a and a contains exactly N 0 (a) zeros, then the number of zeros in u is Case. 0 j w 1. We denote Then, we hae N 0 (u) = (w 1) N 0 (a) +. B = b j b j+1 b j+ b j b j+1 b j+.... b j b j+1 b j+ s j = u + L j (b) = A u + B. If b j+ = 0, then there are N 0 (b) 1 zeros in b j,..., b j+w, and w N 0 (b) ones in b j,..., b j+w. So. N 0 (s j ) = N 0 (a) (N 0 (b) 1) + ( N 0 (a)) (w N 0 (b)) +. If b j+ = 1, then there are N 0 (b) zeros in b j,..., b j+, and w N 0 (b) 1 ones in b j,..., b j+. So N 0 (s j ) = N 0 (a) N 0 (b) + ( N 0 (a)) (w N 0 (b) 1). We obsere that the number of zeros in each sequence from the constructed family is indeendent of the chosen shift sequence e. The urose for adding strong conditions on e is to get the desired correlation roerty. Wang and Qi [8] gie the balance roerty of the interleaed construction with two Legendre sequences of twin rime eriods and +, resectiely. Howeer, there are some tyos in their result. In the next corollary that follows immediately from the reious theorem, we gie the balance roerty to the interleaed construction with two Legendre sequences of any two rime eriods and q which also corrects Wang and Qi s result. We remark that a Legengre sequence of length is balanced and so it contains exactly ( 1)/ zeros.

6 6 JING JANE HE, DANIEL PANARIO, AND QIANG WANG Corollary 4.. Let and q be two rime numbers and S(a, b, e) = {s j j = 0, 1,..., q} be the family of interleaed sequences constructed by Algorithm 3., where the base sequences a and b are Legendre sequences of eriod and q resectiely. Then the number of zeros N 0 (s j ) in one eriod of s j (0 j q) is N 0 (s j ) = (q+1) q+1, j = q; (q+1)+, b j+q 1 = 0, j q 1; (q 1), b j+q 1 = 1, j q Cross Correlation Gong [4] uses two m-sequences of the same eriod to construct a family of long sequences from interleaed structures with the desired roerties. Then she generalizes this idea to use seeral tyes of two-leel autocorrelation sequences of eriod [5]. Also in [5] the criterion for choosing the shift sequence e = (e 0,..., e 1 ) to get maximum correlation alue + 3 is gien like {e j e j+s 0 j < s} = s, for all 1 s < (Theorem in [5]). In Algorithm 3. we can use two sequences with different eriods and w and restrict the shift sequence e to satisfy that e 1 e 0, e e 1,..., e w e is in an arithmetic rogression. Before we roide the roof of the cross correlation alues we need the following results. Proosition 5.1. (Proosition 3 in [5]) Let a and b be two sequences oer F of eriod N. For τ 0, we hae (1) < a, b >= C a,b (0); () < a, L τ (b) >= C a,b (τ); (3) < L i (a), L j+τ (a) >= C Li (a),l j (a)(τ) = C a (j i + τ) where i, j 0; (4) for c, d F, < a + c, b + d >= ( 1) c+d < a, b >. In our construction, the shift sequence e = (e 0,..., e ) lays an imortant role in comuting the alues of the correlation function. To determine the alues of the cross correlation of the constructed sequences in S, we need to study the number of roots of e j+s e j + r 0 (mod ) for 0 j, s < w, 0 r <. We obsere that the index j +s could go beyond the eriod of the shift sequence e. For conenience, we introduce the extended sequence defined by e j+w = 1 + e j. Hence, the extended shift sequence e is (e 0,..., e w,, e w = 1 + e 0,..., e w = 1 + e w, ). For the extended one, we still use the same notation e for notational conenience. In the following when encountering an element of e out of the range of the original shift sequence, we just use the extended shift sequence, i.e. e j+w = 1+e j. The following roosition studies the matrix form of an interleaed sequence. It is a modification of Proosition 4 in [5] from = w to arbitrary and w. Proosition 5.. Let u be a (, w) interleaed sequence associated with (a, e). We extend the sequence (e 0,..., e w, ) to e = (e 0,..., e ) by defining e j+w = 1 + e j, for j = 0,..., w 1. For τ 0, let T = (T 0, T 1,..., T ) be the matrix form of L τ (u). If we write τ = rw + s, 0 r <, 0 s < w then T j = L r+es+j (a). Proof. We use one index k for 0 k < w 1, or two indices (i, j) for 0 i < and 0 j < w, to show the osition of an element in the matrix form of

7 INTERLEAVED CONSTRUCTION 7 an interleaed sequence. Let u 0 u A u =... u ( 1)w u ( 1)w+ = u 0,0 u 0,... u 1,0 u 1, be the matrix form of u. Let A j be the j th column. We obsere that the first entry in the sequence L τ (u) is u r,s in A u. From the definition of the interleaed sequences, we hae u r,+j = u r+1,j for each j with s j < w. So T has the following matrix form T = u r,s u r, u r+1,0 u r+1,s 1 u r+1,s u r+1, u r+,0 u r+,s 1. u 1,s u 1, u 1,0 u 1,s 1. u r 1,s u r 1, u r,0 u r,s 1 Therefore, for 0 j < w s, we recall (5.1) T j = L r (A s+j ) = L r (L es+j (a)) = L r+es+j (a). For w s j < w, we hae (5.) T j = L r+1 (A j (w s) ) = L r+1 (L e j (w s) (a)) = L r+1+e j (w s) (a). For 0 j w 1, define (5.3) e j+w = 1 + e j. Then, the sequence e of eriod w can be exanded to a sequence of eriod w. For simlicity we still use the symbol e for that sequence. Alying (5.3) for w s j w 1 we obtain 1 + e j (w s) = e j (w s)+w = e s+j. Substituting it into (5.), we get that T j = L r+ej+s (a). Together with (5.1), the result follows. From Proosition 5., the following modification of Lemma 1 in [5] is immediate. Lemma 5.3. Let S be a family of sequences constructed using Algorithm 3., and let τ = rw + s with 0 s < w and 0 r <. Then, for s k S \ {s w }, the j th column sequence of L τ (s k ) is gien by L r+es+j (a) + b k+s+j, 0 j < w. Moreoer, the j th column sequence of s w is gien by L r+es+j (a). Remark 5.4. Let s h, s k S be two sequences in the (, w) interleaed sequence family from Algorithm 3.. Let S = (S 0,..., S ) and T = (T 0,..., T ) be the matrix forms of s h and L τ (s k ), resectiely, where τ 0. Proosition 5.1 () and Proosition 5. imly that the cross correlation between s h and s k can be comuted as C h,k (τ) = < S j, T j >..

8 8 JING JANE HE, DANIEL PANARIO, AND QIANG WANG Similarly we hae the following modification of Lemma in [5]. Lemma 5.5. For 0 h, k w, suose that s h and s k are two sequences in S. If τ = rw + s, 0 s < w, 0 r <, then the correlation function between s h and s k is C h,k (τ) = { ( 1)b h+j b k+s+j, 0 h w 1, ( 1)b k+s+j, h = w. Proof. Let S = (S 0,..., S ) and T = (T 0,..., T ) be the matrix forms of s h and L τ (s k ), resectiely. According to Remark 5.4, C h,k (τ) = < S j, T j >. First we consider 0 h w 1. From Lemma 5.3, for 0 j < w, we hae S j = L ej (a) + b k+j, T j = L r+es+j (a) + b k+s+j. Alying Proosition 5.1 (4) and then Proosition 5.1 (3), we get < S j, T j > = < L ej (a) + b h+j, L r+es+j (a) + b k+s+j > The case h = w can be shown similarly. = ( 1) b h+j+b k+s+j < L ej (a), L r+es+j (a) > = ( 1) b h+j b k+s+j. Remark 5.6. We recall that e = (e 0,..., e ), where e =. In order to study e j+s e j for 0 j, s < w, we introduce three elements:, 1, and, and define =, k = 1 and k = for any integer k. Notation 5.7. Let d i = e i+1 e i be in an arithmetic rogression for i = 0,..., w 3. For 0 r <, 1 s < w, let N(r, s) be the number of j with 0 j < w such that e j+s e j + r 0 (mod ). Lemma 5.8. For 0 r <, 1 s < w, let N(r, s) be the number of j with 0 j < w such that e j+s e j + r 0 (mod ), then N(r, s) w + 1. Proof. For the shift sequence e = (e 0,..., e ), we need d 0 = e 1 e 0, d 1 = e e 1,..., d w = e e w to be in an arithmetic rogression with a constant difference d. So we deduce that e i = e 0 + id 0 + i(i 1)d. Since j + s = w gies e j+s e j + r =, we consider the following two cases: (1) j + s < w. We hae (j + s)(j + s 1)d e j+s e j + r = (j + s)d 0 + jd 0 ( ) = sd 0 + r + (s s)d + sdj. j(j 1)d This is a linear equation modulo. It has no solution or one solution for j when j <. Thus in the extended e, there are at most w + 1 j s satisfying e j+s e j + r 0 (mod ), and j < w. + r

9 INTERLEAVED CONSTRUCTION 9 () j < w < j + s. The aboe exression changes into e j+s e j + r = 1 + e (j+s)( mod w) e j + r = 1 + (j + s w)d 0 + (j + s w)(j + s 1)d jd 0 = 1 + (s w)d 0 + r + (s w) (s w) d + (s w)dj. j(j 1)d This equation has at most one solution for j <, and so has at most w + 1 solutions when j < w. We comment that a different roof of the aboe result is in [8]. From Lemma 5.5, we find that the cross correlation of any two sequences in the constructed family is related to C a (e j+s e j +r). Let a be a balanced sequence of eriod with ( 1)/ zeros. Then we can denote C a ( 1 ) = C a ( ) = 1 and C a ( + k) =. Indeed, the sequence a will turn into the zero sequence after shifting it for infinitely many times. Therefore, C a ( 1 ) = 1 i=0 ( 1)ai 0 = 1 since it has ( 1)/ zeros in any eriod. The same reason leads to C a ( ) = 1. Similarly, we get C a ( ) = 1 i=0 ( 1)0 0 =. We comment that these notations are conenient in the case j + s = w 1 or j = w 1, that is, when one of the terms in e j+s e j is. Theorem 5.9. Let a be a two-leel autocorrelation sequence with eriod and b be a balanced low cross correlation sequence of eriod w with the maximal absolute alue of nontriial autocorrelation equal to δ b. The family of sequences S generated by Algorithm 3. is a (w, w + 1, δ 1 ) signal set, where {( w } δ 1 = max + 1 ( + 1) + w, δ b. Proof. We know that the autocorrelation of a is C a (τ) = { 1, }. Case 1. τ = 0. It follows τ = 0 w + 0, that is, r = s = 0. By Lemma 5.5 we hae { C h,k (0) = ( 1)b h+j b k+j C a (0), 0 h w 1, ( 1)b k+j C a (0), h = w. Since C a (τ) = { 1, }, we hae C a (0) ( 1) b h+j b k+j = ( 1) b h+j b k+j = C b (h k). We want to find the nontriial correlation alue. So when τ = 0, the two sequences s h, s j should be different. Hence C h,k (0) δ b. When one of the sequences s h or s k is u, we hae C h,k (τ) = ( 1) b j+k, because the sequence b is balanced. Case. τ = rw + 0 and 0 < r <. In this case since s = 0 and r 0, we hae e j+s e j + r = r (mod ) and so N(r, 0) = {0}. Then, for 0 h, k < w, we hae + r

10 10 JING JANE HE, DANIEL PANARIO, AND QIANG WANG C h,k (τ) = ( 1) b h+j b k+j C a (r) = C a (r) C b (h k) = ( 1) C b (h k). Hence C h,k (τ) w. When one of the sequences s h or s k is u, we hae C h,k (τ) = ( 1) bw+j C a (r) = C a(r) ( 1) bj Ca (r) = 1. Case 3. τ = rw + s (0 r <, 0 < s < w). For 0 h, k w, we hae, e j+s e j + r 0 (mod ), = 1, e j+s e j + r 0 (mod ), 1, e j+s e j + r = 1 or. We only need to categorize the alues of N(r, s) to calculate C h,k (τ). We consider the following cases: Case 3.1: If N(r, s) = 0, there is no j satisfying e j+s e j + r 0 (mod ) and so for 0 h, k < w and 0 j w 1, we hae = 1. Thus, C h,k (τ) = ( 1) C b (k + s h). The maximal absolute alue of the correlation alues is C h,k (τ) Cb (k + s h) = w. When one of the sequences s h or s k is u, the correlation alue is C a(e j+s e j + r) ( 1) bj 1. Case 3.: If N(r, s) = 1, there is one j, say j 0, satisfying e j0+s e j0 + r 0 (mod ), then, for 0 h, k < w, we hae C h,k (τ) = q 1 ( 1) b h+j b k+s+j = ( 1) b h+j 0 b k+s+j0 Ca (0) + j j 0 ( 1) b h+j b k+s+j ( 1) q 1 = ( 1) b h+j 0 b k+s+j0 ( + 1) + ( 1) ( 1) b h+j b k+s+j = ( 1) b h+j 0 b k+s+j0 ( + 1) + ( 1) Cb (k + s h) = ( 1) C b (k + s h) + {±( + 1)}. Therefore C h,k (τ) ( + 1) + w.

11 INTERLEAVED CONSTRUCTION 11 Hence, When one of the sequences s h or s k is u, we get C h,k (τ) = ( 1) b k+s+j = ( 1) b k+s+j 0 Ca (0) + j j 0 ( 1) b k+s+j ( 1) = ( 1) b k+s+j 0 ( + 1) + ( 1) ( 1) b k+s+j. C h,k (τ) ( + 1) + 1. Case 3.3: If N(r, s) =, there are two j s, say j 0 and j 1, satisfying e j+s e j +r 0 (mod ). Then, for 0 h, k < w, Therefore, Thus, C h,k (τ) = ( 1) b h+j b k+s+j = ( 1) b h+j 0 b k+s+j0 Ca (0) + ( 1) b h+j 1 b k+s+j1 Ca (0) + j j 0,j 1 ( 1) bh+j bk+s+j ( 1) = {±( + 1)} + {±( + 1)} + ( 1) C b (k + s h). C h,k (τ) ( + 1) + w. When one of the sequences s h or s k is u, we obtain C h,k (τ) = ( 1) b k+s+j = {±( + 1)} + {±( + 1)} + ( 1) ( 1) b k+s+j. C h,k (τ) ( + 1) + 1. Case 3.4: As we can see from the reious subcases, the maximum magnitude of the correlation alue in our estimations grows as N(r, s) increases. Hence we only gie here the estimates for the largest alue of N(r, s). If N(r, s) = w + 1, then for 0 h, k < w, we hae C h,k (τ) = {±( + 1)} + + {±( + 1)} + ( 1) C b (k + s h), where the number of coies of {±( + 1)} is w + 1. Thus ( w ( w C h,k (τ) + 1 ( + 1) w = + 1 ( + 1) + w. If one of the sequences s h or s k is u, ( w C h,k (τ) + 1 ( + 1) + 1.

12 1 JING JANE HE, DANIEL PANARIO, AND QIANG WANG Recall that δ 1 denotes the maximum magnitude among all the cross correlation alues and nontriial autocorrelation alues of S. Hence we obtain a (w, w +1, δ 1 ) signal set, where {( w } δ 1 = max + 1 ( + 1) + w, δ b. The reious theorem has full generality. We do not require that two sequences a and b hae the same eriod, nor that the sequence b is two-leel autocorrelated. In the next theorem we focus on an imortant case when both two base sequences hae two-leel autocorrelation (and thus they are balanced as well). Theorem If both a and b are two-leel autocorrelation sequences with eriods and w, resectiely, then the family of sequences constructed by Algorithm 3. is a (w, w + 1, δ ) signal set with ( w δ = + 1 ( + 1) + 1. Proof. The roof is similar to Theorem 5.9 but requires further refinements. In articular, in this case δ b = 1 because sequence b is two-leel autocorrelated. When k + s h 0 (mod w), the correlaton function C b (k + s h) = ( 1) b k+s+j b h+j = w. It is equialent to that the sequence L k+s (b) L h (b) is the zero sequence. By Lemma 5.5, we hae C h,k (τ) = = ( 1) b h+j b k+s+j ( w ( w + 1. The last inequality holds because there is at most one j such that e j+s e j + r 0 (mod ). When k + s h 0 (mod w), let j 0,..., j w be the solutions from 0 to such that e j+s e j +r 0 (mod ). We obsere that this is the case that gies the worst ossible cross correlation alue. Then the correlaton

13 INTERLEAVED CONSTRUCTION 13 function satisfies C h,k (τ) = ( 1) b h+j b k+s+j = ( 1) b h+j 0 b k+s+j0 Ca (0) + + ( 1) b h+j w b k+s+j w C a (0) + ( 1) b h+j b k+s+j ( 1) j j 0,...,j w = {±( + 1)} + + {±( + 1)} + ( 1) Cb (k + s h) ( w + 1 ( + 1) + 1. Hence, the maximal absolute alue of the correlation alues is ( w C h,k (τ) + 1 ( + 1) + 1. We emhasize again that it is more desirable to generate more sequences in a family of sequences and thus we can assume w. This means that we use a as the first base sequence and b as the second base sequence. In articular, when = w and the two base sequences are two-leel autocorrelated, we recoer Gong s result (Theorem in [5] or age 364 in [3]). Corollary When and w are equal, the family of sequences generated by Algorithm 3. is a (, + 1, + 3) signal set. Next we obtain a few results when both base sequences a and b are Legendre sequences with the eriod equal to rime number and q, resectiely. We recall that the Legendre sequence a of eriod 3 (mod 4) has ideal two-alued autocorrelation C a (τ) = { 1, }. Corollary 5.1. Fix a rime number 3 (mod 4) and any other rime q. The family of sequences S generated by Algorithm 3. from two Legendre sequences of eriods and q is a (q, q + 1, δ) signal set, where δ = δ 1 = ( q + 1 ) ( + 1) + q. Furthermore, when both and q are congruent to 3 (mod 4) we obtain ( q δ = δ = + 1 ( + 1) + 1. We remark that Wang and Qi s result is the case when taking two Legendre sequences a and b with twin rime eriods 3 (mod 4) and q = +, resectiely. Corollary [8] Let two Legendre sequences of twin rime eriods and +, where 3 (mod 4) be the base sequences under the construction of the algorithm. The maximum magnitude of nontriial cross correlation alues of this constructed family is

14 14 JING JANE HE, DANIEL PANARIO, AND QIANG WANG If Legendre sequence a has the rime eriod 1 (mod 4), then a is not twoleel autocorrelated. In fact, it is three-leel correlated. In this case, we slightly modify this Lengedre sequence so that we hae a two-leel autocorrelation sequence a. Lemma Let a be a Legendre sequence with rime eriod 1 (mod 4). If we let the entries a i = with i 0 (mod ) and a i = a i with i 0 (mod ), then the modified Legendre sequence a has two-alued autocorrelation. Proof. The roof is on age 94 in [6]. Now we estimate the maximal cross correlation of the interleaed construction from a and b by using the maximal cross correlation of the interleaed construction from a and b. Theorem Fix a rime number 1 (mod 4) and any other rime q. The family of sequences S generated by Algorithm 3. from two Legendre ) sequences of eriods and q is a (q, q + 1, δ 3 ) family, where δ 3 = + 1 ( + 1) + 3q. ( q Proof. Fix a rime 1 (mod 4), we use the modified Legendre sequence a = (, a 1,..., a 1 ) as the first base sequence in Algorithm 3.. Then by Theorem 5.9, the family of sequence S constructed from a and b has the maximal correlation alue ( q C h,k (τ) δ 1 = + 1 ( + 1) + q. For each sequence s j S, we hae s j = u +L j (b) and eery column of the matrix form of the interleaed sequence u contains a. Therefore, there are q 1 coies of s in eery sequence s j S for j = 0,..., q. If sequence s j in the family of sequences is constructed by Algorithm 3. from ordinary Legendre sequence a and b, then the difference between s j and the corresonding s j haens exactly at these entries of in s j. Then the correlation function of any two sequences s j = (s j0, s j1,..., s j(q 1) ), s k = (s k0,..., s k(q 1) ) in the family S constructed from the ordinary Legendre sequence a and b is C h,k (τ) C h,k (τ) + (q 1) ( q + 1 ( + 1) + q + (q 1) ( q = + 1 ( + 1) + 3q. 6. Conclusions In this aer we study interleaed constructions of low cross correlation sequences including their balance and cross correlation roerties. In articular, we generalize Gong s results of interleaed sequences of two-leel autocorrelation sequences of the same eriod to the case where the eriods are distinct. Moreoer, we study interleaed sequences constructed from two Legendre sequences of eriods and q, where and q are rime numbers. We gie the balance and cross correlation roerties as well. For further work, it would be interesting to hae a study of the linear comlexity, merit factors, and aeriodic correlation of these constructions.

15 INTERLEAVED CONSTRUCTION 15 Acknowledgements We thank the referee for ery helful suggestions which lead to a significant imroement of the results. References [1] C. Ding, T. Helleseth and W. Shan, On the linear comlexity of Legendre sequences, IEEE Trans. Inform. Theory, ol. 44, , [] S. W. Golomb, Shift Register Sequences, Aegean Park Press, 198. [3] S. W. Golomb and G. Gong, Signal Design for Good Correlation, Cambridge Uniersity Press, 005. [4] G. Gong, Theory and alications of q-ary interleaed sequences, IEEE Trans. Inform. Theory, ol. 41, , [5] G. Gong, New design for signal sets with low cross correlation, balance roerty, and large linear san: GF () case, IEEE Trans. Inform. Theory, ol. 48, , 00. [6] M.R. Schroeder, Number Theory in Science and Communication: with Alications in Crytograhy, Physics, Digital Information, Comuting, and Self-Similarity, Third Edition, Sringer, Berlin, [7] V. M. Sidelniko, On mutual correlation of sequences, So. Math. Doklady, ol. 1, , [8] J.-S. Wang and W.-F. Qi, A new class of binary sequence family with low correlation and large linear comlexity, Proceeding of IWSDA 07, IEEE, 007. [9] L. R. Welch, Lower bounds on the maximum cross correlation of signals, IEEE Trans. Inform. Theory, ol. 0, , School of Mathematics & Statistics, Carleton Uniersity, Ottawa, K1S 5B6, Canada address: jhe@connect.carleton.ca School of Mathematics & Statistics, Carleton Uniersity, Ottawa, K1S 5B6, Canada address: daniel@math.carleton.ca School of Mathematics & Statistics, Carleton Uniersity, Ottawa, K1S 5B6, Canada address: wang@math.carleton.ca