Almost p-ary Perfect Sequences

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1 Almost -Ary Perfect Sequences Yeow Meng Chee 1,YinTan 1,, and Yue Zhou 2, 1 Division of Mathematical Sciences, School of Physical & Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singaore ymchee@ntu.edu.sg, itanyinmath@gmail.com 2 Deartment of Mathematics, Otto-von-Guericke-University Magdeburg, Magdeburg, Germany yue.zhou@st.ovgu.de Abstract. A sequence a =(a 0,a 1,a 2,,a n) is said to be an almost -ary sequence of eriod n +1 if a 0 =0anda i = ζ b i for 1 i n, where ζ is a rimitive -th root of unity and b i {0, 1,, 1}. Such a sequence a is called erfect if all its out-of-hase autocorrelation coefficients are zero; and is called nearly erfect if its out-of-hase autocorrelation coefficients are all 1, or are all 1. In this aer, on the one hand, we construct almost -ary erfect and nearly erfect sequences; on the other hand, we resent results to show they do not exist with certain eriods. It is shown that almost -ary erfect sequences corresond to certain relative difference sets, and almost -ary nearly erfect sequences corresond to certain direct roduct difference sets. Finally, two tables of the existence status of such sequences with eriod less than 100 are given. Keywords: almost -ary sequences, almost -ary erfect sequences, almost -ary nearly erfect sequences, relative difference set, direct roduct difference set. 1 Introduction Let a =(a 0,a 1,a 2,,a n ) be a comlex sequence of eriod n +1. We call a an m-ary sequence if a i = ζm, bi whereζ m is a rimitive m-th root of unity and b i {0, 1,,m 1} for 0 i n. In articular, the sequence a is called an almost m-ary sequence if a 0 =0. For an (almost) m-ary sequence a with eriod n + 1,theautocorrelation coefficients of a are the elements in the set {C t (a) = n a i a i+t :0 t n}, i=0 Research of Yeow Meng Chee and Yin Tan is suorted in art by the National Research Foundation of Singaore under Research Grant NRF-CRP and by the Nanyang Technological University under Research Grant M Research of Yue Zhou is artially suorted by China Scholarshi Council. C. Carlet and A. Pott (Eds.): SETA 2010, LNCS 6338, , c Sringer-Verlag Berlin Heidelberg 2010

2 400 Y.M. Chee et al. where is the comlex conjugate and all subscrits are comuted modulo n +1. For all t 0 mod (n + 1), the C t (a) s are called out-of-hase autocorrelation coefficients, and in-hase autocorrelation coefficients otherwise. Motivated by alications in engineering, sequences with small out-of-hase coefficients are of articular interests. Usually, the comlex sequence a is exected to have a two-level autocorrelation function, i.e. all out-of-hase autocorrelation coefficients are a constant γ. For an almost m-ary sequence a, wecalla erfect if it has a two-level autocorrelation function and γ = 0. Moreover, we call a nearly erfect if out-of-hase autocorrelation coefficients γ all satisfy γ =1, or all they satisfy γ = 1. We refer to [7] for a well-rounded survey on erfect binary sequences, to [9] for results of erfect and nearly erfect -ary sequence, where is an odd rime. In the following we briefly introduce the relationshi between (almost) binary (with entries ±1) erfect sequences and (Conference) Hadamard matrices. AmatrixH with entries ±1 and order v is called a Hadamard matrix if HH T = vi; andamatrixc with entries 0, ±1 and order v is called a Conference matrix if CC T =(v 1)I, wherei is the identity matrix. It is well known that erfect binary (with entries ±1) sequences of eriod v are equivalent to cyclic difference sets (see [7, Section 2]). In articular, when v 0 mod 4, the erfect binary sequences are equivalent to circulant Hadamard matrices, or cyclic Hadamard difference sets (see [12, Section 1.1]). More recisely, let a = (a 0,a 1,...,a v 1 ) be a binary erfect sequence of eriod v. LetH =(h i,j ) v 1 i,j=0 be a circulant matrix (H is called circulant if h i+1,j+1 = h i,j for all i, j) defined by h 0,j = a j for j Z v,thenh is a circulant Hadamard matrix of order v. Similarly, let a =(a 0,a 1,...,a v 1 ) be an almost binary erfect sequence, i.e. a 0 =0anda i = ±1 for1 i v 1. Then the circulant matrix C = (h i,j ) v 1 i,j=0 defined by h 0,j = a j for j Z v is a circulant Conference matrix. The famous circulant Hadamard matrices conjecture is that there do not exist circulant Hadamard matrices if v>4. In contrast to this still oen roblem, an elegant and elementary roof in [13] shows that there do not exist circulant Conference matrices: It seems that the mathematical behavior of binary erfect sequences and almost binary erfect sequences are quite different, which is one motivation of our aer. In this aer, we study the roerties of general almost -ary erfect sequences, where is a rime. It turns out that almost -ary erfect sequences of eriod n+1 are equivalent to (n+1,,n,(n 1)/) relative difference sets in Z n+1 Z relative to Z (Theorem 1). Thelackofexamlesofalmost-ary erfect sequences motivates our research in almost -ary nearly erfect sequences. It is shown that eriodic almost -ary nearly erfect sequences corresond to certain direct roduct difference sets (Theorem 6). This aer is organized as follows. In Section 2, we give necessary definitions and results. The discussions about almost -ary erfect and nearly erfect sequences are given in Section 3 and 4 resectively. In Aendix, we give two tables of the existence status of almost -ary erfect and nearly erfect sequences

3 Almost -Ary Perfect Sequences 401 with eriod less than 100. Our results generalize those about erfect and nearly erfect -ary sequences which have been done by Ma and Ng in [9]. 2 Preliminaries In this section, we give necessary definitions and results used in this aer. 2.1 Relative Difference Sets, Characters and Grou Rings To facilitate the study of difference sets by using grou rings and character theory, we use the multilicatively written grou G = g g n =1 instead of the additively written grou Z n. We refer to [10] for the basic facts of grou rings and [8] for character theory on finite fields. In the following, we identify a subset A of G with a grou ring element a A a of C[G] and we still denote it by A. For any integer t, we define A (t) = a A at. For a grou ring element A = g G a gg C[G], we define A = g G a g. Let G be an abelian grou of order mn and let N be a subgrou of order n. A k-subset R of G is said to be an (m, n, k, λ) relative difference set (RDS) in G relative to N if all elements not in N can be reresented exactly λ times as the form r 1 r2 1, r 1,r 2 R and r 1 r 2, and no element in N can be reresented as this form. In the language of grou rings, R is an (m, n, k, λ) relative difference set in G relative to N if and only if RR ( 1) = k + λ(g N). A k-subset D of a grou G is said to be a (v, k, λ) difference set (DS) if all non-identity elements of G can be reresented exactly λ times as the form d 1 d 1 2, d 1,d 2 D, d 1 d 2. We refer to [2] for details on difference sets. Relative difference sets can be regarded as the lifting of difference sets. Result 1. [11, Lemma ] Let R be an abelian (m, n, k, λ)-rds in G relative to N, wheren is a subgrou of G of order n. LetL be a subgrou of N of order l and let ρ : G G/L be the natural eimorhism. Then ρ(r) is an (m, n l,k,lλ)- RDS in G/L relative to N/L. Moreover, if L = N, thenρ(r) is an (m, k, nλ) difference set in G/L. The following result rovides a useful method to rove a k-subset R of G to be an (m, n, k, λ)-rds. Result 2. Let G be an abelian grou of order mn and let N beasubgrouofg of order n. Ak-subset R is an (m, n, k, λ)-rds in G relative to N if and only if n if χ is not rincial on N, χ(r)χ(r) = k nλ if χ is rincial on N but nonrincial on G, (1) k 2 if χ is rincial on G, where χ is a character of G.

4 402 Y.M. Chee et al. Arime is said to be self-conjugate modulo w if j 1 mod (w )forsome j, wherew is the maximal -free art of w, i.e. the maximal factor of w which is relative rime to. A comosite integer m is said to be self-conjugate modulo w if every rime divisor of m is self-conjugate modulo w. The self-conjugate condition is quite useful in determining the existence of RDSs. The following two results are used later; see [11]. Result 3. Let be a rime and ζ w be a rimitive w-th root of unity in C. 1. If w = e, then the decomosition of the ideal () into rime ideals is () = (1 ζ w ) φ(w). 2. If (w, ) =1, then the rime ideal decomosition of the ideal () is () = π 1 π g,whereπ i s are distinct rime ideals. Furthermore, g = φ(w)/f where f is the order of modulo w. The field automorhism ζ w ζw fixes the ideals π i. 3. If w = e w with (w,)=1, then the rime ideal () decomoses as () = (π 1 π g ) φ(e),whereπ i s are distinct rime ideals and g = φ(w )/f. Ift is an integer not divisible by and t s mod (w ) for a suitable integer s, then the field automorhism ζ w ζw t fixes the ideals π i. 2.2 Two Imortant Lemmas The following two lemmas are crucial to the roof of our results in Section 3 and 4. They deal with the cases whether the self-conjugate condition is satisfied or not. We record them here for the convenience of the reader. Lemma 1. [9] Let q be a rime and α be a ositive integer. Let K be an abelian grou such that either q does not divide K or the Sylow q-subgrou of K is cyclic. Let L be any subgrou of K and Y Z[K] where the coefficients of Y lie between a and b where a<b.suose 1. q is self-conjugate modulo ex(k); 2. q r χ(y )χ(y ) for all χ L and q r+1 χ(y )χ(y ) for some χ L ; 3. χ(y ) 0for some χ L Q where Q = K if q K and Q is the subgrou of K of order q otherwise. Here L denotes the subset of the character grou which is non-rincial on L. Then 1. if q K, r is even and q r 2 b a; and 2. if Sylow q-subgrou of K is cyclic, q r 2 2(b a) when L is a roer subgrou of K and q r 2 b a when L = K. Lemma 2. [1] Let G = α H be an abelian grou of exonent v = uw, where ord(α) =u, ex(h) =w and (u, w) =1.Suosey Z[G] and σ Gal(Q(ζ v )/Q) such that 1. χ(y)χ(y) =n for all characters χ of G such that χ(α) =ζ u,wheren is an integer relative rime to w; and 2. σ fixes every rime ideal divisor of nz[ζ v ].

5 If σ(ζ v )=ζ t v,then y (t) = ±βy + Almost -Ary Perfect Sequences 403 r α u/i x i, where β G, x 1,,x r Z[G] and 1,, r are all rime divisors of u. Furthermore, if u is even, then the sign ± can be chosen arbitrarily by choosing aroriate β. 2.3 Direct Product Difference Sets We conclude this section by introducing the direct roduct difference sets. They were first defined in [4], but studied only the case λ 1 =0,λ 2 = 0. The general definition of direct roduct difference sets is given in [9]. Let G = H N, where the order of H and N are m and n resectively. A k- subset R is said to be an (m, n, k, λ 1,λ 2,μ) direct roduct difference set (DPDS) in G relative to H and N if reresent i=1 r 1 r 1 2, r 1,r 2 R, r 1 r 2 1. all non-identity elements in H exactly λ 1 times; 2. all non-identity elements in N exactly λ 2 times; 3. all non-identity elements in G\H N exactly μ times. In the grou ring language, R is an (m, n, k, λ 1,λ 2,μ)-DPDS in G relative to H and N if and only if RR ( 1) =(k λ 1 λ 2 + μ)+(λ 1 μ)h +(λ 2 μ)n + μg. (2) 3 Almost -Ary Perfect Sequences In this section we construct almost -ary erfect sequences, and rove that they do not exist with certain eriods. First we fix some notations which will be frequently used. Let be a rime and let G = H P,whereH = h,p = g, ord(h) =n +1andord(g) =. Letζ be a rimitive -th root of unity and a = {0,a 1,,a n } be an almost -ary sequence of eriod n +1, wherea i = ζ bi with b i {0, 1,, 1}. For the convenience of exression, we define a 0 =0. Now we consider the following n-subset R of G R = {g bi h i i =1, 2,,n}. (3) Obviously, n RR ( 1) = n + t=1 j t n g bj+t bj h t. (4) j=1 mod (n+1)

6 404 Y.M. Chee et al. Lemma 3. Let χ be a character of P and extend χ : Z[G] Q(ζ )[H] to be a ring eimorhism such that χ(x) =x for all x H. Then { n χ(r)χ(r ( 1) )= t=0 C t(a) σ h t if χ is nonrincial on P, (5) 1+(n 1)H if χ is rincial on P, where σ Gal(Q(ζ )/Q) and σ(ζ )=χ(g). Proof. If χ is rincial on P,thenχ(R)χ(R ( 1) )=(H 1) 2 =1+(n 1)H. Otherwise, suose χ(g) =σ(ζ )forsomeσ Gal(Q(ζ )/Q), the results follow from (4). We remind the reader that a is an almost -ary PS of eriod n + 1 if its out-ofhase autocorrelation coefficients are all zero and C 0 (a) =n. Theorem 1. Let a be an almost -ary sequence of eriod n +1,thena is an almost -ary erfect sequence if and only if R is an (n +1,,n,(n 1)/)-RDS in G relative to P, i.e. RR ( 1) = n + n 1 (G P ), (6) where R is defined in (3). Proof. By Lemma 3, for all characters χ of P, { n if χ is nonrincial on P, χ(rr ( 1) )= 1+(n 1)H if χ is rincial on P. Now the result is followed by Result 2. By Theorem 1, we get a necessary condition for the existence of almost -ary PSs with eriod n +1. Corollary 1. If there exists an almost -ary erfect sequence of eriod n +1, then n 1. In the following we give an examle of almost -ary PSs from the classical affine difference sets. Examle 1. Let F := F q be the field of order q and q = w f be a rime ower. Let α be a rimitive element of the field K = F q 2 and let G be the multilicative grou of K. Clearly G = Z q2 1. It is well known that the subset D = {α i Tr K F (αi )=1} of K is a cyclic (q +1,q 1,q,1)-RDS in G relative to N, wheren G and N = Z q 1 (see [11, Theorem ]). Let be a rime divisor of q 1with gcd(, q +1) = 1 and let M N,M = Z q 1.Letρ : G G/M be the )-RDS in G/M relative to N/M. It is clear that G/M = Z q+1 Z and N/M = Z. By Theorem 1, there exists an almost -ary PS of eriod q + 1 whenever q is a rime ower and q 1, gcd(, q +1)=1. natural eimorhism, then ρ(r) isa(q +1,,q, q 1

7 Almost -Ary Perfect Sequences 405 Next we give several nonexistence results to show that almost -ary PSs do not exist with certain eriods. Result 4. [5] Abelian slitting (n +1, 2, n,(n 1)/2)-relative difference sets do not exist. By Theorem 1, we have the following result. Theorem 2. There do not exist almost binary erfect sequences of eriod n +1 for any n. Result 5. [6] Let R be an abelian (n +1,n 1,n,1)-RDS in G relative to N, then n should be a rime ower for n 10, 000. Using the technique in [9], we have the following result. Note that gcd(, q) =1, where is a rime divisor of n and q is a rime divisor of n +1. We use the notation r n to denote r strictly divide n, namely r n but r+1 n. Theorem 3. Let R be an (n +1,,n, n 1 )-RDS in G relative to P.Assume that there exists a rime divisor q of n and q is self-conjugate modulo u, where u n +1.Letq r n, thenr is even and q r 2 n+1 u. Proof. Let ρ : G K := G/ h u be the natural eimorhism. By (6), ρ(r)ρ(r ( 1) )=n + n 1 ( n +1 K ρ(p )). u It is clear that the coefficients of ρ(r) lie between 0 and n+1 u.letχ be a nonrincial character of K, wehave χ(ρ(r))χ(ρ(r)) = { n if χ is nonrincial on ρ(p ), 1 if χ is rincial on ρ(p ). Now set L = ρ(p )andy = ρ(r), where L and Y aredefinedinlemma1.itis routine to verify that the conditions in Lemma 1 are satisfied for L and Y here. Therefore we comlete the roof. The following two results can be easily followed from Theorem 3. Corollary 2. Assume that there exists a rime divisor q of n such that q is self-conjugate modulo (n +1), then there do not exist (n +1,,n, n 1 )-RDSs in G relative to P. In other words, there do not exist almost -ary erfect sequences of eriod n +1. Corollary 3. Assume that there exists a rime divisor q of n such that q is self-conjugate modulo. Ifq 2s+1 n, then there do not exist (n +1,,n, n 1 )- RDSs in G relative to P. In other words, there do not exist almost -ary erfect sequences of eriod n +1.

8 406 Y.M. Chee et al. The above results deend on the self-conjugate condition. Usually it is difficult to determine the existence status of almost -ary PSs if this condition is not satisfied. However, in some cases we can determine whether the almost -ary PSs exist or not when n is small. We briefly introduce the main idea of this method here. An (m, n, k, λ)-rds is called regular if k 2 λmn. LetD be a RDS in G and let t be an integer with gcd(t, G ) =1.Wecallt a multilier of D if D (t) = Dg for some g G. ItiswellknownthatRg is also a RDS for any g G. The following result tells us that we may assume R satisfying R (t) = R if R is regular. Result 6. [11] Let R be a regular (m, n, k, λ)-rds and let t be a multilier of D. Then there exists at least one translate Rg such that (Rg) (t) = Rg. Let Ω be the set of orbits of G under the grou automorhism x x t.since R (t) = R, weseethatr is the union of elements in Ω, namely R = ω Φ ω, where Φ Ω. A natural way to construct R is to combine the elements in Ω. On the one hand, if there do not exist a subset Φ of Ω such that ω Φ ω = R, then clearly R do not exist. On the other hand, to construct R, we may find suitable Φ with ω Φ ω = R and verify whether ω Φ ω is a RDS. Next result gives a way to find multiliers of RDSs. Theorem 4. Let R be an (n +1,,n, n 1 )-RDS in G = H P = h g = Z n+1 Z relative to P,where is an odd rime. Let n = r1 1 r2 2 r l l be the rime decomosition of n. For1 i l, letσ i Gal(Q(ζ)/Q) defined by σ i (ζ) =ζ i,whereζ is a rimitive (n +1)-th root of unity. Assume that l i=1 σ i {1} and let ϕ l i=1 σ i. Ifϕ(ζ) =ζ α,thenα is a multilier of R. Proof. By (6), we have RR ( 1) = n+ n 1 (G P ). Let χ be a character of G such that χ(g) =ζ,thenχ(r)χ(r) =n. By Result 3, the rime ideal factorization of (n) inz[ζ (n+1) ]is (n) =( r1 1 r2 2 r l l )= l (P 1,i P si,i) ri, where s i = φ((n+1))/f i and f i =ord (n+1) ( i ). By Result 3 (ii) and gcd(n, (n+ 1)) = 1, we know that σ i fixes the rime ideals P j,i for 1 j s i. Therefore, ϕ fixes all rime ideals P j,i for 1 j s i and 1 i l. By Lemma 2, i=1 R (α) = ±βr + Px, where β G and x Z[G]. Let χ 0 be the rincial character of G, then n = χ 0 (R (α) )=χ 0 (±βr + Px)=±n + x.

9 Almost -Ary Perfect Sequences 407 It follows that x = 0 as gcd(, n) = 1. Next we show that x must be 0. Note that R = G/P 1 and we can assume that R = n g bi h i, where 0 b i 1. Therefore we have RP = G P.Now R (α) R ( α) =(βr + Px)(βR + Px) ( 1) i=1 = RR ( 1) + βx ( 1) RP ( 1) + β 1 xr ( 1) P + xx ( 1) P = n +( n 1 + βx ( 1) + β 1 x)g ( n 1 + βx ( 1) xx ( 1) + xβ 1 )P. On the other hand, note that gcd(α, G ) =1,wehave R (α) R ( α) =(RR ( 1) ) (α) =(n + n 1 Therefore, { βx ( 1) + β 1 x =0, βx ( 1) xx ( 1) + xβ 1 =0. (G P )) (α) = n + n 1 (G P ). From above we have xx ( 1) = 0, which imlies that x = 0. It follows that R (α) = β R for some β G. The roof is comleted. Next we give an examle to disrove the existence of an almost -ary PS by alying Theorem 4. Examle 2. There do not exist almost 7-ary PS with eriod 23. Proof. First it can be verified that almost 7-ary PSs with eriod 23 do not satisfy the conditions of Theorem 3. By Theorem 1, it is equivalent to rove that there do not exist (23, 7, 22, 3)-RDS, say R, ing = Z 23 Z 7 relative to Z 7.Itcanbe checked that 2 is a multilier of R by Theorem 4. Using MAGMA[3], we comute the orbits of G under the grou automorhism x x 2. The results are in the following table. We checked that the only ossible combination of orbits such that the cardinality is 22 and find it is not a RDS. Then we finish the roof. Using similar argument, we get the following result. Table 1. Orbits of G under x x 2 Length of orbit number

10 408 Y.M. Chee et al. Theorem 5. There do not exist almost -ary PSs of eriod n +1,where n 1 and n {22, 28, 45, 52}. Remark 1. The method in Examle 2 cannot determine the existence of RDSs if the number of orbits gets larger. Indeed, in this case usually there exist many combinations of the orbits such that the size of the sum of these orbits equal to the cardinality of the RDS, and therefore it is imossible to verify all of them whether is an RDS one by one. For examle, when n =77and =19inTheorem 4, it can be verified that 49 is a multilier of the corresonding (78, 19, 77, 19)- RDS. The orbits of G = Z 78 Z 19 under the grou automorhism x x 49 is ,wherei j imlies that there are j orbits with length i. Itcanbe comuted that the number of the combinations of the orbits such that the size of the sum of them equal to R =77is 12 i=0 ( i )(( )( ) i 2 ( 36 i )( 6 5 )) In Table 2, we have the following oen cases to be determined. They cannot be excluded by using the above method. Question 1. Whether almost -ary PSs exist for n =50, 76, 77, 94, 99, 100, where odd rime n 1. Table 2 in Aendix lists the existence status of -ary PSs of eriod n +1for 3 n 100 and is a rime divisor of n 1. The question mark? in the table is used to denote an undecided case. 4 Almost -Ary Nearly Perfect Sequences Let G, H, P, a be the same as those in the last section. The sequence a is called an almost -ary nearly erfect sequence (NPS) of tye I (II) if the out-of-hase autocorrelation coefficients are all 1(1). Similar to Theorem 1, we have the following result. Theorem 6. Let a =(0,a 1,a 2,, a n ) be an almost -ary sequence of eriod n +1,wherea i = ζ bi and 0 b i 1 for 1 i n. LetR = n g bi h i.then 1. a is an almost -ary NPS of tye I if and only if R is an (n+1,,n, n 1, 0, n ) direct roduct difference set in G relative to H and P. 2. a is an almost -ary NPS tye II if and only if R is an (n +1,,n, n 2 + 1, 0, n 2 ) direct roduct difference set in G relative to H and P. From Theorem 6 we have the following necessary condition for the existence of almost -ary NPSs. Corollary 4. (1) If there exists an almost -ary NPS of eriod n +1 of tye I, then n. (2) If there exists an almost -ary NPS of eriod n +1 of tye II, then n 2. Next we construct a family of almost -ary NPSs of tye I. i=1

11 Almost -Ary Perfect Sequences 409 Examle 3. Let q be a rime and let bearimedivisorofq 1. Let H be the additive grou of the finite field F q and let N be the multilicative grou of F q. Let G = H N = Z q Z q 1. Define R = {(x, x) x =0, 1,,q 2}. Clearly R is a (q, q 1,q 1, 0, 0, 1) direct roduct difference set in G relative to H and N (see [11, Examle 5.3.2]). By (2), RR ( 1) = q +(G H N). (7) Let ρ : G G/M be the natural eimorhism, where M N and M = Z q 1. Then by (7), we have ρ(r)ρ(r ( 1) )=q N/M + q 1 (G/M N/M), which follows that ρ(r) isa(q,, q 1, q 1 relative to H/M = Z q 1, 0, q 1 )-DPDS in G/M = Z q Z and N/M = Z. By Theorem 6 (1), there exists an almost -ary NPS of tye I with eriod q. In the following we resent results to show almost -ary NPSs of tye I do not exist with certain eriods. Lemma 4. Let n be an odd integer and b i {0, 1,,n 1} for 1 i n. Assume that b i b j when i j, then {b i+1 b i i =1, 2,,n 1} <n 1 (b i b j is comuted modulo n). Proof. Let S = {b i+1 b i i =1, 2,,n 1}. Clearly S n 1. Now assume that S = n 1, then S = {1,...,n 1} as b i b j for i j. Therefore, n 1 i=1 (b i+1 b i )= n 1 i=1 i n(n 1) 2 0 mod n as n is odd. On the other hand, n 1 i=1 (b i+1 b i )=b n b 1. The contradiction arises as b i b j mod n. Theorem 7. Let G = H P = h g = Z n+1 Z n and let R be an (n + 1,n,n,0, 0, 1)-DPDS in G relative to H and P.ThenR does not exist if n is an odd integer. Therefore, for any odd rime, there do not exist almost -ary NPSs of tye I with eriod +1. Proof. Since R = G/P 1 and no elements in P can be reresented as the differences of elements in R, we can assume that R = n i=1 h i and b gbi i {0, 1,...,n 1}. Similarly, as there are also no elements in H can be reresented as the differences of elements in R and P = R = n,wehave{b i i =1,...,n} = {0, 1,...,n 1}. Now n n n +1+(G H P )=RR ( 1) = n + g bi+t bi h t. It follows that for each t 0, t=1 i t i=1 mod (n+1) {b i+t b i :1 i n i t mod (n +1)} = {1,...,n 1}. However, by letting t = 1 and we see the above equation cannot hold by Lemma 4. We finish the roof. By Lemma 1, we have the following nonexistence result.

12 410 Y.M. Chee et al. Theorem 8. Let q be a rime divisor of n+1 such that q r n+1,q. Assume that q is self-conjugate modulo u for a divisor u of n +1.LetG = H P = h g = Z n+1 Z and let K = G/ h u = Z u Z. If there exists an almost -ary NPS of tye I with eriod n +1,then 1. If q K, r is even and q r 2 n+1 u ;and 2. If q K, q r 2 2 n+1 u. Proof. By Theorem 6, we can assume that there is an (n +1,,n, n 1, 0, n )- DPDS R in G relative to H and P.ThenRR ( 1) =(n +1) H + n (G P ). Let ρ : G K be the natural eimorhism. Clearly the coefficients of ρ(r) Z[K] lie between 0 and n+1 u.sinceq is self-conjugate modulo ex(k) = u and for any nonrincial character χ of K, n +1if χ is nonrincial on both ρ(p ) and ρ(h), χ(ρ(r))χ(ρ(r)) = 1 if χ is nonrincial on ρ(h), 0 if χ is nonrincial on ρ(p ). Take L = ρ(p )andk = ρ(r) in Lemma 1, then obviously q r χ(ρ(r))(χ(ρ(r))) for χ ρ(p ) and q r+1 χ(ρ(r))(χ(ρ(r))) for χ ρ(h) ρ(p ).Ifq K, it is also easy to see χ(ρ(r)) 0 because χ(ρ(r))χ(ρ(r)) = n +1 for χ ρ(q) ρ(p ),whereq is the subgrou of K of order q. Now the result follows from Lemma 1. Corollary 5. If there exists a rime divisor q of n +1 with q 2s+1 n +1 and q is self-conjugate modulo, then there do not exist almost -ary NPSs of tye I with eriod n +1. Proof. Take u = 1 in Theorem 8 and note that K =. ByTheorem8,the result follows as q for every rime divisor q of n +1. Corollary 6. Let q be a rime divisor of n +1 with q is self-conjugate modulo (n +1). Assume that q r n +1 for r 4 if q =2. Then there do not exist almost -ary NPSs of tye I with eriod n +1. For almost -ary NPSs of tye II with eriod n + 1, we only find the examle with =2andn =2,namelya = {0, 1, 1}. Wehavedoneacomutersearchfor =3andn {2, 5, 8, 11, 14, 17}, and however, no examle is found. We leave it as the following oen question. Question 2. Do almost -ary NPSs of tye II with eriod n +1exist? In Aendix, Table 3 lists the existence status of the almost -ary NPSs of tye I with eriod n+1 for 2 n 100, where is a rime divisor of n. Thequestion mark? in the table is used to denote an undecided case.

13 Almost -Ary Perfect Sequences 411 Acknowledgement The authors are grateful to the invaluable discussions with Professor Alexander Pott which enables this work. They also thank the anonymous referees for their valuable suggestions. Part of this work was done during the second author s visit at the Otto-von- Guericke- University. He would like to thank the hositality of the Institute of Algebra and Geometry. References 1. Arasu, K.T., Ma, S.L.: Abelian difference sets without self-conjugacy. Des. Codes Crytograhy 15(3), (1998) 2. Beth, T., Jungnickel, D., Lenz, H.: Design theory, 2nd edn. Cambridge University Press, New York (1999) 3. Bosma, W., Cannon, J., Playoust, C.: The magma algebra system I: The user language. Journal of Symbolic Comutation 24(3-4), (1997), htt:// 1774bb42b9fe09d31c90f383c419c7d2 4. Ganley, M.J.: Direct roduct difference sets. Journal of Combinatorial Theory, Series A 23(3), (1977), htt:// B6WHS-4D7D14S-XS/2/4f3316fe57bc9e1b6b ca6aa 5. Jungnickel, D.: On automorhism grous of divisible designs, II: grou invariant generalised conference matrices. Archiv der Mathematik 54(2) (February 1990) 6. Jungnickel, D., Pott, A.: Comutational non-existence results for abelian affine difference sets. Congressus Numerantium 68, (1989) 7. Jungnickel, D., Pott, A.: Perfect and almost erfect sequences. Discrete Alied Mathematics 95(1-3), (1999) 8. Lidl, R., Niederreiter, H.: Finite fields, Encycloedia of Mathematics and its Alications, 2nd edn., vol. 20. Cambridge University Press, Cambridge (1997) 9. Ma, S.L., Ng, W.S.: On nonexistence of erfect and nearly erfect sequences. Int. J. Inf. Coding Theory 1(1), (2009) 10. Passman, D.S.: The algebraic structure of grou rings. John Wiley & Sons, New York (1977) 11. Pott, A.: Finite Geometry and Character Theory. LNM, vol Sringer, Heidelberg (1995) 12. Schmidt, B.: Characters and cyclotomic fields in finite geometry. LNM, vol Sringer, Berlin (2002) 13. Stanton, R., Mullin, R.C.: On the nonexistence of a class of circulant balanced weighing matrices. SIAM Journal on Alied Mathematics 30, (1976)

14 412 Y.M. Chee et al. Aendix Table 2. Existence Status of Perfect Sequences n Existence Status n Existence Status 3 2 not exist by Theorem exist by Examle not exist by Theorem not exist by Corollary 3 with q=2 7 2 not exist by Theorem exist by Examle exist by Examle not exist by Theorem not exist by Corollary 3 with q= not exist by Theorem exist by Examle not exist by Result not exist by Theorem exist by Examle not exist by Corollary 3 with q= not exist by Theorem not exist by Corollary 3 with q= exist by Examle exist by Examle not exist by Theorem not exist by Corollary 3 with q= not exist by Theorem exist by Examle not exist by Result not exist by Theorem not exist by Corollary 3 with q= not exist by Corollary 3 with q= not exist by Theorem not exist by Theorem exist by Examle not exist by Result not exist by Theorem exist by Examle not exist by Corollary 3 with q= not exist by Theorem exist by Examle not exist by Theorem not exist by Theorem exist by Examle not exist by Corollary 3 with q= not exist by Theorem exist by Examle exist by Examle exist by Examle not exist by Theorem not exist by Corollary 3 with q= not exist by Corollary 3 with q= not exist by Theorem not exist by Corollary 3 with q= not exist by Corollary 2 with q= not exist by Corollary 2 with q= not exist by Theorem exist by Examle not exist by Corollary 3 with q= not exist by Theorem not exist by Corollary 3 with q= not exist by Corollary 3 with q= not exist by Corollary 3 with q= not exist by Theorem exist by Examle not exist by Corollary 3 with q= not exist by Theorem exist by Examle exist by Examle not exist by Result not exist by Theorem not exist by Theorem not exist by Corollary 3 with q= not exist by Corollary 3 with q= not exist by Theorem exist by Examle not exist by Result not exist by Theorem exist by Examle ? 51 2 not exist by Theorem not exist by Corollary 3 with q= not exist by Theorem not exist by Corollary 3 with q= not exist by Theorem exist by Examle 1

15 Almost -Ary Perfect Sequences 413 Table 2. (continued) not exist by Corollary 3 with q= not exist by Theorem not exist by Corollary 3 with q= not exist by Corollary 3 with q= not exist by Corollary 3 with q= not exist by Theorem not exist by Corollary 3 with q= not exist by Corollary 3 with q= not exist by Corollary 3 with q= not exist by Theorem exist by Examle not exist by Result not exist by Theorem exist by Examle exist by Examle not exist by Corollary 3 with q= not exist by Theorem not exist by Corollary 3 with q= exist by Examle exist by Examle not exist by Theorem not exist by Corollary 3 with q= not exist by Corollary 3 with q= not exist by Theorem exist by Examle exist by Examle notexistbyresult not exist by Theorem not exist by Corollary 3 with q= not exist by Corollary 3 with q= not exist by Corollary 3 with q= not exist by Theorem exist by Examle exist by Examle notexistbyresult not exist by Theorem exist by Examle not exist by Result not exist by Theorem not exist by Corollary 3 with q=3 76 3? 77 2 not exist by Theorem ? 78 7 not exist by Corollary 3 with q= not exist by Corollary 3 with q= not exist by Theorem exist by Examle not exist by Result not exist by Theorem exist by Examle not exist by Corollary 3 with q= not exist by Theorem exist by Examle not exist by Result not exist by Theorem not exist by Corollary 3 with q= not exist by Corollary 3 with q= not exist by Corollary 3 with q= not exist by Corollary 3 with q= not exist by Theorem not exist by Corollary 3 with q= not exist by Corollary 3 with q= not exist by Corollary 3 with q= not exist by Theorem exist by Examle not exist by Corollary 3 with q= not exist by Theorem exist by Examle exist by Examle not exist by Result not exist by Theorem not exist by Corollary 3 with q= not exist by Corollary 3 with q= ? 95 2 not exist by Theorem not exist by Corollary 3 with q= not exist by Corollary 3 with q= not exist by Corollary 3 with q= not exist by Theorem exist by Examle not exist by Corollary 3 with q= not exist by Theorem ? 100 3? ?

16 414 Y.M. Chee et al. Table 3. Existence Status of Nearly Perfect Sequences n Existence Status n Existence Status 2 2 exist by examle not exist by Theorem 7 with q=3 4 2 exist by examle not exist by Corollary 5 with q=2 6 2 exist by examle exist by examle not exist by Theorem 7 with q=7 8 2 not exist by Corollary 6 with q=3 9 3 not exist by Corollary 5 with q= exist by examle exist by examle not exist by Theorem 7 with q= exist by examle exist by examle not exist by Corollary 5 with q= not exist by Corollary 5 with q= not exist by Corollary 5 with q= not exist by Corollary 6 with q= not exist by Corollary 6 with q= exist by examle not exist by Corollary 5 with q= exist by examle exist by examle not exist by Theorem 7 with q= not exist by Corollary 5 with q= not exist by Corollary 5 with q= not exist by Corollary 5 with q=2 21 7? 22 2 exist by examle exist by examle not exist by Theorem 7 with q= not exist by Corollary 6 with q= not exist by Corollary 6 with q= not exist by Corollary 5 with q= not exist by Corollary 5 with q= ? 27 3? 28 2 exist by examle exist by examle not exist by Corollary 5 with q= exist by examle exist by examle exist by examle not exist by Theorem 7 with q= not exist by Corollary 5 with q= not exist by Corollary 5 with q= not exist by Corollary 5 with q= not exist by Corollary 5 with q=5 17 not exist by Corollary 5 with q=5 35 5? 35 7 not exist by Corollary 6 with q= exist by examle exist by examle not exist by Theorem 7 with q= not exist by Corollary 5 with q= not exist by Corollary 5 with q= not exist by Corollary 5 with q= not exist by Corollary 5 with q= exist by examle exist by examle not exist by Corollary 5 with q= exist by examle exist by examle exist by examle not exist by Theorem 7 with q= not exist by Corollary 5 with q= ? 45 3 not exist by Corollary 5 with q= not exist by Corollary 5 with q= exist by examle exist by examle not exist by Theorem 7 with q= not exist by Corollary 6 with q= not exist by Corollary 6 with q= not exist by Corollary 6 with q= not exist by Corollary 5 with q= not exist by Corollary 5 with q=3 51 3? not exist by Corollary 5 with q= exist by examle exist by examle not exist by Corollary 5 with q=2

17 Almost -Ary Perfect Sequences 415 Table 3. (continued) 54 2 not exist by Corollary 5 with q= not exist by Corollary 5 with q= not exist by Corollary 5 with q= not exist by Corollary 5 with q= not exist by Corollary 5 with q= not exist by Corollary 5 with q= not exist by Corollary 5 with q= not exist by Corollary 5 with q= exist by examle exist by examle not exist by Theorem 7 with q= exist by examle exist by examle exist by examle not exist by Corollary 5 with q= not exist by Corollary 5 with q= not exist by Corollary 6 with q= not exist by Corollary 6 with q=2 63 7? 64 2 not exist by Corollary 5 with q= not exist by Corollary 5 with q= not exist by Corollary 5 with q= exist by examle exist by examle exist by examle not exist by Theorem 7 with q= not exist by Corollary 5 with q= not exist by Corollary 5 with q= not exist by Corollary 5 with q= not exist by Corollary 5 with q= exist by examle exist by examle exist by examle not exist by Theorem 7 with q= exist by examle exist by examle not exist by Theorem 7 with q= not exist by Corollary 5 with q= not exist by Corollary 5 with q=3 75 3? 75 5 not exist by Corollary 5 with q= not exist by Corollary 5 with q= ? 77 7? not exist by Corollary 5 with q= exist by examle exist by examle not exist by Theorem 7 with q= not exist by Corollary 6 with q= not exist by Corollary 6 with q= not exist by Corollary 6 with q= exist by examle exist by examle not exist by Theorem 7 with q= not exist by Corollary 5 with q= not exist by Corollary 5 with q= not exist by Corollary 5 with q= not exist by Corollary 5 with q= not exist by Corollary 5 with q= not exist by Corollary 5 with q= not exist by Corollary 5 with q= not exist by Corollary 5 with q= not exist by Corollary 5 with q= exist by examle exist by examle not exist by Corollary 5 with q= exist by examle exist by examle exist by examle not exist by Theorem 7 with q= not exist by Corollary 5 with q= ? 93 3 not exist by Corollary 5 with q= ? 94 2 not exist by Corollary 5 with q= not exist by Corollary 5 with q= not exist by Corollary 5 with q= not exist by Corollary 5 with q= exist by examle exist by examle not exist by Corollary 5 with q= not exist by Corollary 5 with q= ? 99 3 not exist by Corollary 6 with q= ? exist by examle 3