Periodic complementary sets of binary sequences

PERIODI C COMPLEMENT AR Y SETS OF BINAR Y SEQUENCES DRA GOMIR ˇ Z. D – OKO VI ´ C Abstract. Let P C S N p denote a set o f p binary sequences of length N such that the sum of their p erio dic auto-co r relation functions is a δ -function. In the 19 90, B¨ omer and Ant w eiler addr e ssed the problem of constructing P C S N p . They pr esented a table covering the range p ≤ 12, N ≤ 5 0 and showing in which cases it w as k nown at t hat time whether P C S N p exist, do not exist, o r the question of existence is undecided. The n umber of undecided ca s es was rather large. Subsequently the num b er of undecided cases was reduced to 26 by the author. In the present note, several cyclic difference fa milies are constructed and used to obtain new s ets of per io dic bina ry sequences. Thereby the or iginal problem of B¨ omer and Ant w eiler is completely solved. 2000 Mathematics Sub ject Classification 05B20, 05B30 1. Intr oduction Let a = a (0) , a (1) , . . . , a ( N − 1) b e a binary sequence of length N . By this w e mean tha t eac h a ( i ) = ± 1. The p eriodic and non-p erio dic auto-correlation functions (P A CF a nd NA CF) o f a are defined b y ˜ ϕ a ( i ) = N − 1 X j =0 a ( j ) a ( i + j mo d N ) , 0 ≤ i < N , and ϕ a ( i ) = N − 1 − i X j =0 a ( j ) a ( i + j ) , 0 ≤ i < N , resp ectiv ely . By conv en t io n, ϕ a ( i ) = 0 for i ≥ N , and ϕ a ( − i ) = ϕ a ( i ) for all i ’s. A family of p binary se quences { a i } , 1 ≤ i ≤ p , all of length N , is a family of p eriodic resp. ap erio dic complemen tary binary sequenc es Key wor ds and phr ases. Cyclic differe nc e family , supplementary difference s ets, per io dic auto correla tion function, genetic algor ithm. Suppo rted in part b y a n NSERC Discov er y Gra nt . 1 2 D. ˇ Z. D – OKO V I ´ C ( P C S N p resp. AC S N p ) if the sum of their P A CF resp. NA CF is a δ - function. As ˜ ϕ a ( i ) = ϕ a ( i ) + ϕ a ( N − i ) for 0 ≤ i < N , w e ha v e AC S N p ⇒ P C S N p . B¨ omer and Ant w eiler [5] addressed the problem of constructing P C S N p . They presen ted a table cov ering the range p ≤ 12 , N ≤ 50 and show- ing in whic h cases it was known at that time whether P C S N p exist, do not exist, or the question of existence is undecided . The n um b er of undecided cases was rather large. In o ur note [9 ] w e hav e reduced the n umber of undecided cases to 2 6 , see also [13]. In this note w e shall construct P C S N p co vering all undecided cases. The construction uses the approach via difference families, also know n as supplem en t ary difference sets (SDS). The connection is recalled in section 3. In our preprin t [12] w e hav e in tro duced a normal form for SDS’s. All SDS’s pres en ted in the remainder of this note a re written in that normal form. All of them w ere constructed b y using our genetic type algorithm. 2. Base s equences and the case p = 4 Base seque nces, origina lly introduced b y T uryn [1 5 ], a re quadruples ( a ; b ; c ; d ) of binary sequenc es, with a and b of length m and c a nd d of length n , and suc h that the sum o f their NA CF’s is a δ -f unction. W e denote b y B S ( m, n ) the set of suc h sequences . According to [7, p. 321] the B S ( n + 1 , n ) exist ( we sa y “ exist” instead of “is non-empt y”) for 0 ≤ n ≤ 35. In our pap er [11] one can find an extensiv e list of B S ( n + 1 , n ) cov ering the ra nge n ≤ 32. There is a map (2.1) B S ( m, n ) → AC S m + n 4 defined by ( a ; b ; c ; d ) → ( a, c ; a, − c ; b, d ; b, − d ), where a, c denotes the concatenation o f the sequences a and c , and − c denotes the negation of the sequence c , i.e., w e ha v e ( − c )( i ) = − c ( i ) for all i ’s. In particular, for m = n = N w e hav e a map AC S N 4 = B S ( N , N ) → AC S 2 N 4 . It follo ws that AC S N 4 exist for N ≤ 72 . Since AC S N p ⇒ P C S N p , w e ha v e Prop osition 2.1. AC S N p and P C S N p exist if p is divisible by 4 and N ≤ 72 . PERIODIC COMPLEMENT AR Y BINA R Y SEQUENCES 3 3. Supplement ar y difference se ts Let Z N = { 0 , 1 , . . . , N − 1 } b e the cyc lic group of order N with addition mo dulo N as the group op eratio n. F or m ∈ Z N and a sub- set X ⊆ Z N let ν ( X, m ) b e the num b er o f o rdered pairs ( i, j ) with i, j ∈ X suc h that i − j ≡ m (mo d N ). W e sa y that the subsets X 1 , . . . , X p ⊆ Z N are supplemen tary difference sets (SDS) with param- eters ( N ; k 1 , . . . , k p ; λ ) if | X i | = k i for all i and p X i =1 ν ( X i , m ) = λ, ∀ m ∈ Z N \ { 0 } . If also p = 1 then X 1 is called a difference set. If { a i } , 1 ≤ i ≤ p , are P C S N p , then the sets (3.1) X i = { j ∈ Z N : a i ( j ) = − 1 } , 1 ≤ i ≤ p, are SDS with parameters ( N ; k 1 , . . . , k p ; λ ), where k i = | X i | for all i ’s. Moreo ver, if N > 1, the follo wing condition holds: (3.2) 4( k 1 + · · · + k p − λ ) = pN . The con v erse is also true: If X 1 , . . . , X p are SDS with parameters ( N ; k 1 , . . . , k p ; λ ) satisfying the ab o v e condition, then the binary se- quences { a i } , 1 ≤ i ≤ p , defined b y (3.1) are P C S N p . These facts are easy to prov e, see e.g. [2, 5]. One can also sho w easily that if ( N ; k 1 , . . . , k p ; λ ) are para meters of an SDS t hen pN = p X i =1 ( N − 2 k i ) 2 . This is useful in selecting the p ossibilities for the parameter se ts of a h yp othetical SDS. 4. The cases p = 1 and p = 2 It follows from (3.2) that if P C S N 1 exists then N is divisible by 4. The sequence + , + , + , − is a P C S 4 1 . No P C S N 1 are kno wn for N > 4. In fact it is kno wn (see [14]) that they do not exist for 4 < N < 10 12 . The AC S N 2 are kno wn as Golay pairs of length N . W e sa y that N is a Golay n umber if AC S N 2 exist. It is kno wn that if N > 1 is a Gola y n umber t hen N is even and not divisible by any prime congruent to 3 (mo d 4). In particular, N is a sum of t w o squares. The Golay num b ers in the range N ≤ 50 are 1,2,4 ,8 ,10,16,20,2 6 ,32 and 40. See [6, 1 0 ] for more details and a dditional references. If P C S N 2 exist a nd N > 1 then (3.2) implies that N m ust b e even . It is also w ell kno wn t ha t N m ust b e a sum of tw o squares, see e.g. 4 D. ˇ Z. D – OKO V I ´ C [2]. Apart from the Gola y num b ers, the in tegers satisfying t hese con- ditions and b elonging to the range N ≤ 50 are 18 ,3 4,36 and 50 . It is kno wn that P C S 18 2 and P C S 36 2 do not exist [2, 16]. Three non- equiv alent examples of P C S 34 2 and a single example of a P C S 50 2 ha v e b een constructed in our pap ers [8, 9 , 12]. In particular the follo wing holds Prop osition 4.1. In the r ange N ≤ 50 , P C S N 1 exist iff N ∈ { 1 , 4 } , and P C S N 2 exist iff N ∈ { 1 , 2 , 4 , 8 , 10 , 16 , 20 , 26 , 32 , 34 , 40 , 5 0 } . 5. The case p = 3 If P C S N 3 exist a nd N > 1 then (3.2) implies t ha t N is divisible b y 4. Explicit examples of P C S N 3 for N = 4 , 8 , 12 and 16 are giv en in [5]. The non-existence of P C S 20 3 w as first established by a computer searc h in [5] and then theoretically in [2]. In our previous note [8] w e ha v e constructed P C S N 3 for N = 24 , 28 and 32. W e shall no w giv e the SDS’s with parameters (36; 15 , 15 , 15; 18) , (40; 19 , 18 , 15; 22) , (44; 20 , 20 , 17; 24) , (48; 24 , 24 , 18; 30) , Since these parameter sets satisfy the condition (3.2 ), the facts men- tioned in section 3 imply tha t P C S N 3 exist fo r N = 36 , 40 , 44 and 48 . PERIODIC COMPLEMENT AR Y BINA R Y SEQUENCES 5 The fo ur SDS’s are: N = 36 : X 1 = { 0 , 1 , 2 , 3 , 4 , 6 , 7 , 11 , 13 , 15 , 18 , 21 , 23 , 27 , 29 } , X 2 = { 0 , 1 , 2 , 6 , 7 , 8 , 10 , 11 , 13 , 14 , 18 , 23 , 26 , 27 , 29 } , X 3 = { 0 , 1 , 3 , 4 , 6 , 7 , 8 , 13 , 14 , 15 , 18 , 21 , 23 , 27 , 32 } ; N = 40 : X 1 = { 0 , 2 , 3 , 4 , 5 , 6 , 7 , 9 , 10 , 12 , 14 , 18 , 1 9 , 20 , 24 , 28 , 31 , 33 , 34 } , X 2 = { 0 , 2 , 3 , 4 , 5 , 8 , 9 , 12 , 14 , 15 , 20 , 21 , 22 , 25 , 27 , 29 , 31 , 35 } , X 3 = { 0 , 1 , 2 , 3 , 7 , 8 , 10 , 11 , 14 , 18 , 19 , 22 , 25 , 27 , 30 } ; N = 44 : X 1 = { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 9 , 11 , 12 , 16 , 17 , 1 9 , 23 , 24 , 25 , 28 , 32 , 35 , 39 } , X 2 = { 0 , 2 , 3 , 4 , 5 , 7 , 8 , 12 , 13 , 14 , 17 , 18 , 19 , 21 , 27 , 29 , 31 , 34 , 37 , 40 } , X 3 = { 0 , 1 , 4 , 5 , 6 , 7 , 9 , 13 , 14 , 16 , 19 , 24 , 25 , 27 , 31 , 33 , 35 } ; N = 48 : X 1 = { 0 , 1 , 2 , 5 , 6 , 7 , 8 , 12 , 13 , 14 , 15 , 18 , 20 , 23 , 25 , 27 , 28 , 29 , 33 , 36 , 37 , 39 , 41 , 44 } , X 2 = { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 10 , 11 , 13 , 16 , 17 , 19 , 20 , 22 , 25 , 26 , 28 , 29 , 30 , 32 , 34 , 36 , 37 } , X 3 = { 0 , 1 , 4 , 5 , 7 , 9 , 10 , 11 , 15 , 18 , 19 , 20 , 22 , 27 , 29 , 35 , 38 , 45 } . Hence, we ha v e Prop osition 5.1. In the r ange N ≤ 50 , P C S N 3 exist iff N ∈ { 1 , 4 , 8 , 12 , 16 , 24 , 28 , 32 , 36 , 40 , 44 , 48 } . 6. The case p = 5 If P C S N 5 exist a nd N > 1 then (3.2) implies t ha t N is divisible b y 4. Clearly P C S N 5 exist if P C S N 2 and P C S N 3 exist. Hence , it remains to consider t he cases N = 12 , 20 , 24 , 28 , 36 , 44 or 48. A P C S 12 5 is giv en explicitly in [5]. In our previous note [8 ] we hav e constructed a P C S N 5 for N = 20 , 24 , 28 and 36. W e shall no w give the SDS’s with para meters (44; 21 , 20 , 19 , 18 , 17; 40) , (48; 23 , 21 , 21 , 20 , 19; 44) . 6 D. ˇ Z. D – OKO V I ´ C Since these parameter sets satisfy the condition (3.2), the existence of P C S N 5 is established for N = 4 4 and 48. The tw o SDS’s a r e: N = 44 : X 1 = { 0 , 1 , 2 , 3 , 4 , 5 , 7 , 8 , 10 , 11 , 14 , 15 , 1 9 , 22 , 24 , 26 , 29 , 31 , 32 , 37 , 38 } , X 2 = { 0 , 2 , 3 , 4 , 5 , 6 , 8 , 10 , 13 , 14 , 18 , 19 , 21 , 24 , 25 , 30 , 31 , 34 , 38 , 40 } , X 3 = { 0 , 2 , 3 , 4 , 6 , 7 , 10 , 13 , 15 , 19 , 20 , 21 , 27 , 28 , 30 , 32 , 35 , 37 , 39 } , X 4 = { 0 , 1 , 3 , 4 , 5 , 7 , 9 , 11 , 12 , 15 , 17 , 20 , 21 , 26 , 30 , 31 , 32 , 33 } , X 5 = { 0 , 1 , 2 , 3 , 5 , 6 , 9 , 10 , 11 , 13 , 14 , 19 , 24 , 25 , 32 , 34 , 37 } ; N = 48 : X 1 = { 0 , 1 , 2 , 3 , 4 , 5 , 8 , 10 , 11 , 13 , 18 , 19 , 20 , 22 , 24 , 25 , 26 , 28 , 30 , 35 , 38 , 40 , 45 } , X 2 = { 0 , 1 , 2 , 3 , 4 , 7 , 9 , 10 , 13 , 17 , 18 , 19 , 21 , 22 , 26 , 27 , 30 , 32 , 33 , 36 , 41 } , X 3 = { 0 , 1 , 3 , 5 , 6 , 7 , 8 , 9 , 11 , 12 , 13 , 15 , 1 9 , 20 , 21 , 30 , 31 , 34 , 35 , 41 , 42 } , X 4 = { 0 , 1 , 2 , 4 , 5 , 10 , 13 , 14 , 16 , 17 , 19 , 21 , 2 6 , 2 7 , 29 , 31 , 34 , 36 , 37 , 40 } , X 5 = { 0 , 2 , 4 , 6 , 8 , 9 , 12 , 13 , 15 , 19 , 20 , 24 , 25 , 26 , 29 , 32 , 36 , 41 , 43 } . Th us w e ha ve Prop osition 6.1. In the r ange N ≤ 50 , P C S N 5 exist iff N is 1 o r a multiple of 4 . 7. The case p = 6 If P C S N 6 exist and N > 1 then (3.2) implies that N is ev en. Clearly P C S N 5 exist if P C S N 2 or P C S N 3 exist. Hence, it remains to consider the cases 6,14,18 ,22,30,38,4 2 and 46. As men tioned in [5], a P C S 6 6 can b e constructed b y using the row s of a 6 b y 6 p erfect binary array . Suc h arra y has b een constructed in [4]. In our prev ious note [8] we ha v e constructed a P C S N 6 for N = 14 , 18 , 22 and 30. W e shall now giv e the SDS’s with parameters (38; 18 , 17 , 16 , 16 , 16 , 14; 40) , (42; 19 , 18 , 18 , 18 , 17 , 17; 44 ) , (46; 21 , 21 , 21 , 21 , 21 , 16; 52) . PERIODIC COMPLEMENT AR Y BINA R Y SEQUENCES 7 Since these parameter sets satisfy the condition (3.2), the existence of P C S N 6 is established for N = 3 8 , 42 and 4 6. The f our SDS’s are: N = 38 : X 1 = { 0 , 1 , 2 , 6 , 7 , 10 , 11 , 12 , 13 , 17 , 18 , 20 , 2 1 , 2 2 , 25 , 27 , 29 , 33 } , X 2 = { 0 , 1 , 3 , 4 , 7 , 8 , 9 , 10 , 13 , 14 , 16 , 19 , 21 , 22 , 26 , 27 , 29 } , X 3 = { 0 , 1 , 2 , 3 , 4 , 7 , 8 , 9 , 11 , 16 , 18 , 19 , 2 1 , 22 , 25 , 31 } , X 4 = { 0 , 1 , 2 , 4 , 6 , 8 , 10 , 12 , 15 , 17 , 18 , 23 , 25 , 26 , 31 , 34 } , X 5 = { 0 , 2 , 3 , 4 , 5 , 6 , 9 , 14 , 16 , 18 , 20 , 23 , 26 , 27 , 30 , 35 } , X 6 = { 0 , 1 , 3 , 4 , 5 , 10 , 11 , 13 , 14 , 15 , 18 , 23 , 2 5 , 2 9 } ; N = 42 : X 1 = { 0 , 1 , 2 , 3 , 5 , 6 , 7 , 9 , 12 , 14 , 17 , 18 , 1 9 , 24 , 26 , 28 , 32 , 34 , 39 } , X 2 = { 0 , 1 , 2 , 3 , 4 , 5 , 7 , 10 , 15 , 17 , 18 , 19 , 21 , 25 , 26 , 27 , 31 , 37 } , X 3 = { 0 , 1 , 2 , 6 , 9 , 10 , 12 , 13 , 14 , 15 , 18 , 20 , 2 1 , 2 3 , 27 , 28 , 30 , 37 } , X 4 = { 0 , 2 , 4 , 5 , 6 , 8 , 9 , 11 , 15 , 16 , 18 , 19 , 22 , 25 , 27 , 29 , 33 , 35 } , X 5 = { 0 , 1 , 2 , 4 , 5 , 9 , 13 , 14 , 17 , 18 , 20 , 23 , 24 , 25 , 30 , 33 , 36 } , X 6 = { 0 , 1 , 2 , 4 , 7 , 8 , 9 , 12 , 13 , 16 , 20 , 21 , 22 , 23 , 26 , 31 , 34 } ; N = 46 : X 1 = { 0 , 1 , 2 , 3 , 5 , 6 , 8 , 9 , 12 , 14 , 15 , 18 , 2 0 , 24 , 26 , 27 , 32 , 34 , 36 , 37 , 41 } , X 2 = { 0 , 1 , 2 , 3 , 5 , 7 , 8 , 9 , 12 , 14 , 18 , 19 , 2 1 , 22 , 25 , 28 , 29 , 30 , 32 , 34 , 35 } , X 3 = { 0 , 2 , 3 , 4 , 7 , 8 , 9 , 10 , 11 , 14 , 15 , 16 , 19 , 20 , 24 , 25 , 28 , 33 , 35 , 36 , 40 } , X 4 = { 0 , 1 , 2 , 3 , 5 , 8 , 10 , 13 , 14 , 16 , 18 , 20 , 22 , 23 , 24 , 27 , 29 , 31 , 32 , 38 , 41 } , X 5 = { 0 , 1 , 2 , 3 , 5 , 6 , 9 , 11 , 13 , 14 , 15 , 17 , 18 , 21 , 24 , 25 , 29 , 36 , 38 , 40 , 41 } , X 6 = { 0 , 1 , 2 , 3 , 4 , 10 , 11 , 15 , 18 , 20 , 21 , 26 , 2 8 , 3 0 , 33 , 34 } . Th us w e ha ve Prop osition 7.1. In the r an g e N ≤ 50 , P C S N 6 exist iff N = 1 or N is even. 8 D. ˇ Z. D – OKO V I ´ C In the case N = 42 w e hav e found another non-equiv alen t SD S with the same parameter set: N = 42 : X 1 = { 0 , 1 , 3 , 4 , 6 , 8 , 9 , 11 , 12 , 14 , 15 , 17 , 19 , 21 , 23 , 28 , 29 , 31 , 38 } , X 2 = { 0 , 1 , 2 , 3 , 4 , 5 , 9 , 12 , 13 , 14 , 15 , 19 , 20 , 24 , 28 , 29 , 33 , 35 } , X 3 = { 0 , 1 , 2 , 4 , 5 , 6 , 7 , 9 , 14 , 15 , 20 , 2 2 , 25 , 26 , 29 , 32 , 34 , 36 } , X 4 = { 0 , 1 , 2 , 4 , 5 , 7 , 9 , 10 , 13 , 17 , 19 , 20 , 25 , 26 , 30 , 31 , 33 , 37 } , X 5 = { 0 , 1 , 2 , 3 , 4 , 7 , 8 , 11 , 14 , 19 , 20 , 24 , 27 , 28 , 30 , 32 , 35 } , X 6 = { 0 , 1 , 2 , 3 , 4 , 7 , 10 , 12 , 13 , 16 , 17 , 19 , 21 , 2 3 , 25 , 36 , 37 } . 8. Conclusion W e reconsider the problem o f constructing p erio dic complemen tar y sequence s P C S N p ( p sequences, each of length N ) in the ra ng e p ≤ 12, N ≤ 50. This problem has b een addressed b y B¨ o mer and An tw eiler in their pa p er [5], where they presen ted a diagram show ing fo r whic h pairs ( p, N ) they w ere able to construct suc h set of sequences. Man y cases w ere left as undecided. The non-existence w as established in a num b er of cases. Subsequen tly w e ha ve reduced t he n um b er of undecided cases to just 26 , see [9]. F or v arious metho ds of constructing P C S N p one should also consult the pap er [13]. In the presen t note w e hav e eliminated all 26 undecided cases by constructing suitable supplemen tary difference sets. T able 1 sho ws how one can construct a P C S N p in the range p ≤ 1 2 and N ≤ 5 0 when one exists. By Prop osition 2.1, if p is divisible b y 4 then P C S N p exist fo r all 1 ≤ N ≤ 50. Therefore w e omit the ro ws with p divisible by 4. When p is not divisible by 4 and N > 1, then the equation (3 .2) implies t ha t P C S N p ma y exist only for N ev en. F or this reason we o mit the o dd N ’s fro m the table. A blank en try in position ( p, N ) means tha t P C S N p do not exist. A bullet en try means that a P C S N p exists and has to b e constructed directly by using a w ell kno wn tec hnique suc h as one for Golay pairs, p erfect binary arra ys, or an SDS, etc. The references for the bullet en tries ( p , N ) are giv en in the main text. The circle en try means that a P C S N p can b e constructed in a t rivial w ay , i.e., by com bining sev eral P C S N q for q < p corresp onding t o bullet en tries or q = 4. PERIODIC COMPLEMENT AR Y BINA R Y SEQUENCES 9 T able 1: Construction of P C S N p for ev en N ≤ 50 2 4 6 8 10 20 30 40 50 1 • 2 • ◦ • • • • • • • • • 3 ◦ • • • • • • • • • • 5 ◦ ◦ • ◦ • • • ◦ • ◦ • • 6 ◦ ◦ • ◦ ◦ ◦ • ◦ • ◦ • ◦ ◦ ◦ • ◦ ◦ ◦ • ◦ • ◦ • ◦ ◦ 7 ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ 9 ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ 10 ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ 11 ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ 10 D. ˇ Z. 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