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On a Class of Almost Perfect Sequences
192, 641]650 Ž1997. JA976962
JOURNAL OF ALGEBRA ARTICLE NO.
On a Class of Almost Perfect Sequences K. T. Arasu* Department of Mathematics and Statistics, Wright State Uni¨ ersity, Dayton, Ohio 45435
S. L. Ma† Department of Mathematics, National Uni¨ ersity of Singapore, Kent Ridge, Singapore 119260, Republic of Singapore
and N. J. Voss ‡ Department of Mathematics and Statistics, Wright State Uni¨ ersity, Dayton, Ohio 45435 Communicated by Walter Feit Received October 23, 1995
Periodic "1 sequences all but one of whose out-of-phase autocorrelation coefficients are zero are studied by Wolfmann w9x. Using the equivalence of these almost perfect sequences to certain classes of cyclic divisible difference sets Žas noted by Pott and Bradley w7x., we investigate the case u s 2 Žin the terminology of w9x.. Sequences of periods 8, 12, and 28 are given and several nonexistence results are obtained. Our results suggest that it is unlikely to have such sequences for periods greater than 28. Q 1997 Academic Press
1. INTRODUCTION Let S s Ž si . be an n-periodic sequence with entries "1, i.e., si s siqn and si s 1 or y1 for i s 0, 1, 2, . . . . The autocorrelation coefficients are 1 defined by Ct Ž S . s Ý ny is0 s i s iqt . The sequence S is said to be almost *The research was supported by AFOSR Grant F49620-96-1-0328 and NSF Grant NCR9200265. † This work was done when the author visited Wright State University. E-mail: matmasl@ nus.sg. ‡ The research was supported by NSF Grant NCR-9200265 under REU-supplement. 641 0021-8693r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
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ARASU, MA, AND VOSS
perfect if Ct Ž S . s 0 for all t / 0 Žmod n. with exactly one exception. Almost perfect sequences were introduced by Wolfmann w9x. As subset D of the additive group Z m m9 is said to be a Ž cyclic . Ž m, m9, k, l1 , l2 . di¨ isible difference set if < D < s k and the list of differences Ž d y d9 < d, d9 g D . contains each number im Ž i s 1, 2, . . . , m9 y 1. exactly l1 times and each number w g Z m m9 , w / im Ž i s 1, 2, . . . , m9 y 1. exactly l2 times. For details on divisible difference sets, see Jungnickel w4x. Recently, Pott and Bradley w7x proved the following theorem. THEOREM 1.1. An n-periodic sequence S s Ž si . with entries "1 is almost perfect if and only if D s i < 0 F i - n and si s q14 is a cyclic
ž
n 2
, 2,
n y 2u 2
, u Ž u y 1. ,
n y 4u 4
/
di¨ isible difference set where Ž n y 2 u .r2 is the number of entries q1 in a generating cycle. Ž Note that n must be di¨ isible by 4.. The case u s 1 was investigated in w7, 9x. These divisible difference sets correspond to so-called affine difference sets; see Pott w6x. Furthermore, by the prime power conjecture for affine difference sets Žsee w6x., it is believed that Ž n y 2.r2 must be an odd prime power. In this paper, we study the case u s 2. We obtain almost perfect sequences of periods 8, 12, and 28. Some nonexistence results are proved and the results seem to suggest that the only almost perfect sequences with u s 2 are the ones of periods 8, 12, and 28. We verify this for periods up to 20000 with the only undecided cases being 348, 4908, and 16572.
2. PRELIMINARIES In this section, we shall list some preliminary results. First, we have a particular case of a theorem by Bose and Connor w1x; see also w6x. THEOREM 2.1. If an Ž m, m9, k, l1 , l2 . di¨ isible difference set exists and if m ' 2 mod 4 and m9 is e¨ en, then k y l1 is a sum of two squares. To facilitate the study of divisible difference sets using group rings and characters, we use the multiplicative group G s ² g < g m m9 s 1: instead of the additive group Z m m9. Let N s ² g m : be the subgroup of G of order m9. A subset D of G is an Ž m, m9, k, l1 , l2 . divisible difference set if and only if DD Žy1. s Ž k y l1 . q Ž l1 y l2 . N q l2 G,
Ž 1.
ALMOST PERFECT SEQUENCES
643
where we identify a subset A of G with the element Ý h g A h of Zw G x and write AŽ t . s h t < h g A4 for any integer t. By applying the characters of G on Ž1., we have the following well-known result; see w6x. THEOREM 2.2. Let G be a cyclic group of order mm9 and N be the subgroup of G of order m9. A subset D of G is an Ž m, m9, k, l1 , l2 . di¨ isible difference set if and only if, for e¨ ery character x of G,
¡k y l
1
x Ž D. x Ž D. s
~k
¢k
2
2
if x is nonprincipal on N,
y mm9l2 if x is principal on N but nonprincipal on G, if x is principal on G.
Ž 2. Let zw denote the complex wth root of unity expŽ2p'y 1 rw .. By Theorem 2.2, we translate the properties of divisible difference sets into equations of algebraic integers in the ring Zw zm m9 x. By studying these equations, we can obtain some nonexistence results of divisible difference sets Žsee w6x for this approach.. In order to do so, we need the following standard result concerning algebraic integers; see w3, 8x. LEMMA 2.3. Let p be a prime and ¨ s p s w with Ž w, p . s 1. Then the s ideal pZw z¨ x decomposes as pZw z¨ x s Ž P1 P2 ??? Pr . f Ž p . where f Ž x . is the Euler f-function and P1 , P2 , . . . , Pr are prime ideals in Zw z¨ x. If t is an integer relati¨ ely prime to p and t ' p j mod w for some integer j, then the ring automorphism z¨ ¬ z¨t fixes e¨ ery Pi . The next lemma is a variation of a result by Ma w5x. LEMMA 2.4. Let ¨ s uw with Ž u, w . s 1, q an integer relati¨ ely prime to ¨ , and H an abelian group of order ¨ which contains an element h of order w. If y g Zw z q xw H x satisfies x Ž y . g f Ž zw .Zw z q ¨ x, for all characters x of H with x Ž h. s zw , where f Ž X . is a polynomial in Zw z q xw X x such that f Ž zw .Zw z q ¨ x and uZw z q ¨ x are relati¨ ely prime, then r
y s f Ž h. x0 q
Ý ² hw r p : x i , i
is1
where x 0 , x 1 , . . . , x r g Zw z q xw H x and p1 , p 2 , . . . , pr are all prime di¨ isors of w. Proof. Let K be the subgroup of H of order u and t : Zw z q xw H x ª Zw z q w xw K x be a ring homomorphism such that t Ž h. s zw and t Žg . s g for all g g K. For any character x 9 of K, we have x 9Žt Ž y .. g f Ž zw .Zw z q ¨ x. By the Fourier inversion formula, we obtain ut Ž y . s f Ž zw . z for some z g Zw z q w xw K x. Let n be the norm of f Ž zw . with respect to the field extension
644
ARASU, MA, AND VOSS
QŽ z q w . over Q. Since Ž n, u. s 1, there exist integers a and b such that an q bu s 1. Then
t Ž y . s ant Ž y . q but Ž y . s f Ž zw . z9 for some z9 g Zw z q w xw K x. Finally, the lemma follows because the kernel of t is Ý ris1² hw r p i : x i < x i g Zw z q xw H x4. 3. MAIN RESULTS We begin by giving the known examples of almost perfect sequences with u s 2: sequences of periods 8, 12, and 28. The sequence of period 8 is trivial. A sequence of period 12 had already been mentioned in w7x and was first given by Clatworthy w2x. EXAMPLE 3.1. 0, 44 is a cyclic Ž4, 2, 2, 2, 0. divisible difference set Z 8 . The corresponding almost perfect sequence of period 8 is given by generating cycle: q] ] ]q] ] ]. EXAMPLE 3.2. 1, 2, 4, 84 is a cyclic Ž6, 2, 4, 2, 1. divisible difference set in Z 12 . The corresponding almost perfect sequence of period 12 is given by the generating cycle: ]qq]q] ] ]q] ] ]. E XAMPLE 3.3. 0, 1, 3, 6, 9, 10, 14, 19, 21, 25, 26, 274 is a cyclic Ž14, 2, 12, 2, 5. divisible difference set in Z 28 . The corresponding almost perfect sequence of period 28 is given by the generating cycle: qq]q] ]q] ]qq] ] ]q] ] ] ]q]q] ] ]qqq. We proceed to nonexistence results. From now on, G is a multiplicative group or order n, where n ' 0 mod 4, and D is a cyclic Ž nr2, 2, Ž n y 4.r2, 2, Ž n y 8.r4. divisible difference set in G. Let N be the subgroup of G of order 2. By Theorem 2.2, for every nonprincipal character x of G,
x Ž D. x Ž D. s
½
Ž n y 8 . r2 4
if x is nonprincipal on N, if x is principal on N.
First, let us study the intersection numbers < D l Ng < for g g G. It is obvious that < D l Ng < s 0, 1, or 2 for each coset Ng in G. In fact, < D l Ng < s 2 for exactly one coset Ng and < D l Ng < s 1 for exactly Ž n y 8.r2 cosets Ng. Translating D is necessary, we may assume N ; D. Thus D s N j S, where S ; Gr 14 , < S < s Ž n y 8.r2, and no two elements of S lie in the same coset of N.
645
ALMOST PERFECT SEQUENCES
LEMMA 3.4.
Let r : G ª GrN be the natural epimorphism. Then either
Ži. Ž GrN . _ Ž r Ž S . j 14. s b, b 2 , b 4 4 , where b g GrN and oŽ b . s 7, or Žii. Ž GrN . _ Ž r Ž S . j 14. s a, c, ac4 , where a, c g GrN, oŽ c . s 2, and c / 1, a. Proof. Let E s Ž GrN . _ Ž r Ž S . j 14.. Note that < E < s 3 and r Ž D . s GrN q 1 y E. Applying r to Ž1., we obtain
ž
G N
q1yE
/ž
G N
q 1 y E Žy1. s 4 q
/
ž
ny8 G 2
/
N
and hence EE Žy1. s 3 q E q E Žy1.. Write E s x, y, z 4 where x, y, z g Ž GrN . _ 14 . Then
Ž xyy1 q yxy1 . q Ž xzy1 q zxy1 . q Ž yzy1 q zyy1 . s Ž x q xy1 . q Ž y q yy1 . q Ž z q zy1 . . There are six possible cases: Ži. zqz ; Žii. y q yy1 ; Žiii. z q zy1 ; Živ. x q xy1 ; Žv. y q yy1 ; Žvi. x q xy1 . y1
xyy1 q yxy1 s x q xy1 , xzy1 q zxy1 s y q yy1 , yzy1 q zyy1 s xyy1 q yxy1 s x q xy1 , xzy1 q zxy1 s z q zy1, yzy1 q zyy1 s xyy1 q yxy1 s y q yy1 , xzy1 q zxy1 s x q xy1 , yzy1 q zyy1 s xyy1 q yxy1 s y q yy1 , xzy1 q zxy1 s z q zy1, yzy1 q zyy1 s xyy1 q yxy1 s z q zy1 , xzy1 q zxy1 s x q xy1, yzy1 q zyy1 s xyy1 q yxy1 s z q zy1 , xzy1 q zxy1 s y q yy1, yzy1 q zyy1 s
Case Ži.. xyy1 q yxy1 s x q xy1 and yzy1 q zyy1 s z q zy1 imply y s x 2 and y s z 2 . Let a g GrN such that oŽ a. s 2. Then we have x, y, z 4 s x, x 2 , ax4 . Substituting into xzy1 q zxy1 s y q yy1 , we obtain 2 a s a q ay1 s x 2 q xy2 . Hence x 2 s a and E s a, c, ac4 where c s x. Case Žii.. xyy1 q yxy1 s x q xy1 and xzy1 q zxy1 s z q zy1 imply y s x 2 and x s z 2 . Substituting into yzy1 q zyy1 s y q yy1 , we obtain z 3 q zy3 s z 4 q zy4 . Hence z 7 s 1 and E s b, b 2 , b 4 4 where b s z. Similarly, we get the same results for all the remaining cases.
646
ARASU, MA, AND VOSS
Let n s 2 s w, where s G 2 and w is odd, and let a be an element of G of order 2 s and K the subgroup of G of order w. By Lemma 3.4, we have
where either
D s ²a 2
sy 1
:qXqa2
sy 1
s ²a 2
sy 1
: q Ž1 y a 2
Ž M y X y A.
sy 1
sy 1
. X q a 2 Ž M y A. ,
X ; M _ A, M s K j a K j a 2 K j a 3 K j ??? ja 2
sy 1
y1
K and
Ži. A s 1, h, h 2 , h 4 4 , where h g K and oŽ h. s 7, or sy 2 sy 2 sy 2 Žii. A s 1, a 2 , g, a 2 g 4 , where g g G and g f ² a 2 :. Note that, in Zw G x, M s K Ž 1 q a q a 2 q a 3 q ??? qa 2
sy 1
y1
sy2 i
. s K Ł Ž1 q a 2 . . is0
Thus, for any character x of G which is nonprincipal on ² a 2
¡2 x Ž X . q x Ž A.
~
sy 1
:,
if x is nonprincipal on K , sy2
x Ž D . s 2 x Ž X . q x Ž A. y w Ł 1 q x Ž a 2 . i
¢
THEOREM 3.5.
Ž 3.
is0
if x is principal on K .
If A s 1, h, h 2 , h4 4 , then n s 28.
Proof. First, we show that 8 ¦ n. Assume 8 < n, i.e., s G 3. Let x be any sy 1 character of G which is nonprincipal on ² a 2 : and x Ž h. s z 7 . Then
x Ž A . s 1 q z 7 q z 72 q z 74 s y Ž z 73 q z 75 q z 76 . . Since Ž z 73 q z 75 q z 76 . Ž z 73 q z 75 q z 76 . Zw z 7 x s 2Zw z 7 x, we can write 2Z w zn x s P1 P2 ??? Pr P1 P2 ??? Pr
ž
2 sy 1
/
,
where P1 , P2 , . . . , Pr are prime ideals in Zw zn x, Ž P1 P2 ??? Pr . 2 s Ž z 73 q z 75 q z 76 .Zw zn x,, and Pi / Pj for all i, j. As x Ž D .x Ž D . ' 0 mod 4, we have sy 1
x Ž D . g P1m 1 P2m 2 ??? Prm r P1n1 P2n 2 ??? Prn r , where m i q n i s 2 s for all i. By Ž3., it is obvious that n i s 0 for all i.
647
ALMOST PERFECT SEQUENCES
Hence 2s
x Ž D . g Ž P1 P2 ??? Pr . s Ž z 73 q z 75 q z 76 . Z w zn x . 2
Since this is true for all characters x nonprincipal on ² a 2 x Ž h. s z 7, by Lemma 2.4, we have 2
D s Ž h 3 q h 5 q h6 . x 0 q ² h : x 1 q ² a 2
sy 1
2y 1
: and
: x2 ,
Ž 4.
where x 0 , x 1 , x 2 g Zw G x. Let B s D l ²a 2
sy 1
, h: s 1, a 2
½
sy 1
, Ža2
sy 1
e1
e2
sy 1
sy 1
e3
. h 3 , Ž a 2 . h 5 , Ž a 2 . h6 5 , Ž 5.
where e i s 0 or 1. By Ž4., we have 2
B s Ž h3 q h5 q h6 . xX0 q ² h: xX1 q ² a 2
sy 1
: xX2 ,
Ž 6.
where xX0 , xX1 , xX2 g Zw a 2 , h x. Let x 9 be a character of ² a 2 , h:, with sy 1 x 9Ž a 2 . s y1 and x 9Ž h. s z 7 . By Ž5., x 9Ž B . s "z 73 " z 75 " z 76 s Ž z 73 q z 75 q z 76 .u , where sy 1
¡1
us
sy 1
if x 9 Ž B . s z 73 q z 75 q z 76 ,
yz 7 Ž 1 q z 72 .
if x 9 Ž B . s z 73 q z 75 y z 76 ,
yz 76 Ž 1 q z 73 .
if x 9 Ž B . s z 73 y z 75 q z 76 ,
~yz
4 7
Ž1 q z7 .
if x 9 Ž B . s yz 73 q z 75 q z 76 ,
z 7 Ž 1 q z 72 .
if x 9 Ž B . s yz 73 y z 75 q z 76 ,
z 76 Ž 1 q z 73 .
if x 9 Ž B . s yz 73 q z 75 y z 76 ,
z 74 Ž 1 q z 7 .
if x 9 Ž B . s z 73 y z 75 y z 76 ,
¢y1
if x 9 Ž B . s yz 73 y z 75 y z 76 .
Note that u is a unit in Zw z 7 x. On the other hand, by Ž6., x 9Ž B . s Ž z 73 q z 75 q z 76 . 2x 9Ž xX0 .. So Ž z 73 q z 75 q z 76 . x 9Ž xX0 . s u is a unit, a contradiction. So 8 ¦ n. Suppose n / 28. Let x 1 be a character which is principal on sy 1 ² h: but nonprincipal on both ² a 2 : and K. By Ž3.,
x 1Ž D . s 2 x 1Ž X . q 4 ' 0
mod 2.
But x 1Ž D .x 1 Ž D . s Ž n y 8.r2 ' 2 mod 4, a contradiction. THEOREM 3.6. If n ) 8 and A s 1, a 2 s s 2. and oŽ g . s nr4.
sy 2
, g, a 2
sy 2
g 4 , then 8 ¦ n Ž i.e.,
648
ARASU, MA, AND VOSS sy 1
Proof. Let x be a character of G of order n. For any g g G _ ² a 2 :, if oŽg . s 2 p d for an odd prime p, then Ž1 q x Žg ..Zw zn x is a divisor of pZw zn x and hence is relatively prime to 2Zw zn x; if oŽg . is not equal to 2 p d for any prime p, then 1 q x Žg . is a unit in Zw zn x; and if oŽg . s 2 d where 2 F d F s, then Ž1 q x Žg ..Zw zn x s Ž1 q z 2 d .Zw zn x is a divisor of 2Zw zn x. ŽSee w9x for the details.. Let P1 , P2 , . . . , Pr be all prime ideal divisors of 2Zw zn x. Then 2Zw zn x s sy 1 Ž P1 P2 ??? Pr . 2 and Ž1 q z 2 d .Zw zn x s Ž P1 P2 ??? Pr .2 sy d for 2 F d F s. By Ž3., for each Pi , we have
x Ž D . g Pi2
sy 2
qe
_ Pi2
sy 2
qe q1
,
where e s 0 if oŽ g . is not a power of 2 and e s 2 sy f if oŽ g . s 2 f where 3 F f F s. Thus x Ž D .x Ž D . k 0 mod 4, i.e., 8 ¦ n. Hence A s 1, a , g, a g 4 and g g K. Assume oŽ g . / nr4. Let x 1 be a character of G which is principal on ² g : but nonprincipal on both ² a 2 : and K. By Ž3.,
x 1 Ž D . s 2 x 1 Ž X . q 2 Ž 1 " 'y 1 . k 0
mod 2,
a contradiction. The following is a consequence of Theorems 2.1, 3.5, and 3.6. COROLLARY 3.7. n y 8 is a sum of two squares. The next result can be classified as a multiplier theorem. For any integer t and y s Ý g g H a g g g Rw H x, where H is a group, R is a ring, and a g g R, we define y Ž t . s S g g H a g g t. THEOREM 3.8. Suppose n ) 8 and A s 1, a , g, a g 4 . Let u be a di¨ isor of nr4, U a subgroup of G of order nr4u, and r : G ª GrU the natural epimorphism. If there exists an integer t relati¨ ely prime to < GrU < s 4 u such that for each prime di¨ isor p of Ž n y 8.r4, t ' p j p mod 4 u where j p is an integer, then
Ž r Ž D . y ² r Ž a . 2 :.
Žt.
2 s b Ž r Ž D . y ² r Ž a . :. ,
where either b g ² r Ž a .: and t ' 1 mod u or b g ² r Ž a .:r Ž g . t and t ' y1 mod u. Proof. Let t : Zw GrU x ª Zw'y 1 xw KrU x be a ring homomorphism such that t Ž a . s 'y 1 and t Žg . s g for all g g KrU. Let y 1 s t Ž r Ž D .. and y 2 s t Ž r Ž D .Ž t . .. Let x be any character of KrU. By Ž3., we have x Ž y 1 . s Ž1 q 'y 1 .u where u g Zw z4 u x and uu s Ž n y 8.r4. Since s : z4 u ¬ z4tu fixes every prime ideal divisor of u Žsee Lemma 2.3. and Ž1
649
ALMOST PERFECT SEQUENCES
q 'y 1 . s Zw z4 u x s Ž1 q 'y 1 .Zw z4 u x, we have s
x Ž y 2 . j Ž y1 . s x Ž y1 . x Ž y1 . g
ž
ny8 2
/
Z w z4 u x .
So by Lemma 2.4, y 2 y 1Žy1. s x Ž n y 8.r2 for some x g Zw'y 1 xw KrU x. Since y 1 y 1Žy1. s Ž n y 8.r2, we have xx Žy1. s 1. Let x s Ý g g K r U a g g where a g g Zw'y 1 x. By considering the coefficient of the identity element of the equation xx s 1, we have Ý g g K r U a g a g s 1 and hence only c one nonzero a g . So x s 'y 1 g for some g g KrU. This implies y 2 c s 'y 1 g y 1. Since the kernel of t is ² r Ž a . 2 : z < z g Zw GrU x4 , we have
r Ž D.
Žt.
2 s br Ž D . q ² r Ž a . : z,
where b s r Ž a . cg and z g Z w GrU x. Let h : GrU ª H s Ž GrU .r² r Ž a . 2 : be the natural epimorphism. Then h Ž r Ž D ..Ž t . s b 1h Ž r Ž D .. q 2h Ž z . where b 1 s h Ž b .. Since h Ž r Ž D .. s Ž nr2 u. H q 1 y a 1 y g 1 y a 1 g 1 where a 1 s h Ž r Ž a .. and g 1 s h Ž r Ž g .., we have y Ž 1 q a 1 q g 1t q a 1 g 1t . q b 1 Ž 1 q a 1 q g 1 q a 1 g 1 . s 2 h Ž z . y Ž 1 y b1 . . Since 1, a 1 , g 1 , a 1 g 1 are distinct elements in H,
1, a 1 , g 1t , a 1 g 1t 4 s b1 , b1 a 1 , b1 g 1 , b1 a 1 g 1 4 . Note that h Ž z . s 1 y b 1 and hence
Ž r Ž D . y ² r Ž a . 2 :.
Žt.
2 s b Ž r Ž D . y ² r Ž a . :. .
Assume b 1 g 1, a 14 . Then g 1t , a 1 g 1t 4 s g 1 , a 1 g 1 4 . Since oŽ g 1 . s u is odd, we have g 1 s g 1t and hence t g 1 mod u. Assume b 1 g g 1t , a 1 g 1t 4 . Then 1, a 1 4 s g 1tq1, a 1 g 1tq1 4 . Thus g 1tq1 s 1 and hence t ' y1 mod u. COROLLARY 3.9. Let n / 8, 28 and let u be a di¨ isor of nr4. For e¨ ery prime di¨ isor p of Ž n y 8.r4, define Op s p j mod 4 u < j g Z4 ; Z 4 u . Then
F
½
Op < P is a prime di¨ isor of
ny8 4
5
; "1, u " 1, 2 u " 1, 3u " 1 4 .
In particular, we have the following result. COROLLARY 3.10.
If n / 8, 12, 28, then Ž n y 8.r4 is not a prime power.
Proof. Assume n / 8, 28. Then A s 1, a , g, a g 4 . Let Ž n y 8.r4 s p r where p is an odd prime. By Theorem 3.7 with t s p r and u s nr4, we obtain p r ' "1 mod nr4 which is impossible unless n s 12 and p r s 1.
650
ARASU, MA, AND VOSS
We have done a computer search for n such that Ža. Žb. Žc. Žd.
12 - n F 20000; n ' 4 mod 8; n y 8 is a sum of two squares; and n satisfies Corollary 3.9.
Three number are obtained: 348, 4908, and 16572. This means that for n F 20000, there are no almost perfect sequences with u s 2 and period n unless n s 8, 12, 28, 348, 4908, 16572. We do not know whether there are any such sequences of periods 348, 4908, 16572.
REFERENCES 1. R. C. Bose and W. S. Connor, Combinatorial properties of group divisible incomplete block designs, Ann. Math. Statist. 23 Ž1952., 367]383. 2. W. H. Clatworthy, ‘‘Tables of Two Associate Class Partially Balanced Designs,’’ NBS Applied Mathematics Series, vol. 63, 1973. 3. K. Ireland and M. Rosen, ‘‘A Classical Introduction to Modern Number Theory,’’ Springer-Verlag, BerlinrNew York, 1990. 4. D. Jungnickel, On automorphism groups of divisible designs, Canad. J. Math. 34 Ž1982., 257]297. 5. S. L. Ma, Planar functions, relative difference sets and character theory, J. Algebra, 185 Ž1996., 342]356. 6. A. Pott, ‘‘Finite Geometry and Character Theory,’’ Springer-Verlag, BerlinrNew York, 1995. 7. A. Pott and S. P. Bradley, Existence and non-existence of almost perfect autocorrelation sequence, IEEE Trans. Inform. Theory 41 Ž1995., 301]304. 8. E. Weiss, ‘‘Algebraic Number Theory,’’ McGraw]Hill, New York, 1963. 9. J. Wolfmann, Almost perfect autocorrelation sequences, IEEE Trans. Inform. Theory 38 Ž1992., 1412]1418.
JOURNAL OF ALGEBRA ARTICLE NO.
On a Class of Almost Perfect Sequences K. T. Arasu* Department of Mathematics and Statistics, Wright State Uni¨ ersity, Dayton, Ohio 45435
S. L. Ma† Department of Mathematics, National Uni¨ ersity of Singapore, Kent Ridge, Singapore 119260, Republic of Singapore
and N. J. Voss ‡ Department of Mathematics and Statistics, Wright State Uni¨ ersity, Dayton, Ohio 45435 Communicated by Walter Feit Received October 23, 1995
Periodic "1 sequences all but one of whose out-of-phase autocorrelation coefficients are zero are studied by Wolfmann w9x. Using the equivalence of these almost perfect sequences to certain classes of cyclic divisible difference sets Žas noted by Pott and Bradley w7x., we investigate the case u s 2 Žin the terminology of w9x.. Sequences of periods 8, 12, and 28 are given and several nonexistence results are obtained. Our results suggest that it is unlikely to have such sequences for periods greater than 28. Q 1997 Academic Press
1. INTRODUCTION Let S s Ž si . be an n-periodic sequence with entries "1, i.e., si s siqn and si s 1 or y1 for i s 0, 1, 2, . . . . The autocorrelation coefficients are 1 defined by Ct Ž S . s Ý ny is0 s i s iqt . The sequence S is said to be almost *The research was supported by AFOSR Grant F49620-96-1-0328 and NSF Grant NCR9200265. † This work was done when the author visited Wright State University. E-mail: matmasl@ nus.sg. ‡ The research was supported by NSF Grant NCR-9200265 under REU-supplement. 641 0021-8693r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
642
ARASU, MA, AND VOSS
perfect if Ct Ž S . s 0 for all t / 0 Žmod n. with exactly one exception. Almost perfect sequences were introduced by Wolfmann w9x. As subset D of the additive group Z m m9 is said to be a Ž cyclic . Ž m, m9, k, l1 , l2 . di¨ isible difference set if < D < s k and the list of differences Ž d y d9 < d, d9 g D . contains each number im Ž i s 1, 2, . . . , m9 y 1. exactly l1 times and each number w g Z m m9 , w / im Ž i s 1, 2, . . . , m9 y 1. exactly l2 times. For details on divisible difference sets, see Jungnickel w4x. Recently, Pott and Bradley w7x proved the following theorem. THEOREM 1.1. An n-periodic sequence S s Ž si . with entries "1 is almost perfect if and only if D s i < 0 F i - n and si s q14 is a cyclic
ž
n 2
, 2,
n y 2u 2
, u Ž u y 1. ,
n y 4u 4
/
di¨ isible difference set where Ž n y 2 u .r2 is the number of entries q1 in a generating cycle. Ž Note that n must be di¨ isible by 4.. The case u s 1 was investigated in w7, 9x. These divisible difference sets correspond to so-called affine difference sets; see Pott w6x. Furthermore, by the prime power conjecture for affine difference sets Žsee w6x., it is believed that Ž n y 2.r2 must be an odd prime power. In this paper, we study the case u s 2. We obtain almost perfect sequences of periods 8, 12, and 28. Some nonexistence results are proved and the results seem to suggest that the only almost perfect sequences with u s 2 are the ones of periods 8, 12, and 28. We verify this for periods up to 20000 with the only undecided cases being 348, 4908, and 16572.
2. PRELIMINARIES In this section, we shall list some preliminary results. First, we have a particular case of a theorem by Bose and Connor w1x; see also w6x. THEOREM 2.1. If an Ž m, m9, k, l1 , l2 . di¨ isible difference set exists and if m ' 2 mod 4 and m9 is e¨ en, then k y l1 is a sum of two squares. To facilitate the study of divisible difference sets using group rings and characters, we use the multiplicative group G s ² g < g m m9 s 1: instead of the additive group Z m m9. Let N s ² g m : be the subgroup of G of order m9. A subset D of G is an Ž m, m9, k, l1 , l2 . divisible difference set if and only if DD Žy1. s Ž k y l1 . q Ž l1 y l2 . N q l2 G,
Ž 1.
ALMOST PERFECT SEQUENCES
643
where we identify a subset A of G with the element Ý h g A h of Zw G x and write AŽ t . s h t < h g A4 for any integer t. By applying the characters of G on Ž1., we have the following well-known result; see w6x. THEOREM 2.2. Let G be a cyclic group of order mm9 and N be the subgroup of G of order m9. A subset D of G is an Ž m, m9, k, l1 , l2 . di¨ isible difference set if and only if, for e¨ ery character x of G,
¡k y l
1
x Ž D. x Ž D. s
~k
¢k
2
2
if x is nonprincipal on N,
y mm9l2 if x is principal on N but nonprincipal on G, if x is principal on G.
Ž 2. Let zw denote the complex wth root of unity expŽ2p'y 1 rw .. By Theorem 2.2, we translate the properties of divisible difference sets into equations of algebraic integers in the ring Zw zm m9 x. By studying these equations, we can obtain some nonexistence results of divisible difference sets Žsee w6x for this approach.. In order to do so, we need the following standard result concerning algebraic integers; see w3, 8x. LEMMA 2.3. Let p be a prime and ¨ s p s w with Ž w, p . s 1. Then the s ideal pZw z¨ x decomposes as pZw z¨ x s Ž P1 P2 ??? Pr . f Ž p . where f Ž x . is the Euler f-function and P1 , P2 , . . . , Pr are prime ideals in Zw z¨ x. If t is an integer relati¨ ely prime to p and t ' p j mod w for some integer j, then the ring automorphism z¨ ¬ z¨t fixes e¨ ery Pi . The next lemma is a variation of a result by Ma w5x. LEMMA 2.4. Let ¨ s uw with Ž u, w . s 1, q an integer relati¨ ely prime to ¨ , and H an abelian group of order ¨ which contains an element h of order w. If y g Zw z q xw H x satisfies x Ž y . g f Ž zw .Zw z q ¨ x, for all characters x of H with x Ž h. s zw , where f Ž X . is a polynomial in Zw z q xw X x such that f Ž zw .Zw z q ¨ x and uZw z q ¨ x are relati¨ ely prime, then r
y s f Ž h. x0 q
Ý ² hw r p : x i , i
is1
where x 0 , x 1 , . . . , x r g Zw z q xw H x and p1 , p 2 , . . . , pr are all prime di¨ isors of w. Proof. Let K be the subgroup of H of order u and t : Zw z q xw H x ª Zw z q w xw K x be a ring homomorphism such that t Ž h. s zw and t Žg . s g for all g g K. For any character x 9 of K, we have x 9Žt Ž y .. g f Ž zw .Zw z q ¨ x. By the Fourier inversion formula, we obtain ut Ž y . s f Ž zw . z for some z g Zw z q w xw K x. Let n be the norm of f Ž zw . with respect to the field extension
644
ARASU, MA, AND VOSS
QŽ z q w . over Q. Since Ž n, u. s 1, there exist integers a and b such that an q bu s 1. Then
t Ž y . s ant Ž y . q but Ž y . s f Ž zw . z9 for some z9 g Zw z q w xw K x. Finally, the lemma follows because the kernel of t is Ý ris1² hw r p i : x i < x i g Zw z q xw H x4. 3. MAIN RESULTS We begin by giving the known examples of almost perfect sequences with u s 2: sequences of periods 8, 12, and 28. The sequence of period 8 is trivial. A sequence of period 12 had already been mentioned in w7x and was first given by Clatworthy w2x. EXAMPLE 3.1. 0, 44 is a cyclic Ž4, 2, 2, 2, 0. divisible difference set Z 8 . The corresponding almost perfect sequence of period 8 is given by generating cycle: q] ] ]q] ] ]. EXAMPLE 3.2. 1, 2, 4, 84 is a cyclic Ž6, 2, 4, 2, 1. divisible difference set in Z 12 . The corresponding almost perfect sequence of period 12 is given by the generating cycle: ]qq]q] ] ]q] ] ]. E XAMPLE 3.3. 0, 1, 3, 6, 9, 10, 14, 19, 21, 25, 26, 274 is a cyclic Ž14, 2, 12, 2, 5. divisible difference set in Z 28 . The corresponding almost perfect sequence of period 28 is given by the generating cycle: qq]q] ]q] ]qq] ] ]q] ] ] ]q]q] ] ]qqq. We proceed to nonexistence results. From now on, G is a multiplicative group or order n, where n ' 0 mod 4, and D is a cyclic Ž nr2, 2, Ž n y 4.r2, 2, Ž n y 8.r4. divisible difference set in G. Let N be the subgroup of G of order 2. By Theorem 2.2, for every nonprincipal character x of G,
x Ž D. x Ž D. s
½
Ž n y 8 . r2 4
if x is nonprincipal on N, if x is principal on N.
First, let us study the intersection numbers < D l Ng < for g g G. It is obvious that < D l Ng < s 0, 1, or 2 for each coset Ng in G. In fact, < D l Ng < s 2 for exactly one coset Ng and < D l Ng < s 1 for exactly Ž n y 8.r2 cosets Ng. Translating D is necessary, we may assume N ; D. Thus D s N j S, where S ; Gr 14 , < S < s Ž n y 8.r2, and no two elements of S lie in the same coset of N.
645
ALMOST PERFECT SEQUENCES
LEMMA 3.4.
Let r : G ª GrN be the natural epimorphism. Then either
Ži. Ž GrN . _ Ž r Ž S . j 14. s b, b 2 , b 4 4 , where b g GrN and oŽ b . s 7, or Žii. Ž GrN . _ Ž r Ž S . j 14. s a, c, ac4 , where a, c g GrN, oŽ c . s 2, and c / 1, a. Proof. Let E s Ž GrN . _ Ž r Ž S . j 14.. Note that < E < s 3 and r Ž D . s GrN q 1 y E. Applying r to Ž1., we obtain
ž
G N
q1yE
/ž
G N
q 1 y E Žy1. s 4 q
/
ž
ny8 G 2
/
N
and hence EE Žy1. s 3 q E q E Žy1.. Write E s x, y, z 4 where x, y, z g Ž GrN . _ 14 . Then
Ž xyy1 q yxy1 . q Ž xzy1 q zxy1 . q Ž yzy1 q zyy1 . s Ž x q xy1 . q Ž y q yy1 . q Ž z q zy1 . . There are six possible cases: Ži. zqz ; Žii. y q yy1 ; Žiii. z q zy1 ; Živ. x q xy1 ; Žv. y q yy1 ; Žvi. x q xy1 . y1
xyy1 q yxy1 s x q xy1 , xzy1 q zxy1 s y q yy1 , yzy1 q zyy1 s xyy1 q yxy1 s x q xy1 , xzy1 q zxy1 s z q zy1, yzy1 q zyy1 s xyy1 q yxy1 s y q yy1 , xzy1 q zxy1 s x q xy1 , yzy1 q zyy1 s xyy1 q yxy1 s y q yy1 , xzy1 q zxy1 s z q zy1, yzy1 q zyy1 s xyy1 q yxy1 s z q zy1 , xzy1 q zxy1 s x q xy1, yzy1 q zyy1 s xyy1 q yxy1 s z q zy1 , xzy1 q zxy1 s y q yy1, yzy1 q zyy1 s
Case Ži.. xyy1 q yxy1 s x q xy1 and yzy1 q zyy1 s z q zy1 imply y s x 2 and y s z 2 . Let a g GrN such that oŽ a. s 2. Then we have x, y, z 4 s x, x 2 , ax4 . Substituting into xzy1 q zxy1 s y q yy1 , we obtain 2 a s a q ay1 s x 2 q xy2 . Hence x 2 s a and E s a, c, ac4 where c s x. Case Žii.. xyy1 q yxy1 s x q xy1 and xzy1 q zxy1 s z q zy1 imply y s x 2 and x s z 2 . Substituting into yzy1 q zyy1 s y q yy1 , we obtain z 3 q zy3 s z 4 q zy4 . Hence z 7 s 1 and E s b, b 2 , b 4 4 where b s z. Similarly, we get the same results for all the remaining cases.
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ARASU, MA, AND VOSS
Let n s 2 s w, where s G 2 and w is odd, and let a be an element of G of order 2 s and K the subgroup of G of order w. By Lemma 3.4, we have
where either
D s ²a 2
sy 1
:qXqa2
sy 1
s ²a 2
sy 1
: q Ž1 y a 2
Ž M y X y A.
sy 1
sy 1
. X q a 2 Ž M y A. ,
X ; M _ A, M s K j a K j a 2 K j a 3 K j ??? ja 2
sy 1
y1
K and
Ži. A s 1, h, h 2 , h 4 4 , where h g K and oŽ h. s 7, or sy 2 sy 2 sy 2 Žii. A s 1, a 2 , g, a 2 g 4 , where g g G and g f ² a 2 :. Note that, in Zw G x, M s K Ž 1 q a q a 2 q a 3 q ??? qa 2
sy 1
y1
sy2 i
. s K Ł Ž1 q a 2 . . is0
Thus, for any character x of G which is nonprincipal on ² a 2
¡2 x Ž X . q x Ž A.
~
sy 1
:,
if x is nonprincipal on K , sy2
x Ž D . s 2 x Ž X . q x Ž A. y w Ł 1 q x Ž a 2 . i
¢
THEOREM 3.5.
Ž 3.
is0
if x is principal on K .
If A s 1, h, h 2 , h4 4 , then n s 28.
Proof. First, we show that 8 ¦ n. Assume 8 < n, i.e., s G 3. Let x be any sy 1 character of G which is nonprincipal on ² a 2 : and x Ž h. s z 7 . Then
x Ž A . s 1 q z 7 q z 72 q z 74 s y Ž z 73 q z 75 q z 76 . . Since Ž z 73 q z 75 q z 76 . Ž z 73 q z 75 q z 76 . Zw z 7 x s 2Zw z 7 x, we can write 2Z w zn x s P1 P2 ??? Pr P1 P2 ??? Pr
ž
2 sy 1
/
,
where P1 , P2 , . . . , Pr are prime ideals in Zw zn x, Ž P1 P2 ??? Pr . 2 s Ž z 73 q z 75 q z 76 .Zw zn x,, and Pi / Pj for all i, j. As x Ž D .x Ž D . ' 0 mod 4, we have sy 1
x Ž D . g P1m 1 P2m 2 ??? Prm r P1n1 P2n 2 ??? Prn r , where m i q n i s 2 s for all i. By Ž3., it is obvious that n i s 0 for all i.
647
ALMOST PERFECT SEQUENCES
Hence 2s
x Ž D . g Ž P1 P2 ??? Pr . s Ž z 73 q z 75 q z 76 . Z w zn x . 2
Since this is true for all characters x nonprincipal on ² a 2 x Ž h. s z 7, by Lemma 2.4, we have 2
D s Ž h 3 q h 5 q h6 . x 0 q ² h : x 1 q ² a 2
sy 1
2y 1
: and
: x2 ,
Ž 4.
where x 0 , x 1 , x 2 g Zw G x. Let B s D l ²a 2
sy 1
, h: s 1, a 2
½
sy 1
, Ža2
sy 1
e1
e2
sy 1
sy 1
e3
. h 3 , Ž a 2 . h 5 , Ž a 2 . h6 5 , Ž 5.
where e i s 0 or 1. By Ž4., we have 2
B s Ž h3 q h5 q h6 . xX0 q ² h: xX1 q ² a 2
sy 1
: xX2 ,
Ž 6.
where xX0 , xX1 , xX2 g Zw a 2 , h x. Let x 9 be a character of ² a 2 , h:, with sy 1 x 9Ž a 2 . s y1 and x 9Ž h. s z 7 . By Ž5., x 9Ž B . s "z 73 " z 75 " z 76 s Ž z 73 q z 75 q z 76 .u , where sy 1
¡1
us
sy 1
if x 9 Ž B . s z 73 q z 75 q z 76 ,
yz 7 Ž 1 q z 72 .
if x 9 Ž B . s z 73 q z 75 y z 76 ,
yz 76 Ž 1 q z 73 .
if x 9 Ž B . s z 73 y z 75 q z 76 ,
~yz
4 7
Ž1 q z7 .
if x 9 Ž B . s yz 73 q z 75 q z 76 ,
z 7 Ž 1 q z 72 .
if x 9 Ž B . s yz 73 y z 75 q z 76 ,
z 76 Ž 1 q z 73 .
if x 9 Ž B . s yz 73 q z 75 y z 76 ,
z 74 Ž 1 q z 7 .
if x 9 Ž B . s z 73 y z 75 y z 76 ,
¢y1
if x 9 Ž B . s yz 73 y z 75 y z 76 .
Note that u is a unit in Zw z 7 x. On the other hand, by Ž6., x 9Ž B . s Ž z 73 q z 75 q z 76 . 2x 9Ž xX0 .. So Ž z 73 q z 75 q z 76 . x 9Ž xX0 . s u is a unit, a contradiction. So 8 ¦ n. Suppose n / 28. Let x 1 be a character which is principal on sy 1 ² h: but nonprincipal on both ² a 2 : and K. By Ž3.,
x 1Ž D . s 2 x 1Ž X . q 4 ' 0
mod 2.
But x 1Ž D .x 1 Ž D . s Ž n y 8.r2 ' 2 mod 4, a contradiction. THEOREM 3.6. If n ) 8 and A s 1, a 2 s s 2. and oŽ g . s nr4.
sy 2
, g, a 2
sy 2
g 4 , then 8 ¦ n Ž i.e.,
648
ARASU, MA, AND VOSS sy 1
Proof. Let x be a character of G of order n. For any g g G _ ² a 2 :, if oŽg . s 2 p d for an odd prime p, then Ž1 q x Žg ..Zw zn x is a divisor of pZw zn x and hence is relatively prime to 2Zw zn x; if oŽg . is not equal to 2 p d for any prime p, then 1 q x Žg . is a unit in Zw zn x; and if oŽg . s 2 d where 2 F d F s, then Ž1 q x Žg ..Zw zn x s Ž1 q z 2 d .Zw zn x is a divisor of 2Zw zn x. ŽSee w9x for the details.. Let P1 , P2 , . . . , Pr be all prime ideal divisors of 2Zw zn x. Then 2Zw zn x s sy 1 Ž P1 P2 ??? Pr . 2 and Ž1 q z 2 d .Zw zn x s Ž P1 P2 ??? Pr .2 sy d for 2 F d F s. By Ž3., for each Pi , we have
x Ž D . g Pi2
sy 2
qe
_ Pi2
sy 2
qe q1
,
where e s 0 if oŽ g . is not a power of 2 and e s 2 sy f if oŽ g . s 2 f where 3 F f F s. Thus x Ž D .x Ž D . k 0 mod 4, i.e., 8 ¦ n. Hence A s 1, a , g, a g 4 and g g K. Assume oŽ g . / nr4. Let x 1 be a character of G which is principal on ² g : but nonprincipal on both ² a 2 : and K. By Ž3.,
x 1 Ž D . s 2 x 1 Ž X . q 2 Ž 1 " 'y 1 . k 0
mod 2,
a contradiction. The following is a consequence of Theorems 2.1, 3.5, and 3.6. COROLLARY 3.7. n y 8 is a sum of two squares. The next result can be classified as a multiplier theorem. For any integer t and y s Ý g g H a g g g Rw H x, where H is a group, R is a ring, and a g g R, we define y Ž t . s S g g H a g g t. THEOREM 3.8. Suppose n ) 8 and A s 1, a , g, a g 4 . Let u be a di¨ isor of nr4, U a subgroup of G of order nr4u, and r : G ª GrU the natural epimorphism. If there exists an integer t relati¨ ely prime to < GrU < s 4 u such that for each prime di¨ isor p of Ž n y 8.r4, t ' p j p mod 4 u where j p is an integer, then
Ž r Ž D . y ² r Ž a . 2 :.
Žt.
2 s b Ž r Ž D . y ² r Ž a . :. ,
where either b g ² r Ž a .: and t ' 1 mod u or b g ² r Ž a .:r Ž g . t and t ' y1 mod u. Proof. Let t : Zw GrU x ª Zw'y 1 xw KrU x be a ring homomorphism such that t Ž a . s 'y 1 and t Žg . s g for all g g KrU. Let y 1 s t Ž r Ž D .. and y 2 s t Ž r Ž D .Ž t . .. Let x be any character of KrU. By Ž3., we have x Ž y 1 . s Ž1 q 'y 1 .u where u g Zw z4 u x and uu s Ž n y 8.r4. Since s : z4 u ¬ z4tu fixes every prime ideal divisor of u Žsee Lemma 2.3. and Ž1
649
ALMOST PERFECT SEQUENCES
q 'y 1 . s Zw z4 u x s Ž1 q 'y 1 .Zw z4 u x, we have s
x Ž y 2 . j Ž y1 . s x Ž y1 . x Ž y1 . g
ž
ny8 2
/
Z w z4 u x .
So by Lemma 2.4, y 2 y 1Žy1. s x Ž n y 8.r2 for some x g Zw'y 1 xw KrU x. Since y 1 y 1Žy1. s Ž n y 8.r2, we have xx Žy1. s 1. Let x s Ý g g K r U a g g where a g g Zw'y 1 x. By considering the coefficient of the identity element of the equation xx s 1, we have Ý g g K r U a g a g s 1 and hence only c one nonzero a g . So x s 'y 1 g for some g g KrU. This implies y 2 c s 'y 1 g y 1. Since the kernel of t is ² r Ž a . 2 : z < z g Zw GrU x4 , we have
r Ž D.
Žt.
2 s br Ž D . q ² r Ž a . : z,
where b s r Ž a . cg and z g Z w GrU x. Let h : GrU ª H s Ž GrU .r² r Ž a . 2 : be the natural epimorphism. Then h Ž r Ž D ..Ž t . s b 1h Ž r Ž D .. q 2h Ž z . where b 1 s h Ž b .. Since h Ž r Ž D .. s Ž nr2 u. H q 1 y a 1 y g 1 y a 1 g 1 where a 1 s h Ž r Ž a .. and g 1 s h Ž r Ž g .., we have y Ž 1 q a 1 q g 1t q a 1 g 1t . q b 1 Ž 1 q a 1 q g 1 q a 1 g 1 . s 2 h Ž z . y Ž 1 y b1 . . Since 1, a 1 , g 1 , a 1 g 1 are distinct elements in H,
1, a 1 , g 1t , a 1 g 1t 4 s b1 , b1 a 1 , b1 g 1 , b1 a 1 g 1 4 . Note that h Ž z . s 1 y b 1 and hence
Ž r Ž D . y ² r Ž a . 2 :.
Žt.
2 s b Ž r Ž D . y ² r Ž a . :. .
Assume b 1 g 1, a 14 . Then g 1t , a 1 g 1t 4 s g 1 , a 1 g 1 4 . Since oŽ g 1 . s u is odd, we have g 1 s g 1t and hence t g 1 mod u. Assume b 1 g g 1t , a 1 g 1t 4 . Then 1, a 1 4 s g 1tq1, a 1 g 1tq1 4 . Thus g 1tq1 s 1 and hence t ' y1 mod u. COROLLARY 3.9. Let n / 8, 28 and let u be a di¨ isor of nr4. For e¨ ery prime di¨ isor p of Ž n y 8.r4, define Op s p j mod 4 u < j g Z4 ; Z 4 u . Then
F
½
Op < P is a prime di¨ isor of
ny8 4
5
; "1, u " 1, 2 u " 1, 3u " 1 4 .
In particular, we have the following result. COROLLARY 3.10.
If n / 8, 12, 28, then Ž n y 8.r4 is not a prime power.
Proof. Assume n / 8, 28. Then A s 1, a , g, a g 4 . Let Ž n y 8.r4 s p r where p is an odd prime. By Theorem 3.7 with t s p r and u s nr4, we obtain p r ' "1 mod nr4 which is impossible unless n s 12 and p r s 1.
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ARASU, MA, AND VOSS
We have done a computer search for n such that Ža. Žb. Žc. Žd.
12 - n F 20000; n ' 4 mod 8; n y 8 is a sum of two squares; and n satisfies Corollary 3.9.
Three number are obtained: 348, 4908, and 16572. This means that for n F 20000, there are no almost perfect sequences with u s 2 and period n unless n s 8, 12, 28, 348, 4908, 16572. We do not know whether there are any such sequences of periods 348, 4908, 16572.
REFERENCES 1. R. C. Bose and W. S. Connor, Combinatorial properties of group divisible incomplete block designs, Ann. Math. Statist. 23 Ž1952., 367]383. 2. W. H. Clatworthy, ‘‘Tables of Two Associate Class Partially Balanced Designs,’’ NBS Applied Mathematics Series, vol. 63, 1973. 3. K. Ireland and M. Rosen, ‘‘A Classical Introduction to Modern Number Theory,’’ Springer-Verlag, BerlinrNew York, 1990. 4. D. Jungnickel, On automorphism groups of divisible designs, Canad. J. Math. 34 Ž1982., 257]297. 5. S. L. Ma, Planar functions, relative difference sets and character theory, J. Algebra, 185 Ž1996., 342]356. 6. A. Pott, ‘‘Finite Geometry and Character Theory,’’ Springer-Verlag, BerlinrNew York, 1995. 7. A. Pott and S. P. Bradley, Existence and non-existence of almost perfect autocorrelation sequence, IEEE Trans. Inform. Theory 41 Ž1995., 301]304. 8. E. Weiss, ‘‘Algebraic Number Theory,’’ McGraw]Hill, New York, 1963. 9. J. Wolfmann, Almost perfect autocorrelation sequences, IEEE Trans. Inform. Theory 38 Ž1992., 1412]1418.