TWO BINARY SEQUENCE FAMILIES WITH LARGE MERIT FACTOR. Kai-Uwe Schmidt. Jonathan Jedwab. Matthew G. Parker. (Communicated by Aim Sciences)

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1 Volume X, No. 0X, 00X, X XX Web site: TWO BINARY SEQUENCE FAMILIES WITH LARGE MERIT FACTOR Kai-Uwe Schmidt Department of Mathematics Simon Fraser University 8888 University Drive Burnaby, BC, Canada V5A S6 Jonathan Jedwab Department of Mathematics Simon Fraser University 8888 University Drive Burnaby, BC, Canada V5A S6 Matthew G. Parker Department of Informatics High Technology Center in Bergen University of Bergen Bergen 500, Norway Communicated by Aim Sciences) Abstract. We calculate the asymptotic merit factor, under all rotations of sequence elements, of two families of binary sequences derived from Legendre sequences. The rotation is negaperiodic for the first family, and periodic for the second family. In both cases the maximum asymptotic merit factor is 6. As a consequence, we obtain the first two families of skew-symmetric sequences with known asymptotic merit factor, which is also 6 in both cases.. Introduction We consider a sequence A of length n to be an n-tuple a 0, a,..., a n ) of real numbers. The aperiodic autocorrelation of A at shift u is n u a j a j+u for 0 u < n C A u) := C A u) for n < u < 0, and its energy EA) is C A 0). factor of A is defined to be F A) := Provided that 0< [C Au)] > 0, the merit [EA)] [C A u)]. 0< 000 Mathematics Subject Classification: Primary: 94A55, 68P30; Secondary: 05B0. Key words and phrases: binary sequence, merit factor, asymptotic, Legendre sequence, rotation, construction, periodic, negaperiodic, skew-symmetric. K.-U. Schmidt is supported by Deutsche Forschungsgemeinschaft German Research Foundation). J. Jedwab is supported by NSERC of Canada. c 00X AIMS-SDU

2 Kai-Uwe Schmidt, Jonathan Jedwab and Matthew G. Parker The sequence A = a 0, a,..., a n ) of length n is called binary if each a j takes the value + or, in which case EA) = n. Let F n be the maximum value of the merit factor over the set of all n binary sequences of length n. The Merit Factor Problem, which is to determine the value of lim sup n F n, is important not only in digital communications engineering but also in complex analysis, theoretical physics, and theoretical chemistry see [5] for a survey, and [6] for background on related problems). The current state of knowledge on this problem can be summarised as 6 lim sup n F n, where the lower bound arises from an analysis due to Høholdt and Jensen [4] of a periodically rotated Legendre sequence see Theorem ). There is also considerable numerical evidence, though currently no proof, that an asymptotic merit factor greater than 6.34 can be achieved for a family of binary sequences related to Legendre sequences []. In this paper we determine the asymptotic merit factor, at all rotations of sequence elements, of two families of binary sequences constructed from a Legendre sequence of length n. The first family is derived from a negaperiodic construction described by Parker [8], and its sequences have length n; the second family is derived from a periodic construction described by Yu and Gong [], and its sequences have length 4n. For both families, the maximum asymptotic merit factor over all rotations is 6, equal to the best proven result for lim sup n F n. For the negaperiodic construction, the maximum value of 6 was previously proved by Xiong and Hall [] for two specific negaperiodic rotations, following numerical work in [8], but no asymptotic merit factor values at other negaperiodic rotations were previously known. For the periodic construction, the general form of the merit factor over all periodic rotations was determined numerically for large n by Yu and Gong [], but no asymptotic values were previously known. A skew-symmetric sequence is a binary sequence a 0, a,..., a m ) of odd length m + for which ) a m+j = ) j a m j for 0 < j m. By slight modification of the negaperiodic and periodic constructions we obtain two families of skew-symmetric sequences, each with asymptotic merit factor 6 see Corollaries 6 and 9). To our knowledge, these are the first constructions of families of skew-symmetric sequences for which the asymptotic merit factor has been calculated. Historically, skew-symmetric sequences were considered to be good candidates for a large merit factor, in part because half of their aperiodic autocorrelation coefficients are guaranteed to be zero see [5, Section 3.] for background). While skew-symmetric sequences of length n that attain the optimal merit factor value F n are known for n 3, 5, 7, 9,, 3, 5, 7,, 7, 9, 39, 4, 43, 45, 47, 49, 5, 53, 55, 57, 59}, they are not known for any larger values of n, because the value of F n itself is not known for n > 60 [5]. Golay [3] argued heuristically that the supremum limit of the merit factor of the set of skew-symmetric sequences is equal to lim sup n F n, which would imply that nothing is lost by restricting attention to the subset of binary sequences that are skew-symmetric see [5, Section 4.7] for a discussion). In view of this, it is particularly interesting that the asymptotic merit factor of the skew-symmetric sequence families constructed here is equal to the best proven result for lim sup n F n, namely 6. Volume X, No. X 00X), X XX

3 Binary sequences with large merit factor 3. Definitions and Notation In this section we introduce further definitions and notation for the paper. Let A = a 0, a,..., a n ) and B = b 0, b,..., b n ) be sequences of equal length n. The aperiodic crosscorrelation between A and B at shift u is n u a j b j+u for 0 u < n C A,B u) := C B,A u) for n < u < 0, and the periodic crosscorrelation between A and B at shift u is n R A,B u) := a j b j+u) mod n for 0 u < n. By adopting the convention that C A,B n) = 0, we can write ) R A,B u) = C A,B u) + C A,B u n) for 0 u < n. The aperiodic autocorrelation C A u) defined in Section equals C A,A u) for u < n. From the definition of C A u) and F A) we have the relation 3) F A) = + [EA)] [C A u)] for the reciprocal merit factor /F A). The periodic autocorrelation of A at shift u is R A u) := R A,A u) for 0 u < n. Given a sequence A = a 0, a,..., a n ) of length n, we write [A] j to denote the sequence element a j. Let A = a 0, a,..., a n ) and B = b 0, b,..., b m ) be sequences of length n and m, respectively. The concatenation A; B of A and B is the length n + m sequence given by a j for 0 j < n [A; B] j := b j n for n j < n + m. Provided gcdm, n) =, the product sequence A B of length mn is defined by [A B] j := a j mod n b j mod m for 0 j < mn, and the m-decimation of A is the length n sequence C defined by [C] j := a mj mod n for 0 j < n. The periodic rotation A r of A by a fraction r of its length for any real r) is the length n sequence given by 4) [A r ] j := a j+ nr ) mod n for 0 j < n, and the negaperiodic rotation A er of A by the fraction r is the length n sequence given by a j+ nr ) mod n for 0 j < n nr 5) [A er ] j := a j+ nr ) mod n for n nr j < n. Volume X, No. X 00X), X XX

4 4 Kai-Uwe Schmidt, Jonathan Jedwab and Matthew G. Parker The sequence A er can be viewed as the first n elements of the length n sequence A; A) r. For example, take r = 7 and A = +, +, +,, +,, ), where + and represent sequence elements + and, respectively. Then we have A; A) r A r = +,, +,,, +, +), A er = +,, +,,,, ). = +,, +,,,,,, +,, +, +, +, +). 3. Legendre Sequences This section describes Legendre sequences and their asymptotic merit factor properties. See [0], for example, for background on number-theoretic properties of Legendre sequences.) Given a prime n and an integer j, the Legendre symbol j n) is defined as 0 for j 0 mod n) j n) := for j not a square modulo n + otherwise. The Legendre symbol is multiplicative, so that 6) a n) b n) = ab n), and 7) a b mod n) implies a n) = b n). A Legendre sequence X = x 0, x,..., x n ) of prime length n is defined by for j = 0 x j := j n) for 0 < j < n. In our analysis it will often be convenient to change the initial element in a Legendre sequence to be zero, so that x j = j n) for 0 j < n. Then we will call X a modified Legendre sequence. The periodic autocorrelation function of a modified Legendre sequence X is given [0, p. 94] by 8) R X u) = for 0 < u < n. We shall make repeated use of the following result, under which up to o n) elements of a sequence of length n can be changed by a bounded amount without altering the asymptotic reciprocal merit factor: Proposition. Let An)} and Bn)} be sets of sequences, where each of An) and Bn) has length n. Suppose that, for each n, all elements of An) and Bn) are bounded in magnitude by a constant independent of n. Suppose further that, as n, the number of nonzero elements of Bn) is o n) and that F An)) = O) and EAn)) = Ωn). Then, as n, the elementwise sequence sums We use the notation o, O, and Ω to compare the growth rates of functions fn) and gn) from N to R + in the following standard way: fn) = ogn)) means that fn)/gn) 0 as n ; fn) = Ogn)) means that there is a constant c, independent of n, for which fn) cgn) for all sufficiently large n; and fn) = Ωgn)) means that gn) = Ofn)). Volume X, No. X 00X), X XX

5 Binary sequences with large merit factor 5 An) + Bn)} satisfy F An) + Bn)) = + o)). F An)) Proof. For each n, by the definition of aperiodic autocorrelation and crosscorrelation we have C An)+Bn) u) = C An) u) + C Bn) u) + C An),Bn) u) + C Bn),An) u) for u < n. Since, by assumption, all elements of An) and Bn) are bounded in magnitude by a constant independent of n, and the number of nonzero elements of Bn) is o n) as n, it follows that 9) C An)+Bn) u) = C An) u) + o n) as n. This implies by the definition of F An) + Bn)) that, as n, F An) + Bn)) = [EAn) + Bn))] [C An) u) + o n)] 0< = [EAn) + Bn))] [C An) u)] 0) since, by the Cauchy Schwarz inequality, o n)c An) u) 0< 0< 0< + on) 0< ) on) 0< [C An) u)] + on ) [C An) u)] ). Now by setting u = 0 in 9) and dividing by EAn)) we obtain, as n, EAn) + Bn)) EAn)) = + o n) EAn)) = + o), since EAn)) = Ωn) by assumption. The square of the reciprocal of this relation is [EAn))] = + o) as n. [EAn) + Bn))] Substitute in 0) and use the definition of F An)) to show that, as n, ) F An) + Bn)) = F An)) + o)) + on) + on ) F An)) F An)) EAn)) [EAn))] = + o)), F An)) since F An)) = O) and EAn)) = Ωn) by assumption. A result similar to Proposition was stated in Corollary 6.3 of [], but with conditions on the asymptotic growth of the sequence elements An) and their merit factor F An)) mistakenly omitted. ) Volume X, No. X 00X), X XX

6 6 Kai-Uwe Schmidt, Jonathan Jedwab and Matthew G. Parker The asymptotic merit factor of a Legendre sequence has been calculated at all periodic rotations: Theorem Høholdt and Jensen [4]). Let X be a Legendre sequence of prime length n >, and let r be a real number satisfying r. Then lim F X r) = r 4). n The constraint r in Theorem is for notational convenience only, since by definition X r is the same as X r+ for any real r. By Proposition, Theorem also holds for the modified Legendre sequence X of length n, since X r differs from X r in exactly one element for each n. The exact, rather than the asymptotic, value of F X r ) in Theorem has been calculated [] by refining the analysis of [4], but the exact value is not required here. The following generalisation of Theorem is a key tool of this paper: Theorem 3. Let X be a modified Legendre sequence of prime length n >, and let r, s, and t be real numbers satisfying r+ s+t, s, and t. Let rn)}, sn)}, and tn)} be sets of real numbers such that nsn) and ntn) are integer for each n, and such that, as n, rn) = r + On ), sn) = s + On ), and tn) = t + On ). Then, as n, n C Xrn), X rn)+sn) u) C Xrn)+tn), X rn)+sn)+tn) u) = 6 Proof. See Appendix. + 8 r + s+t 4 ) + s ) + t ) + On log n) ). We can recover Theorem from Theorem 3 by setting rn) = r and sn) = tn) = 0 for each n, applying 3), using Proposition to alter the initial element and thereby change a modified Legendre sequence into a Legendre sequence, and then taking the limit as n. There is no loss of generality in Theorem 3 from the restrictions r + s+t, s, and t, since X r+ = X r for all r. Nonetheless, we will find it useful to define, for real r, s, t, c, ) and ) φr, c) := ψr, s, t) := 6 r c) for r φr +, c) for all r, c + 8φr + s+t, 4 ) + φs, ) + φt, ), so that the conclusion of Theorem 3 can be rephrased to hold for unrestricted r, s, t as n : n C Xrn), X rn)+sn) u) C Xrn)+tn), X rn)+sn)+tn) u) 3) = ψr, s, t) + On log n) ). It is easily verified that φr +, 4 ) = φr, 4 ) for all r, which implies that 4) φr, 4 ) = φr, 4 ) provided r = r + a for some integer a. Volume X, No. X 00X), X XX

7 Binary sequences with large merit factor 7 4. The Negaperiodic Construction In this section we present the first of our two constructions, involving a negaperiodic rotation Y er as defined in 5)) of a sequence Y of length n that is in turn derived from a sequence X of odd length n. Lemma 4 analyses this construction for the case r = ρ, where ρ is subject to the constraint that nρ is integer which forces the number nr of rotated elements in the negaperiodic rotation Y er to be even). We then show in Theorem 5 that, in the specific case that X is a Legendre sequence, Lemma 4 can be used to determine the asymptotic merit factor of Y er for all real r, regardless of whether nr is even or odd. Lemma 4. Let X be a sequence of odd length n, each of whose elements is bounded in magnitude by a constant independent of n, and let Z be the -decimation of X. Let r be a real number, and write ρ := nr /n and δ := n+ n. Define Y to be the first n elements of the sequence X +, +,, ). Then, as n, [C Y eρ u)] = [CZr u) + C Zr+δ u)] + [C Zr,Z r+δ u) C Z r+δ,z r+δ u) + O)]). Proof. We firstly determine an explicit expression for [Y eρ ] j, and deduce an expression for C Y eρ u). By analysis of the cases u even and u odd, we then express C Y eρ u) in terms of crosscorrelations of periodically rotated versions of the sequence Z r, for both positive and negative values of u. Squaring and summing over u then gives the required result. Write X = x 0, x,..., x n ), Y = y 0, y,..., y n ), and Z = z 0, z,..., z n ). Now element j mod 4 of the sequence +, +,, ) can be written as ) j j)/, for any integer j. By the definition of Y we therefore have 5) y j = ) j j x j mod n for 0 j < n. Write k := j+nρ n. Since nρ = nr is integer, the definition 5) of Y eρ gives, for 0 j < n, [Y eρ ] j = y j+nρ) mod n y j+nρ) mod n = ) k y j+nρ) mod n = ) k y j+nρ nk) mod n for 0 j < n nρ for n nρ j < n = ) k+ j+nρ nk) j+nρ nk) x j+nρ) mod n from 5). Since n is odd, this implies that [Y eρ ] j = ) j j nρ x j+nρ) mod n for 0 j < n. Volume X, No. X 00X), X XX

8 8 Kai-Uwe Schmidt, Jonathan Jedwab and Matthew G. Parker Therefore, for 0 u < n, C Y eρ u) = ) u u = ) u u = ) u u n u n u n u ) ju x j+nρ) mod n x j+u+nρ) mod n ) ju [X; X] j+nρ) mod n [X; X] j+u+nρ) mod n ) ju w j w j+u, where W = w 0, w,..., w n ) := X; X) ρ. Take the absolute value, and separate both the summation index j and the argument u into even and odd values, giving CY eρ u) n u n u 6) = w j w j+u) + w j+ w j+u)+ for 0 u < n, 7) C Y eρ u + ) = n u n u w j w j+u)+ w j+ w j+u)+ for 0 u < n. We next express 6) and 7) in terms of crosscorrelations of periodically rotated versions of the sequence Z r. Now, for all integer i, j satisfying 0 j + i < n we have w j+i = [X; X] j+i+nρ) mod n = x j+i+nρ) mod n = x j+n+)i+nρ) mod n = x j+niδ+ nr ) mod n, by definition of δ and ρ. Since nδ = n+ is integer because n is odd, we therefore have, for all integer i, j satisfying 0 j + i < n, w j+i = z j+niδ+ nr ) mod n = [Z r+iδ ] j. Substitution in 6) and 7) then shows that CY eρ u) 8) = CZr u) + C Zr+δ u) for 0 u < n, and CY eρ u + ) = CZr,Z r+δ u) C Zr+δ,Z r+δ u) [Z r+δ ] n u [Z r+δ ] n ) 9) = CZr,Z r+δ u) C Zr+δ,Z r+δ u) + O) for 0 u < n, since, by assumption, the elements of X are bounded in magnitude by a constant independent of n and so the term [Z r+δ ] n u [Z r+δ ] n is O) as n. We lastly find expressions corresponding to 8) and 9) for negative values of the argument. For any sequence A, by definition 0) C A u) = C A u) for 0 < u < n, Volume X, No. X 00X), X XX

9 Binary sequences with large merit factor 9 and so 8) implies that C Y eρ u) = C Zr u) + C Zr+δ u) for 0 < u < n. By combining this with 8), we obtain [ ) CY u)] [ eρ = CZr u) + C u)] Zr+δ. Furthermore, by applying 0) to 9) we also have, for 0 u < n, CY eρ u ) = CZr,Z r+δ u) C Zr+δ,Z r+δ u) + O) = R Zr,Z r+δ u) C Zr,Z r+δ u n) ) R Zr+δ,Z r+δ u) C Zr+δ,Z r+δ u n) ) + O) by ), using the convention that C A,B n) = 0 for sequences A and B of length n. But rotation of each of the sequences Z r and Z r+δ by nδ = n+ elements to produce sequences Z r+δ and Z r+δ ) does not change their periodic crosscorrelation function, so R Zr,Z r+δ u) R Zr+δ,Z r+δ u) is identically zero. Therefore C Y eρ u ) = C Zr,Z r+δ u n) C Zr+δ,Z r+δ u n) + O) for 0 u < n. By combining this with 9), we obtain n u= n [ CY eρ u + )] = which, after addition of ), gives the required result. [ CZr,Z r+δ u) C Z r+δ,z r+δ u) + O)], We now take the sequence X of Lemma 4 to be a Legendre sequence and, using Theorem 3, derive the asymptotic merit factor of the resulting sequence Y at all negaperiodic rotations. Theorem 5. Let X be a Legendre sequence of prime length n >, and let r be a real number satisfying r. Define Y to be the first n elements of the sequence X +, +,, ). Then lim F Y er) = 6 + 8r for r 4 n r ) for 4 r. Proof. Write X = x 0, x,..., x n ) and let X be the modified Legendre sequence 0, x,..., x n ). The sequence Y, obtained using X instead of X in the definition of Y, differs from Y in exactly two elements. By Proposition, it is therefore sufficient to show that /F Y er) has the asymptotic behaviour claimed for /F Y er ). To simplify notation, we continue to work with X and Y but set x 0 := 0. Write ρ := nr /n so that, by the definition 5), y j+ nr ) mod n for 0 j < n nr [Y er ] j = y j+ nr ) mod n for n nr j < n, y j+ nr ) mod n for 0 j < n nr [Y eρ ] j = y j+ nr ) mod n for n nr j < n. For each n, either nr = nr, in which case the length n sequences Y er and Y eρ are identical, or else nr = nr +, in which case Y er and Y eρ share a common Volume X, No. X 00X), X XX

10 0 Kai-Uwe Schmidt, Jonathan Jedwab and Matthew G. Parker subsequence of length n. By Proposition, it is therefore sufficient to show that /F Y eρ ) has the asymptotic behaviour claimed for /F Y er ). Let Z be the -decimation of X. For 0 j < n, this gives [Z] j = x j mod n = j n) by 7). Then by 6) and the definition of a modified Legendre sequence, so that Z = n) X. Therefore [Z] j = n) [X] j for 0 j < n, C Zs,Z t u) = C Xs,X t u) for all integer u satisfying u < n and all real s, t. Application of Lemma 4 to the modified Legendre sequence X then gives n [ CY eρ u)] = n [CXr u) + C Xr+δ u)] ) + [ C Xr,X r+δ u) C Xr+δ,X r+δ u) + O) ] ), where 3) δ = δn) := n + n. We complete the proof by using Theorem 3 to evaluate the expansion of the right hand side of ) as n. When n [C X r,x r+δ u) C Xr+δ,X r+δ u)] is expanded into three sums, by Theorem 3 each of the sums is O) as n. Therefore, by the Cauchy-Schwarz inequality, the additional contribution arising from the inclusion of the term O) in the second square bracket of ) is On / ), which can be neglected. By applying Theorem 3 and 3) to the six resulting sums in the expansion of the right hand side of ) noting from 3) that δn) = +On ) as n ), and then taking the limit, we obtain lim n 4) n [ CY eρ u)] = ψr, 0, 0) + ψr +, 0, 0) + ψr, 0, ) + ψr,, 0) + ψr +,, 0) ψr,, ) = 3 + 3φr + 4, 4 ) + 6φ0, ) by the definition ) of ψ and by 4). By the definition ) of φ, in the given range r we have φ0, ) = 4, 5) r φr + 4, 4 ) = for r 4 r ) for 4 r. Volume X, No. X 00X), X XX

11 Binary sequences with large merit factor Now, since x 0 = 0, the definition of Y gives [EY eρ )] = n ). Substitution of 5) into 4), together with the relation 3), then yields as required. lim F Y eρ) = 6 + 8r for r 4 n r ) for 4 r, The maximum asymptotic merit factor obtained under the construction of Theorem 5 is 6, which occurs at r = 0 and at r =. The special cases r = 0 and r = of Theorem 5 were proved by Xiong and Hall [] in response to numerical evidence presented by Parker [8, Figs.,]. Corollary 6. Let X = x 0, x,..., x n ) be a Legendre sequence of prime length n mod 4), and define Y to be the first n elements of the sequence X +, +,, ). Then the sequence Y ; ) is skew-symmetric of length n + and has asymptotic merit factor 6. Proof. Take r = 0 in Theorem 5, and apply Proposition to show that lim F Y ; )) = lim F Y ) = 6. n n It remains to show that Y ; ) is skew-symmetric. Since n mod 4), we know that n) = see [0, p. 84], for example), and it follows from 6) and 7) that 6) x n j = x j for 0 < j < n. Now by writing Y ; ) = y 0, y,..., y n ), from 5) we find that, for 0 < j < n, y n+j = ) n+j) n+j) x j by 6). Then using 5) again we obtain = ) j+ n j) n j) x n j y n+j = ) j y n j for 0 < j < n, and by construction the same equation also holds for j = n since y 0 = x 0 =. Therefore the sequence Y ; ) is skew-symmetric, by the definition ). To our knowledge, Corollary 6, together with Corollary 9 to be proved in Section 5, are the first known asymptotic results on the merit factor of a family of skewsymmetric sequences. In the case n 3 mod 4), the sequence constructed in Corollary 6 does not satisfy the skew-symmetry condition ). However, in that case we can instead form the length n + sequence Y ; +) and then set the central element of this sequence to be 0. The resulting ternary sequence a 0, a,..., a n ) also has asymptotic merit factor 6, and satisfies the following modification of the skew-symmetry condition: a n+j = ) j+ a n j for 0 < j n. Volume X, No. X 00X), X XX

12 Kai-Uwe Schmidt, Jonathan Jedwab and Matthew G. Parker 5. The Periodic Construction In this section we present the second of our two constructions, involving a periodic rotation Y r of a sequence Y of length 4n derived from a sequence X of odd length n. Lemma 7 analyses this construction for the case r = ρ, where nρ is integer which forces the number 4nr of rotated elements in Y r to be a multiple of 4). We then show in Theorem 8 that, in the case that X is a Legendre sequence, Lemma 7 can be used to determine the asymptotic merit factor of Y r for all real r. Lemma 7. Let X be a sequence of odd length n, each of whose elements is bounded in magnitude by a constant independent of n, and let Z be the 4-decimation of X. Let r be a real number, and write ρ := nr /n and 3n+ 4n for n mod 4) δ := n+ 4n for n 3 mod 4). Let Y be the length 4n sequence X +, +, +, ). Then, as n, [ CYρ u) ] 3 3 = ) iki+k+) C Zr+iδ,Z r+i+k)δ u) + O)). u <4n k=0 i=0 Proof. This proof is closely modelled on that of Lemma 4, and so is presented in less detail. Write X = x 0, x,..., x n ) and Y = y 0, y,..., y 4n ). By definition of Y we can write 7) y j = ) j3 j x j mod n for 0 j < 4n. Since nρ = nr, the definition 4) of Y ρ then gives [Y ρ ] j = ) j3 j x j+4nρ) mod n for 0 j < 4n. Therefore, for 0 u < 4n, C Yρ u) = ) u3 u = ) u3 u 4n u ) juj+u+) x j+4nρ) mod n x j+u+4nρ) mod n 4n u ) juj+u+) w j w j+u, where W = w 0, w,..., w 4n ) := X; X; X; X) ρ. Take the absolute value, and separate the summation index j according to its value mod 4 to obtain 3 4n u i C Yρ u) = ) iui+u+) 4 w 4j+i w 4j+i+u for 0 u < 4n. i=0 Since nδ is integer by definition of δ, a similar argument to that used in the proof of Lemma 4 shows that w 4j+i = [Z r+iδ ] j for all integer i, j satisfying 0 4j + i < 4n. Volume X, No. X 00X), X XX

13 Binary sequences with large merit factor 3 Therefore, for 0 u < n and 0 k 3, 3 4n 4u k i C Yρ 4u + k) = ) iki+k+) 4 [Z r+iδ ] j [Z r+i+k)δ ] j+u i=0 3 8) = ) iki+k+) C Zr+iδ,Z r+i+k)δ u) + O). i=0 We deal with negative values of the argument of the left hand side of 8) as in the proof of Lemma 4, using 0). For 0 < u < n and k = 0 this gives 3 9) C Yρ 4u) = C Zr+iδ u) + O), i=0 while for 0 u < n and k,, 3} we have C Yρ 4u k) 3 = i=0 3 30) = i=0 ) iki+k+) ) iki+k+) RZr+iδ,Z u) C r+i+k)δ Z r+iδ,z u n)) r+i+k)δ + O) C Zr+iδ,Z r+i+k)δ u n) + O), where the terms involving the periodic crosscorrelation R cancel since, for each k,, 3}, ) iki+k+) takes the value + twice and the value twice as i ranges over 0 i 3. By combining 9) and 30) with 8) we obtain the required result. We now take the sequence X of Lemma 7 to be a Legendre sequence and, using Theorem 3, derive the asymptotic merit factor of the resulting sequence Y at all periodic rotations. Theorem 8. Let X be a Legendre sequence of prime length n >, and let r be a real number satisfying r. Let Y be the length 4n sequence X +, +, +, ). Then lim F Y r) = 6 + 8r for r 4 n r ) for 4 r. Proof. Write X = x 0, x,..., x n ) and ρ := nr /n. By Proposition, we can take x 0 = 0. Furthermore, the length 4n sequences Y r and Y ρ share a common subsequence of length at least 4n 3, and so by Proposition it is sufficient to show that /F Y ρ ) has the asymptotic behaviour claimed for /F Y r ). Following the method of proof of Theorem 5, the 4-decimation Z of X is given by Z = 4 n) X, so by 6) we have Z = n) X = X. Application of Lemma 7 then gives n where u <4n [ CYρ u) ] = n 3 k=0 3) δ = δn) := 3 i=0 ) iki+k+) C Xr+iδ,X r+i+k)δ u) + O)), 3n+ 4n for n mod 4) n+ 4n for n 3 mod 4). Volume X, No. X 00X), X XX

14 4 Kai-Uwe Schmidt, Jonathan Jedwab and Matthew G. Parker As n we can neglect the O) term, as in the proof of Theorem 5, so that [ lim n n CYρ u) ] u <4n = lim ) iki+k+)+jkj+k+) n k=0 i=0 = 0 i,j,k 3 n C Xr+iδ,X r+i+k)δ u)c Xr+jδ,X r+j+k)δ u) ) iki+k+)+jkj+k+) ψr + i, k, j i) ) by Theorem 3 and 3), where from 3) we have δn) = + On ) as n and 3 4 for n mod 4) 3) := 0 i,j,k 3 4 for n 3 mod 4). By the definition ) of ψ we then obtain [ lim n n CYρ u) ] u <4n = ) iki+k+)+jkj+k+) 6 + 8φ r + i+j+k 33) = 3 l=0 + 0 i,j,k 3 i+j+k l mod 4) 0 i,j,k 3 0 i,j,k 3 i+j+k l mod 4) ) iki+k+)+jkj+k+) ) iki+k+)+jkj+k+), 4 + φ ) k, ) ) + φ j i), ) 6 + 8φ r + l 8, 4)) ) )) φ k, + φ j i), by 4), for both possible values of given in 3). By direct calculation we find that ) iki+k+)+jkj+k+) 6 for l = = 0 for l 0,, 3}, and that 0 i,j,k 3 ) iki+k+)+jkj+k+) φ ) k, = 0 i,j,k 3 ) iki+k+)+jkj+k+) φ j i), ) = 4 for both possible values of. Substitution in 33) then gives [ lim n n CYρ u) ] = φ r + 4, )) 4, u <4n Volume X, No. X 00X), X XX

15 Binary sequences with large merit factor 5 and then, since [EY ρ )] = 4n 4), by 3) and 5) we conclude that as required. lim F Y ρ) = 6 + 8r for r 4 n r ) for 4 r, The maximum asymptotic merit factor obtained under the construction of Theorem 8 is 6, which occurs at r = 0 and at r =. Theorem 8 explains the numerical results obtained by Yu and Gong [, Fig. ], based on the sequence Y = X +, +, +, ), for all values of r. It can readily be modified to explain also their numerical results based on the related sequence X, +, +, +). Corollary 9. Let X = x 0, x,..., x n ) be a Legendre sequence of prime length n mod 4), and let Y be the length 4n sequence X +, +, +, ). Then the sequence Y ; +) is skew-symmetric of length 4n + and has asymptotic merit factor 6. Proof. The proof is similar to that of Corollary 6. Take r = 0 in Theorem 8, and apply Proposition to show that lim F Y ; +)) = 6. n To show that Y ; +) is skew-symmetric, write Y ; +) = y 0, y,..., y 4n ) and note that 6) holds since n mod 4). Then from 7) we have, for 0 < j < n, y n+j = ) n+j)3 n+j) x j mod n = ) j+ n j)3 n j) x n j) mod n = ) j y n j, and by construction the same equation also holds for j = n since y 0 = x 0 =. 6. Conclusion Theorems 5 and 8 establish the asymptotic merit factor of a negaperiodic construction described in [8] and a periodic construction described in [], based in both cases on Legendre sequences. Corollaries 6 and 9 provide families of skewsymmetric sequences having asymptotic merit factor 6, equal to the current best proven result for lim sup n F n. However there is considerable numerical evidence, though currently no proof, that an asymptotic merit factor greater than 6.34 can be achieved for a family of binary sequences constructed via periodic appending of a Legendre sequence []. Numerical results displayed by Yu and Gong [, Fig. 3] for the periodic construction, and reported by Parker [9, p. ] for the negaperiodic construction, indicate that both constructions also attain a merit factor greater than 6.34 under suitable appending. A possible direction for future research is to attempt to prove that the asymptotic merit factor of these appended sequences is equal to the as yet undetermined) asymptotic merit factor of the periodically appended Legendre sequences described in []. Volume X, No. X 00X), X XX

16 6 Kai-Uwe Schmidt, Jonathan Jedwab and Matthew G. Parker Appendix: Proof of Theorem 3 This appendix contains a proof of Theorem 3. Our starting point is the method of Høholdt and Jensen [4], which calculates the reciprocal merit factor of an arbitrary sequence of odd length n as the sum of expressions involving complex nth roots of unity. We generalise this method in Lemma 0 in order to calculate the sum of products of crosscorrelations of periodic rotations of an arbitrary odd-length sequence. We then prove Theorem 3 by applying Lemma 0 to the specific case of a modified Legendre sequence. Write ɛ j := e πij/n for integer j, where i :=. Given a sequence A = a 0, a,..., a n ) of length n, we define the z-transform of A to be the function Q A : C C given by and define 34) n Q A z) := a k z k, k=0 n Λ A j, k, l) := Q A ɛ a )Q A ɛ a+j )Q A ɛ a+k )Q A ɛ a+l ) for integer j, k, l. a=0 Lemma 0. Let X be a sequence of odd length n. Let S and T be integers, and write s := S/n and t := T/n. Then 35) where n B = n [ ɛ T n 5 k + ɛ S k C = n 5 k= D = 4 n n 5 k= k,l<n k l C X,Xs u) C Xt,X s+t u) = n + 3n 5 Λ X 0, 0, 0) + B + C + D, ) ΛX 0, 0, k) + S+T ) ) ] ɛk + ɛ k ΛX 0, 0, k) [ ɛ S+T ) k + ) ɛ T l + ɛ S ) l ΛX 0, k, l) + ɛ S k ɛ T l Λ X k, 0, l) +ɛ S+T ) k ɛ l Λ X k, 0, l) [ ɛ S k + ɛ T ) S+T k ΛX 0, k, k) + ɛk Λ X k, 0, k) ] ] ɛ k. + ɛ k ɛ k ), ɛ k ɛ k ) ɛ l ), Proof. Let U, V, W, Y, Z be any sequences of length n. Straightforward manipulation shows that Q U z)q V z ) = C U,V u)z u for all z 0. Volume X, No. X 00X), X XX

17 Binary sequences with large merit factor 7 Since Q U ɛ j ) = Q U ɛ j ), we then have n n 3 Q U ɛ j )Q V ɛ j ) Q W ɛ j )Q Z ɛ j ) + n n 3 Q U ɛ j )Q V ɛ j ) Q W ɛ j )Q Z ɛ j ) 36) = n n 3 = n v <n = n u+v 0,n, n} C U,V u)c Z,W u), because n is odd and by definition C W,Z u) = C Z,W u). Let j be an integer and let r be real. By definition of Y r, 37) Q Yr ɛ j ) = ɛ nr j Q Y ɛ j ) and then, by the interpolation formula see [4, p. 6], for example), we have 38) and therefore 39) Q Y ɛ j ) = n ɛ k Q Y ɛ k ) n ɛ k + ɛ j Q Yr ɛ j ) = n Q Yr ɛ j ) = n k=0 + ) u+v)) C U,V u)c W,Z v)ɛ u+v) j n ɛ nr k=0 n ɛ nr k=0 + ) u+v)) C U,V u)c W,Z v) ɛ k k Q Y ɛ k ), ɛ k + ɛ j ɛ j k Q Y ɛ k ). ɛ k + ɛ j Take U = X, V = X s, W = X s+t, and Z = X t in 36) and substitute from 37), 38) and 39) to obtain n C X,Xs u)c Xt,X s+t u) = n n 3 Q X ɛ j ) 4 + n 3 ) 4 n n 0 a, b, c, d<n ɛ S b ɛ S+T ) c ɛ T d 40) Q X ɛ a )Q X ɛ b )Q X ɛ c )Q X ɛ d ) ɛ a + ɛ j ɛ b + ɛ j ɛ c + ɛ j ɛ d + ɛ j = n 3 Λ X0, 0, 0) + 8 n 7 0 a, b, c, d<n ɛ a ɛ j ɛ S b ɛ S+T c ) ɛ T d ɛ a ɛ c n ɛ j Q X ɛ a )Q X ɛ b )Q X ɛ c )Q X ɛ d ) ɛ a + ɛ j )ɛ b + ɛ j )ɛ c + ɛ j )ɛ d + ɛ j ), ɛ c ɛ j Volume X, No. X 00X), X XX

18 8 Kai-Uwe Schmidt, Jonathan Jedwab and Matthew G. Parker by definition 34) of Λ X. Høholdt and Jensen [4] calculated the value of the sum over j in 40) according to which of a, b, c, d in the range 0 a, b, c, d < n) are equal and which are distinct, distinguishing five cases: n ɛ j ɛ a + ɛ j )ɛ b + ɛ j )ɛ c + ɛ j )ɛ d + ɛ j ) 0 for a, b, c, d distinct = n n + ) 48 n 8 ɛ a ɛ a + ɛ d ɛ a ɛ a ɛ d ) for a = b = c = d for a = b = c d n 4 for a = b, and a, c, d distinct ɛ a ɛ c )ɛ a ɛ d ) n ɛ a ɛ c ) for a = b c = d. We therefore partition the sum over a, b, c, d in 40) into sums of the same four types as those giving a non-zero result above, written in the same order, noting that each type requires multiple contributions because of the asymmetry with respect to a, b, c, d of the ɛ terms. This gives 4) n C X,Xs u)c Xt,X s+t u) = n 3 Λ X0, 0, 0) + A + B + C + D, where, after abbreviating Q X to Q, the sums A, B, C, D are given by A = n + 6n 5 B = n 5 C = n 5 D = 4 n 5 n Qɛ a ) 4, a=0 0 a,b<n a b 0 a,b,c<n a,b,c distinct 0 a,b<n a b By the definition 34), [ ɛ T b a + ɛ S ) b a Qɛa ) ) Qɛa ) Qɛ b ) + S+T ) ) ɛb a + ɛ b a Qɛa ) ) ] Qɛa )Qɛ b ) ɛaɛ a + ɛ b ) ɛ a ɛ b ), [ ɛ S+T ) b a ɛ T c a + ɛ S+T ) b a ɛ S c a + ɛ T c a + ɛ S c a) Qɛa ) Qɛ b )Qɛ c ) + ɛ S b a ɛt c a Qɛa ) ) Qɛb ) Qɛ c ) + ɛ S+T ) b a ɛ c a Qɛa ) ) Qɛb )Qɛ c ) [ ɛ S b a + ɛ T ) b a Qɛa ) Qɛ b ) + ɛ S+T b a ] Qɛa ) ) Qɛb ) ) ] ɛ a ɛ b ɛ a ɛ b )ɛ a ɛ c ), ɛ a ɛ b ɛ a ɛ b ). A = n + 6n 5 Λ X 0, 0, 0), and substitution in 4) gives the required term in Λ X 0, 0, 0). We complete the proof by showing that the remaining sums B, C, D in 4) have the required form. Volume X, No. X 00X), X XX

19 Binary sequences with large merit factor 9 For the sum B, replace the sum over b where 0 b < n and b a) by a sum over k := b a) mod n where k < n) to give B = n n [ ɛ T n 5 k + ɛ S ) k Qɛa ) ) Qɛa ) Qɛ a+k ) a=0 k= ) S+T ) Qɛa + ɛk + ɛ k ) ) ] + ɛ k ) Qɛa ) Qɛ a+k ) ɛ k ), and then use 34). For the sum D, use the same procedure together with the identity ɛ a ɛ b ) = ɛ a ɛ b ɛ b a. For the sum C, replace the sum over b and c by a sum over k := b a) mod n and l := c a) mod n to give C = n n 5 a=0 k,l<n k l and then use 34). [ ɛ ɛ S+T ) T k + ) l + ɛ S ) l Qɛa ) Qɛ a+k )Qɛ a+l ) Qɛa ) ) Qɛa+k ) Qɛ a+l ) + ɛ S k ɛ T l +ɛ S+T ) k ɛ l Qɛa ) ) Qɛa+k ) Qɛ a+l ) ] ɛ k ɛ k ) ɛ l ), The method of Høholdt and Jensen [4] deals with the special case S = T = 0 of Lemma 0. Proof of Theorem 3. We apply Lemma 0 to the sequence X rn), setting S := nsn) and T := ntn). Since S and T are integer for each n by assumption, the sequences X s, X t and X s+t appearing in 35) map to X rn)+sn), X rn)+tn), and X rn)+sn)+tn), respectively, so that the left hand side of 35) becomes n C Xrn), X rn)+sn) u) C Xrn)+tn), X rn)+sn)+tn) u). We will prove the desired result by finding the asymptotic form of the right hand side of 35) as n, evaluating the sum D and the term involving Λ Xrn) 0, 0, 0), and bounding the sums B and C. Write R := nrn) and X = x 0, x,..., x n ). Since the modified Legendre sequence X satisfies x j n for n mod 4) Q X ɛ j ) = ix j n for n 3 mod 4) see [0, p. 8], for example), where i :=, the definition 34) together with 37) gives 4) n Λ Xrn) j, k, l) = ɛ R j k+l n x a x a+j x a+k x a+l. The term involving Λ Xrn) 0, 0, 0). Since X is a modified Legendre sequence, we have 43) x 0 for j = 0 j = for 0 < j < n. a=0 Volume X, No. X 00X), X XX

20 0 Kai-Uwe Schmidt, Jonathan Jedwab and Matthew G. Parker Therefore 4) gives n + 3n 5 Λ Xrn) 0, 0, 0) = n + 3n 5 n n ) 44) = 3 + On ) as n. The sums B and C. We have B n n 5 k= = 8 n n 5 k= 8 Λ Xrn) 0, 0, k) ɛ k n R X k) ɛ k by 4), 43) and the definition of R X in Section. Then by 8), 45) B 8 n n 3 ɛ k. k= Since Λ Xrn) 0, k, l) = Λ Xrn) k, 0, l) by 4), we also have C n 5 = n 5 k,l<n k l k,l<n k l 6 Λ Xrn) 0, k, l) ɛ k ɛ l n R X k l) mod n) x k x l ɛ k ɛ l by 4) and 43). Then using 8) and the inequality x k x l, C 4 n 3 k,l<n k l Combining with 45), we obtain ɛ k ɛ l 46) 47) 3 B + C 4 n ) n 3 ɛ k k= = On log n) ) as n, since n k= ɛ k n log n see [4, p. 6], for example). The sum D. By 4), for k < n we have Λ Xrn) 0, k, k) = n n ) and Λ Xrn) k, 0, k) = ɛ R k n n ), so that D = 4n ) n 3 n k= We wish to apply the identity n ɛ j k ɛ k = n j n ) n + 4 k= ɛ S k + ɛt k ɛ k. + ɛr+s+t k for integer j satisfying j n Volume X, No. X 00X), X XX

21 Binary sequences with large merit factor see, [7, p. 6], for example). By the definition of R, S, and T and the assumptions rn) = r + On ), sn) = s + On ), and tn) = t + On ), as n we have 48) R = nr + O), S = ns + O), and T = nt + O). Then, since s and t by assumption, we know that S n and T n for all sufficiently large n. Case : R + S + T n for all but finitely many n. In this case, we apply 47) to show that, for all sufficiently large n, [ S 4n ) n 49) D = n 3 n ) T + n ) R + S + T + ) ] ) n +. n 8 50) By 48), as n we find that D = s ) + t ) + 8 r + s+t 4 ) + On ). Case : R + S + T > n for infinitely many n. In this case 48), together with the assumption r + s+t, implies that 5) r + s + t =. For each sufficiently large n, either R+S+T n, so that 49) holds; or else n < R + S + T n, in which case by applying 47) with j = R+S +T ±n, choosing the appropriate sign for ± so that j n) equation 49) again holds but with the term n R + S + T ) replaced by. n R + S + T ) 3 By 48) and 5), the contribution of this term as n is 3 ) = ), so the same conclusion 50) applies as in Case. The result now follows by substituting the asymptotic forms 44), 46) and 50) in 35). References [] P. Borwein and K.-K.S. Choi, Explicit merit factor formulae for Fekete and Turyn polynomials, Trans. Amer. Math. Soc., ), [] P. Borwein, K.-K.S. Choi, and J. Jedwab, Binary sequences with merit factor greater than 6.34, IEEE Trans. Inform. Theory, ), [3] M.J.E. Golay, The merit factor of long low autocorrelation binary sequences, IEEE Trans. Inform. Theory, IT-8 98), [4] T. Høholdt and H.E. Jensen, Determination of the merit factor of Legendre sequences, IEEE Trans. Inform. Theory, ), [5] J. Jedwab, A survey of the merit factor problem for binary sequences, in Sequences and Their Applications Proceedings of SETA 004 eds. T. Helleseth et al.), Springer-Verlag, Berlin Heidelberg, Lecture Notes in Computer Science, ), [6] J. Jedwab, What can be used instead of a Barker sequence?, Contemporary Math., ), [7] J.M. Jensen, H.E. Jensen, and T. Høholdt, The merit factor of binary sequences related to difference sets, IEEE Trans. Inform. Theory, 37 99), Volume X, No. X 00X), X XX

22 Kai-Uwe Schmidt, Jonathan Jedwab and Matthew G. Parker [8] M.G. Parker, Even length binary sequence families with low negaperiodic autocorrelation, in Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, AAECC-4 Proceedings eds. S. Boztaş and I.E. Shparlinski), Springer-Verlag, Lecture Notes in Computer Science, 7 00), [9] M.G. Parker, Univariate and multivariate merit factors, in Sequences and Their Applications Proceedings of SETA 004 eds. T. Helleseth et al.), Springer-Verlag, Berlin Heidelberg, Lecture Notes in Computer Science, ), [0] M.R. Schroeder, Number Theory in Science and Communication: with Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity, Springer, Berlin, 3rd edition, 997. [] T. Xiong and J.I. Hall, Construction of even length binary sequences with asymptotic merit factor 6, IEEE Trans. Inform. Theory, ), [] N.Y. Yu and G. Gong, The perfect binary sequence of period 4 for low periodic and aperiodic autocorrelations, in Sequences, Subsequences, and Consequences eds. S.W. Golomb et al.), Springer-Verlag, Berlin, Lecture Notes in Computer Science, ), Received November 008; revised March address: kuschmidt@sfu.ca address: jed@sfu.ca address: matthew.parker@ii.uib.no Volume X, No. X 00X), X XX