Hindawi Publishing Corporation
International Journal of Combinatorics
Volume 2010, Article ID 851857, 21 pages
doi:10.1155/2010/851857
Research Article
Classification of Base Sequences BSn 1, n
Dragomir Ž. −
D oković
Department of Pure Mathematics, University of Waterloo, Waterloo, ON, Canada N2L 3G1
Correspondence should be addressed to Dragomir Ž. −D oković, djokovic@uwaterloo.ca
Received 6 February 2010; Revised 8 April 2010; Accepted 12 April 2010
Academic Editor: Masaaki Harada
Copyright q 2010 Dragomir Ž. −D oković. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
Base sequences BSn 1, n are quadruples of {±1}-sequences A; B; C; D, with A and B of length
n 1 and C and D of length n, such that the sum of their nonperiodic autocor-relation functions
is a δ-function. The base sequence conjecture, asserting that BSn 1, n exist for all n, is stronger
than the famous Hadamard matrix conjecture. We introduce a new definition of equivalence for
base sequences BSn 1, n and construct a canonical form. By using this canonical form, we have
enumerated the equivalence classes of BSn 1, n for n ≤ 30. As the number of equivalence classes
grows rapidly but not monotonically with n, the tables in the paper cover only the cases n ≤ 13.
1. Introduction
Base sequences BSm, n are quadruples A; B; C; D of binary sequences, with A and B of
length m and C and D of length n, such that the sum of their nonperiodic autocorrelation
functions is a δ-function. In this paper we take m n 1.
Sporadic examples of base sequences BSn 1, n have been constructed by many
authors during the last 30 years; see, for example, 1–4 and the survey paper 5 and its
references. A more systematic approach has been taken by the author in 6, 7. The BSn1, n
are presently known to exist for all n ≤ 38 ibid and for Golay numbers n 2a 10b 26c ,
where a, b, and c are arbitrary nonnegative integers. However the genuine classification of
BSn 1, n is still lacking. Due to the important role that these sequences play in various
combinatorial constructions such as that for T -sequences, orthogonal designs, and Hadamard
matrices 1, 5, 8, it is of interest to classify the base sequences of small length. Our main goal
is to provide such classification for n ≤ 30.
In Section 2 we recall the basic properties of base sequences of BSn 1, n. We also
recall the quad decomposition and our encoding scheme for this particular type of base
sequences.
2
International Journal of Combinatorics
Table 1: Number of equivalence classes of BSn 1, n.
n
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Equ.
1
1
1
1
3
4
5
17
27
44
98
84
175
475
331
491
Nor.
1
1
1
1
2
1
0
6
14
4
10
3
8
5
0
2
n
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
Equ.
1721
2241
1731
4552
3442
3677
15886
6139
10878
19516
10626
22895
31070
18831
19640
?
Nor.
104
0
2
2
72
0
0
0
0
4
4
0
0
2
0
0
In Section 3 we enlarge the collection of standard elementary transformations of BSn
1, n by introducing a new one. Thus we obtain new notion of equivalence and equivalence
classes. Throughout the paper, the words “equivalence” and “equivalence class” are used in
this new sense. We also introduce the canonical form for base sequences. By using it, we are
able to compute the representatives of the equivalence classes.
In Section 4 we introduce an abstract group, GBS , of order 212 which acts naturally on
all sets BSn 1, n. Its definition depends on the parity of n. The orbits of this group are just
the equivalence classes of BSn 1, n.
In Section 5 we tabulate some of the results of our computations those for n ≤ 13
giving the list of representatives of the equivalence classes of BSn 1, n. The representatives
are written in the encoded form which is explained in the next section. For n ≤ 8 we
also include the values of the nonperiodic autocorrelation functions of the four constituent
sequences. We also raise the question of characterizing the binary sequences having the same
nonperiodic autocorrelation function. A class of examples is constructed, showing that the
question is interesting.
The column “Equ.” in Table 1 gives the number of equivalence classes in BSn 1, n
for n ≤ 30. The column “Nor.” gives the number of normal equivalence classes see Section 5
for their definition.
2. Quad Decomposition and the Encoding Scheme
We denote finite sequences of integers by capital letters. If, say, A is such a sequence of length
n then we denote its elements by the corresponding lower case letters. Thus
A a1 , a2 , . . . , an .
2.1
International Journal of Combinatorics
3
Table 2: Equivalence classes of BSn 1, n, n ≤ 8.
ABCD
1
0
0
1
1
1
2
3
2
3
4
1
2
3
4
5
1
2
3
4
2, −1
n2
NC & ND
1
1
03
3, −2, 1
2, 1
1
3, 0, −1
n3
2, 1
06
4, −1, 0, 1
3, 2, 1
11
4, 1, −2, −1
n4
5, 0, 1, 0, 1
5, 2, −1, −2, −1
5, 0, −1, −2, 1
5, −2, 1, 0, −1
5, 0, −1, −2, 1
3, −2, 1
060
16
082
12
083
4, −1, 0, 1
4, −1, 0, 1
4, 3, 2, 1
4, −1, −2, 1
4, −1, 0, 1
4, −1, 0, 1
016
5, 2, 1, 0, −1
n5
6, 1, 0, 1, 2, 1
5, 2, −1, −2, −1
640
6, −1, 2, −1, 0, −1
5, −2, −1, 2, −1
017
613
064
160
6, 1, −4, −1, 2, 1
6, 3, 2, 1, 0, −1
6, 1, −2, −1, 0, 1
6, −1, 0, 1, −2, −1
5, −2, 1, 0, −1
5, −2, 1, 0, −1
5, 0, 1, 0, 1
5, 0, 1, 0, 1
065
6, −3, 2, −1, 0, 1
5, 0, −1, 2, 1
113
6, 3, 0, −3, −2, −1
n6
5, 0, −1, 2, 1
0612
127
0820
188
0861
162
0872
126
0882
164
7, −2, 3, −2, 1, 0, 1
7, 4, 1, 0, −1, −2, −1
7, −2, 1, 0, 3, −2, 1
7, 0, −1, 2, 1, 0, −1
7, −2, −3, 2, 1, −2, 1
7, 0, 3, 0, −1, 0, −1
7, 2, 1, 0, −1, −2, 1
7, 0, −1, −2, 1, 0, −1
7, 2, 1, 0, −1, −2, 1
7, 0, −1, 2, 1, 0, −1
n7
8, −1, 2, −1, 0, 1, 2, 1
8, 1, 0, 1, −2, −1, 0, −1
8, −1, 2, −1, 0, 1, 2, 1
8, 1, 0, 1, −2, −1, 0, −1
8, 3, −2, −1, 0, 1, 2, 1
8, 1, 0, 1, −2, −1, 0, −1
8, −1, −2, 1, −2, −1, 2, 1
8, 1, 0, −1, 4, 1, 0, −1
6, 1, −4, −1, 2, 1
6, −3, 0, 3, −2, 1
6, 1, 0, −1, −2, 1
6, 1, 0, −1, −2, 1
6, 1, 2, 1, 0, 1
6, 1, −2, −3, 0, 1
6, 1, 0, 1, 2, 1
6, −3, 0, 1, −2, 1
6, 1, −2, −1, 0, 1
6, −3, 2, −1, 0, 1
16
1
N A & NB
n1
2, 1
0165
6123
0165
6141
0166
6122
0173
6161
7, 0, −1, 2, 1, 0, −1
7, 0, −1, −2, 1, 0, −1
7, 0, 3, 0, −1, 0, −1
7, 0, −5, 0, 3, 0, −1
7, 0, −1, 2, 1, 0, −1
7, −4, 3, −2, 1, 0, −1
7, 0, 3, 0, −1, 0, −1
7, 0, −1, 0, −1, 0, −1
4
International Journal of Combinatorics
Table 2: Continued.
5
6
7
8
9
10
11
12
13
14
15
16
17
1
2
3
4
5
6
7
8
9
10
ABCD
0173
6411
0183
6121
0613
1623
0614
1641
0615
1263
0615
1272
0616
1262
0618
1261
0635
1621
0638
1620
0641
1622
0646
1222
0646
1260
06113
1638
06122
1644
06141
1663
06142
1624
06151
1618
06152
1264
06183
1271
06183
1613
06310
1686
06380
1661
N A & NB
8, −1, −2, 1, −2, −1, 2, 1
8, 1, 0, −1, 4, 1, 0, −1
8, −1, −2, 1, −2, −1, 2, 1
8, −3, 0, −1, 0, 1, 0, −1
8, −1, 0, 3, 0, 1, 0, 1
8, 1, −2, 1, 2, −1, −2, −1
8, −1, 4, −1, 0, 1, 0, 1
8, 1, −2, 1, 2, −1, −2, −1
8, −1, 0, 3, 0, 1, 0, 1
8, 1, 2, 1, −2, −1, −2, −1
8, −1, 0, 3, 0, 1, 0, 1
8, 1, 2, 1, −2, −1, −2, −1
8, −1, 4, −1, 0, 1, 0, 1
8, 1, 2, 1, −2, −1, −2, −1
8, −1, 0, −1, −2, 1, 0, 1
8, 1, −2, 1, 0, −1, −2, −1
8, −1, −4, 1, 2, −1, 0, 1
8, −3, 2, −1, 0, 1, −2, −1
8, −1, 0, −3, 0, −1, 0, 1
8, 1, −2, −1, 2, 1, −2, −1
8, 3, 0, 1, 0, −1, 0, 1
8, 1, −2, −1, 2, 1, −2, −1
8, 3, 0, −3, −2, −1, 0, 1
8, −3, 2, −1, 0, 1, −2, −1
8, 3, 0, −3, −2, −1, 0, 1
8, −3, 2, −1, 0, 1, −2, −1
n8
9, 0, 1, 4, −1, 2, 1, 0, 1
9, 2, −1, 2, 1, 0, −1, −2, −1
9, 0, 1, 4, −1, 2, 1, 0, 1
9, −2, 3, −2, 1, 0, −1, −2, −1
9, 0, 5, 0, 1, 0, 1, 0, 1
9, 2, −5, −2, 3, 2, −1, −2, −1
9, 0, 1, 0, −3, 0, 1, 0, 1
9, 2, −1, −2, 3, 2, −1, −2, −1
9, 0, 1, 0, 5, 0, 1, 0, 1
9, 2, −1, 2, −1, −2, −1, −2, −1
9, 0, −3, 0, 1, 0, 1, 0, 1
9, 2, 3, 2, −1, −2, −1, −2, −1
9, 0, 1, −4, −1, −2, 1, 0, 1
9, 2, −1, −2, 1, 0, −1, −2, −1
9, 0, 1, −4, −1, −2, 1, 0, 1
9, 2, −1, −2, 1, 0, −1, −2, −1
9, 0, 1, 2, 1, 4, −1, 0, 1
9, 2, −1, 0, −1, 2, 1, −2, −1
9, −4, 1, −2, 1, 0, −1, 0, 1
9, −2, −1, 0, −1, 2, 1, −2, −1
NC & ND
7, 4, 1, 0, −1, −2, −1
7, −4, 1, 0, −1, 2, −1
7, 4, 3, 2, 1, 0, −1
7, 0, −1, −2, 1, 0, −1
7, −2, 3, −2, 1, 0, 1
7, 2, −1, −2, −3, 0, 1
7, 2, −1, 0, −1, 0, 1
7, −2, −1, 0, −1, 0, 1
7, 2, −3, −2, 1, 2, 1
7, −2, 1, −2, 1, −2, 1
7, 2, −3, −4, −1, 2, 1
7, −2, 1, 0, 3, −2, 1
7, 2, −3, −2, 1, 2, 1
7, −2, −3, 2, 1, −2, 1
7, 2, 1, 2, 1, 2, 1
7, −2, 1, −2, 1, −2, 1
7, 2, 3, 2, 1, 0, 1
7, 2, −1, −2, −3, 0, 1
7, 2, 3, 2, 1, 0, 1
7, −2, −1, 2, −3, 0, 1
7, −2, 3, −2, 1, 0, 1
7, −2, −1, 2, −3, 0, 1
7, 2, 1, 0, 3, 2, 1
7, −2, −3, 4, −1, −2, 1
7, 2, 1, 2, 1, 2, 1
7, −2, −3, 2, 1, −2, 1
8, −1, 0, −3, 0, −1, 0, 1
8, −1, 0, −3, 0, −1, 0, 1
8, 3, 0, −3, −2, −1, 0, 1
8, −1, −4, 1, 2, −1, 0, 1
8, −1, 0, 1, −2, −1, 0, 1
8, −1, 0, 1, −2, −1, 0, 1
8, −1, 4, −1, 0, 1, 0, 1
8, −1, −4, 3, 0, −3, 0, 1
8, −1, 0, −1, −2, 1, 0, 1
8, −1, 0, −1, −2, 1, 0, 1
8, 3, −2, −1, 0, 1, 2, 1
8, −5, 2, −1, 0, 1, −2, 1
8, −1, −2, 5, 0, −1, 2, 1
8, −1, 2, 1, 0, 3, −2, 1
8, −1, 0, 3, 0, 1, 0, 1
8, −1, 0, 3, 0, 1, 0, 1
8, −1, 0 − 1, 0, −3, 0, 1
8, −1, 0 − 1, 0, −3, 0, 1
8, 3, 0, 1, 0, −1, 0, 1
8, 3, 0, 1, 0, −1, 0, 1
International Journal of Combinatorics
5
Table 2: Continued.
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
ABCD
06382
1641
06412
1632
06580
1127
06633
1163
06852
1163
06860
1163
08110
1866
08350
1822
08383
1866
08630
1661
08640
1282
08642
1641
08660
1271
08660
1613
08833
1686
08862
1626
08863
1638
N A & NB
9, 0, 1, −2, −3, 0, −1, 0, 1
9, −2, −1, 0, −1, 2, 1, −2, −1
9, 0, 1, 2, −3, 0, −1, 0, 1
9, 2, −1, 0, −1, 2, 1, −2, −1
9, −4, 1, −2, 1, 0, −1, 0, 1
9, 2, 3, 0, −1, −2, −3, −2, −1
9, 0, −3, 2, −1, −2, −1, 0, 1
9, 2, −1, 0, 1, 0, −3, −2, −1
9, 0, −3, 0, 1, 0, −3, 0, 1
9, 2, −1, 2, −1, −2, −1, −2, −1
9, 0, −3, 0, 1, 0, −3, 0, 1
9, 2, −1, 2, −1, −2, −1, −2, −1
9, 0, 3, 2, 1, 0, 3, −2, 1
9, 2, 1, 0, −1, −2, 1, 0, −1
9, 0, −1, −4, 1, 0, 1, −2, 1
9, −2, −3, 2, −1, −2, 3, 0, −1
9, 0, 3, 0, −1, −2, 1, −2, 1
9, 2, 1, 2, 1, 0, 3, 0, −1
9, −4, −1, 0, 1, 0, 1, −2, 1
9, −2, 1, −2, −1, 2, −1, 0, −1
9, 0, −1, −4, 1, 0, 1, −2, 1
9, −2, 1, −2, −1, 2, −1, 0, −1
9, 0, −1, 0, −3, 0, 1, −2, 1
9, −2, 1, −2, −1, 2, −1, 0, −1
9, 0, −1, −4, 1, 0, 1, −2, 1
9, 2, 1, −2, −1, −2, −1, 0, −1
9, 0, −1, −4, 1, 0, 1, −2, 1
9, 2, 1, −2, −1, −2, −1, 0, −1
9, 0, −1, 2, −1, 2, −1, −2, 1
9, 2, 1, 0, 1, 4, 1, 0, −1
9, 0, −1, 2, −1, 2, −1, −2, 1
9, 2, −3, 0, 1, 0, 1, 0, −1
9, 0, −1, 2, −1, 2, −1, −2, 1
9, 2, 1, 4, 1, 0, 1, 0, −1
NC & ND
8, 3, 0, 1, 0, −1, 0, 1
8, −1, 0, 1, 4, −1, 0, 1
8, −1, 0, 1, 4, −1, 0, 1
8, −1, 0, −3, 0, −1, 0, 1
8, 3, −2, −3, 0, 3, 2, 1
8, −1, −2, 5, 0, −1, 2, 1
8, −1, 2, −1, 0, 1, 2, 1
8, −1, 2, −1, 0, 1, 2, 1
8, −1, 2, −1, 0, 1, 2, 1
8, −1, 2, −1, 0, 1, 2, 1
8, −1, 2, −1, 0, 1, 2, 1
8, −1, 2, −1, 0, 1, 2, 1
8, −1, −2, −1, 0, 1, −2, 1
8, −1, −2, −1, 0, 1, −2, 1
8, −1, 2, 1, 0, 3, −2, 1
8, 3, 2, 1, 0, −1, −2, 1
8, −1, −2, −1, 0, 1, −2, 1
8, −1, −2, −1, 0, 1, −2, 1
8, 3, 0, 1, 0, −1, 0, 1
8, 3, 0, 1, 0, −1, 0, 1
8, −1, −2, 5, 0, −1, 2, 1
8, 3, 2, 1, 0, −1, −2, 1
8, 3, 0, 1, 0, −1, 0, 1
8, −1, 0, 1, 4, −1, 0, 1
8, −1, −2, 5, 0, −1, 2, 1
8, −1, 2, 1, 0, 3, −2, 1
8, −1, 0, 3, 0, 1, 0, 1
8, −1, 0, 3, 0, 1, 0, 1
8, −1, 0, −1, 0, −3, 0, 1
8, −1, 0, −1, 0, −3, 0, 1
8, −1, 4, −1, 0, 1, 0, 1
8, −1, 0, −1, 0, −3, 0, 1
8, −1, 0, −3, 0, −1, 0, 1
8, −1, 0, −3, 0, −1, 0, 1
To this sequence we associate the polynomial
Ax a1 a2 x · · · an xn−1 ,
2.2
viewed as an element of the Laurent polynomial ring Zx, x−1 . As usual, Z denotes the ring
of integers. The nonperiodic autocorrelation function NA of A is defined by
NA i j∈Z
aj aij ,
i ∈ Z,
2.3
6
International Journal of Combinatorics
Table 3: Equivalence classes of BS10, 9.
1
5
9
13
17
21
25
29
33
37
41
AB
01235
01627
01652
01675
01735
01783
06147
06175
06382
06413
06481
CD
66450
64130
64313
61430
61640
61411
16450
12670
16460
16460
16161
2
6
10
14
18
22
26
30
34
38
42
AB
01324
01633
01654
01682
01764
01867
06152
06175
06388
06451
06581
CD
66181
64140
61163
61180
61821
61311
12763
16143
16340
16163
11622
3
7
11
15
19
23
27
31
35
39
43
AB
01618
01642
01655
01684
01765
06124
06164
06187
06412
06458
06583
CD
64150
64560
61180
61122
61281
16282
16133
16131
16273
12612
11631
4
8
12
16
20
24
28
32
36
40
44
AB
01624
01652
01672
01734
01767
06136
06172
06351
06412
06481
06875
CD
64183
61453
61281
64160
61831
16640
12681
16460
16381
12623
11622
where ak 0 for k < 1 and for k > n. Note that NA −i NA i for all i ∈ Z and NA i 0 for
i ≥ n. The norm of A is the Laurent polynomial NA AxAx−1 . We have
NA NA ixi .
i∈Z
2.4
The negation, −A, of A is the sequence
−A −a1 , −a2 , . . . , −an .
2.5
The reversed sequence A and the alternated sequence A∗ of the sequence A are defined by
A an , an−1 , . . . , a1 ,
A∗ a1 , −a2 , a3 , −a4 , . . . , −1n−1 an .
2.6
Observe that N−A NA NA and NA∗ i −1i NA i for all i ∈ Z. By A, B we
denote the concatenation of the sequences A and B.
A binary sequence is a sequence whose terms belong to {±1}. When displaying such
sequences, we will often write for 1 and − for −1. The base sequences consist of four binary
sequences A; B; C; D, with A and B of length m and C and D of length n, such that
NA NB NC ND 2m n.
2.7
NA i NB i NC i ND i 0.
2.8
Thus, for i /
0, we have
We denote by BSm, n the set of such base sequences with m and n fixed. From now
on we will consider only the case m n 1.
International Journal of Combinatorics
7
Table 4: Equivalence classes of BS11, 10.
1
4
7
10
13
16
19
22
25
28
31
34
37
40
43
46
49
52
55
58
61
64
67
70
73
76
79
82
85
88
91
94
97
AB
061173
061450
061463
061563
061740
063140
063513
063810
063840
063873
064130
064170
064413
064510
065843
068560
068580
068641
082661
083510
086231
086310
086421
086473
086640
086840
087110
087130
087323
087372
087683
088651
088771
CD
16456
16267
12826
16134
12684
16862
16441
16616
16262
16217
16465
16344
16271
12638
11276
11263
11643
11634
18642
18226
16248
16613
16427
16217
12631
12682
12863
16461
16282
12864
12637
16424
16264
2
5
8
11
14
17
20
23
26
29
32
35
38
41
44
47
50
53
56
59
62
65
68
71
74
77
80
83
86
89
92
95
98
AB
061253
061450
061463
061582
061870
063413
063550
063821
063842
063881
064141
064313
064480
064510
068110
068571
068611
068752
083110
083521
086243
086333
086432
086483
086643
086860
087110
087131
087343
087383
087732
088673
088771
CD
16246
16443
16271
12631
12286
16822
16414
16445
16242
16342
16423
16282
16213
12676
11863
11632
11634
11276
18863
18642
16277
16616
16242
16344
16134
12671
16273
16262
16282
16382
12684
16434
16424
3
6
9
12
15
18
21
24
27
30
33
36
39
42
45
48
51
54
57
60
63
66
69
72
75
78
81
84
87
90
93
96
AB
061350
061460
061553
061633
061870
063510
063583
063833
063870
064122
064141
064413
064480
064870
068383
068580
068632
068771
083383
083850
086263
086343
086463
086532
086740
086870
087120
087221
087361
087663
088651
088762
CD
16645
16434
12716
12671
16144
16382
16341
16613
16322
16277
16452
12826
16321
12616
11863
11627
11632
11645
18863
18622
16332
16228
16217
16142
12642
12671
16461
16284
16422
16146
16264
16246
Let A; B; C; D ∈ BSn 1, n. For convenience we fix the following notation. For n
even odd we set n 2m n 2m 1. We decompose the pair A; B into quads
ai an2−i
,
bi bn2−i
i 1, 2, . . . ,
n1
,
2
2.9
am1 . Similar decomposition is valid for the pair C; D.
and, if n is even, the central column bm1
Recall the following basic and well-known property 3, Theorem 1.
8
International Journal of Combinatorics
Table 5: Equivalence classes of BS12, 11.
1
4
7
10
13
16
19
22
25
28
31
34
37
40
43
46
49
52
55
58
61
64
67
70
73
76
79
82
AB
011823
013753
016472
016535
016542
016634
016653
016756
017262
017375
017664
018265
018767
061246
061264
061462
061473
061547
061624
061774
063512
063541
063811
063825
063842
063882
064146
064826
CD
661422
663120
645121
612770
614242
614123
612313
611280
641243
641243
614520
612772
613123
166323
162433
164322
164251
127621
126332
126343
162642
162660
164261
162772
164261
163423
162431
126262
AB
012356
016426
016525
016542
016546
016634
016724
016774
017356
017632
017674
018342
018767
061256
061284
061463
061476
061575
061644
063144
063515
063541
063824
063828
063858
063884
064146
064838
2
5
8
11
14
17
20
23
26
29
32
35
38
41
44
47
50
53
56
59
62
65
68
71
74
77
80
83
CD
661422
641272
614232
612463
612440
614231
643121
612363
616123
614620
612462
614620
613230
164341
164231
162262
162431
126232
126341
168281
162441
164420
162660
162273
163320
163421
164320
126451
AB
013682
016445
016525
016542
016572
016643
016727
016817
017374
017646
017674
018382
061186
061264
061462
061472
061476
061618
061764
063412
063515
063551
063824
063828
063877
064143
064615
064842
3
6
9
12
15
18
21
24
27
30
33
36
39
42
45
48
51
54
57
60
63
66
69
72
75
78
81
84
CD
663120
616230
643511
612843
614320
612271
612440
614320
616242
612640
614272
612770
164231
162273
162452
128681
164320
126232
126242
168481
164321
163241
164420
162433
128160
164251
122632
126451
Theorem 2.1. For A; B; C; D ∈ BSn 1, n, the sum of the four quad entries is 2 mod 4 for the
first quad of the pair A; B and is 0 mod 4 for all other quads of A; B and also for all quads of the
pair C; D.
Thus there are 8 possibilities for the first quad of the pair A; B:
1 5 − −
− −
,
2 ,
6 −
− − −
,
3 ,
−
7 − −
− ,
4 ,
− 8 − −
−
,
2.10
.
These eight quads occur in the study of Golay sequences see, e.g., 9, and we refer to them
as the Golay quads.
International Journal of Combinatorics
9
Table 6: Equivalence classes of BS13, 12.
1
4
7
10
13
16
19
22
25
28
31
34
37
40
43
46
49
52
55
58
61
64
67
70
73
76
79
82
85
88
91
94
97
100
103
106
109
112
115
118
121
124
127
130
133
136
AB
0611863
0612573
0612870
0614670
0615230
0615733
0616733
0617220
0617811
0631412
0635152
0635483
0635550
0635872
0638212
0638781
0641370
0641471
0643822
0646430
0648213
0648433
0658463
0663640
0664360
0685871
0686240
0688613
0812661
0817883
0826783
0836121
0838521
0838751
0862270
0862383
0862650
0863353
0864121
0864382
0865261
0865382
0866443
0868761
0872231
0873470
CD
164521
164215
162424
162327
126876
126452
126143
161883
126428
168644
164341
162246
163214
162134
164242
162167
164265
164217
164522
126143
161644
126452
112645
116342
116245
112766
116273
116245
182668
181128
182641
186621
186627
182777
164413
162748
163341
162642
164324
164413
126237
126238
126314
161267
162838
162842
2
5
8
11
14
17
20
23
26
29
32
35
38
41
44
47
50
53
56
59
62
65
68
71
74
77
80
83
86
89
92
95
98
101
104
107
110
113
116
119
122
125
128
131
134
137
AB
0611871
0612760
0614230
0614671
0615230
0616433
0617212
0617623
0617820
0634170
0635340
0635483
0635810
0638112
0638220
0641282
0641470
0641481
0643880
0648112
0648363
0658112
0661363
0663853
0685733
0686130
0686433
0688671
0812883
0826620
0826851
0837383
0838652
0838871
0862312
0862441
0862651
0863353
0864311
0864463
0865310
0865382
0867740
0871130
0872833
0873470
CD
166143
128287
164864
128266
161883
126452
126847
126413
161624
168382
164621
162432
163422
163822
164278
162424
162732
164215
162424
126283
126461
116273
116245
116324
116245
116245
116245
112764
186247
186247
186444
182278
182777
186644
164226
164265
162138
164612
164226
162451
126452
161267
126174
162747
164432
166413
3
6
9
12
15
18
21
24
27
30
33
36
39
42
45
48
51
54
57
60
63
66
69
72
75
78
81
84
87
90
93
96
99
102
105
108
111
114
117
120
123
126
129
132
135
138
AB
0612360
0612760
0614533
0614830
0615643
0616533
0617220
0617651
0618833
0635120
0635440
0635513
0635810
0638121
0638241
0641363
0641471
0643513
0644833
0648212
0648382
0658363
0661453
0663880
0685860
0686230
0687623
0811283
0816621
0826782
0835382
0837383
0838673
0862230
0862322
0862483
0863220
0863482
0864343
0864781
0865343
0866310
0868761
0871332
0873111
0873481
CD
164844
162443
164226
162624
161234
126143
126876
122818
126413
164558
164242
164132
164142
164423
164423
164522
128642
164242
163214
161624
126716
116342
112634
116245
112645
116636
112645
182788
181128
182266
182266
182771
182668
164278
166128
128674
164287
166144
164226
162167
161642
126413
126238
162624
166361
162862
10
International Journal of Combinatorics
Table 6: Continued.
139
142
145
148
151
154
157
160
163
166
169
172
175
AB
0873581
0873681
0876221
0876570
0876581
0876750
0877382
0878663
0881211
0883671
0886471
0886613
0887780
CD
164413
162268
127763
122864
161267
126238
161883
127647
168382
168422
162842
164521
162748
AB
0873583
0873711
0876510
0876570
0876612
0876750
0877871
0878861
0882762
0886261
0886471
0886671
140
143
146
149
152
155
158
161
164
167
170
173
CD
162445
164226
126342
161247
126341
161267
126748
161886
168242
162874
166413
162741
AB
0873670
0873750
0876510
0876581
0876663
0877382
0878631
0878870
0883571
0886280
0886560
886760
141
144
147
150
153
156
159
162
165
168
171
174
CD
162248
164413
126451
126238
126314
126876
127778
161884
168242
162867
164215
164215
There are also 8 possibilities for each of the remaining quads of A; B and all quads of
C; D:
1
,
− 5
,
−
2
− −
6
−
−
,
3
,
− − 7
− −
4
,
− 8
,
−
− −
− −
,
2.11
.
We will refer to these eight quads as the BS-quads. We say that a BS-quad is symmetric if its
two columns are the same, and otherwise we say that it is skew. The quads 1, 2, 7, and 8 are
symmetric and 3, 4, 5, and 6 are skew. We say that two quads have the same symmetry type if
they are both symmetric or both skew.
There are 4 possibilities for the central column:
0
,
1
,
−
2
−
,
3
−
.
−
2.12
We encode the pair A; B by the symbol sequence
p1 p2 · · · pm pm1 ,
2.13
where pi is the label of the ith quad except in the case where n is even and i m 1 in which
case pm1 is the label of the central column.
Similarly, we encode the pair C; D by the symbol sequence
q1 q2 · · · qm respectively q1 q2 · · · qm qm1
2.14
International Journal of Combinatorics
11
Table 7: Equivalence classes of BS14, 13.
1
4
7
10
13
16
19
22
25
28
31
34
37
40
43
46
49
52
55
58
61
64
67
70
73
76
79
82
85
88
91
94
97
100
103
106
109
112
115
118
121
124
127
130
133
136
AB
0116455
0116734
0117663
0118324
0118345
0123672
0131657
0131735
0131842
0132354
0133154
0134174
0134246
0134314
0136764
0161327
0161655
0161762
0162328
0162617
0163174
0163325
0163375
0163562
0164133
0164234
0164327
0164481
0165174
0165327
0165413
0165716
0165826
0166173
0166413
0167125
0167162
0167286
0167356
0167416
0167457
0167583
0168171
0168286
0168465
0171655
CD
6616380
6641822
6618450
6618241
6614212
6614413
6618273
6641851
6618222
6614153
6612743
6641273
6645113
6641851
6634111
6441863
6456330
6418430
6418650
6451282
6168441
6168241
6412612
6451883
6456613
6164450
6412623
6411811
6142582
6142412
6144831
6143581
6123142
6128371
6127832
6142682
6127643
6122162
6183881
6144581
6435150
6112740
6118282
6114531
6112422
6416273
2
5
8
11
14
17
20
23
26
29
32
35
38
41
44
47
50
53
56
59
62
65
68
71
74
77
80
83
86
89
92
95
98
101
104
107
110
113
116
119
122
125
128
131
134
137
AB
0116536
0117653
0118176
0118327
0123628
0123827
0131657
0131743
0132157
0132427
0133414
0134174
0134273
0134416
0137536
0161533
0161742
0162173
0162556
0162766
0163255
0163352
0163417
0163822
0164143
0164314
0164367
0164624
0165243
0165347
0165428
0165816
0166137
0166317
0166423
0167134
0167162
0167345
0167365
0167426
0167465
0167584
0168171
0168286
0168476
0171656
CD
6645183
6618273
6618441
6612712
6645153
6611811
6618422
6644513
6641822
6641810
6618451
6641853
6612412
6641822
6631223
6418643
6418643
6162842
6451340
6413151
6164143
6162760
6414653
6455630
6168242
6162862
6164311
6451133
6141643
6141622
6142152
6114242
6128182
6127481
6141830
6148463
6142763
6144161
6144311
6142280
6141272
6123240
6143841
6123142
6123420
6416851
3
6
9
12
15
18
21
24
27
30
33
36
39
42
45
48
51
54
57
60
63
66
69
72
75
78
81
84
87
90
93
96
99
102
105
108
111
114
117
120
123
126
129
132
135
138
AB
0116546
0117653
0118323
0118327
0123644
0131554
0131676
0131745
0132284
0132463
0133425
0134234
0134274
0136167
0138324
0161633
0161755
0162283
0162582
0163157
0163287
0163372
0163462
0163844
0164156
0164317
0164471
0165173
0165274
0165367
0165523
0165826
0166153
0166324
0166543
0167156
0167238
0167356
0167385
0167455
0167466
0167817
0168276
0168425
0168486
0171831
CD
6645153
6618422
6618222
6614122
6641450
6618272
6612763
6645560
6614580
6618843
6611813
6641513
6612280
6636440
6631411
6414853
6412743
6415680
6451343
6168433
6162131
6414161
6412681
6451220
6456550
6162871
6418541
6142832
6122463
6142312
6143513
6114531
6127433
6124841
6112471
6434350
6128222
6141461
6116270
6431311
6124512
6112763
6114521
6143411
6114530
6416481
12
International Journal of Combinatorics
Table 7: Continued.
139
142
145
148
151
154
157
160
163
166
169
172
175
178
181
184
187
190
193
196
199
202
205
208
211
214
217
220
223
226
229
232
235
238
241
244
247
250
253
256
259
262
265
268
271
274
277
AB
0172647
0173413
0173523
0173612
0173843
0176421
0176515
0176542
0176583
0176744
0176834
0176865
0177683
0178283
0178356
0178377
0178686
0182652
0182662
0182665
0182766
0183262
0183533
0183624
0183734
0183753
0183773
0186553
0186827
0187667
0611455
0611645
0611763
0611853
0612356
0612528
0612548
0612586
0612744
0612753
0612764
0613515
0613753
0614271
0614527
0614577
0614675
CD
6411422
6441863
6414151
6414863
6412141
6146273
6434350
6142280
6145310
6431413
6141241
6121730
6128280
6141231
6124281
6183422
6123123
6124842
6127481
6124712
6141223
6126473
6141650
6142283
6128421
6124411
6141272
6132311
6131223
6131422
1648422
1643851
1624852
1642523
1648241
1638242
1644511
1286420
1642280
1644131
1286620
1663843
1628263
1663481
1622843
1643513
1623412
140
143
146
149
152
155
158
161
164
167
170
173
176
179
182
185
188
191
194
197
200
203
206
209
212
215
218
221
224
227
230
233
236
239
242
245
248
251
254
257
260
263
266
269
272
275
278
AB
0172656
0173474
0173554
0173658
0173843
0176421
0176526
0176554
0176636
0176756
0176843
0176873
0177686
0178325
0178365
0178384
0178737
0182655
0182663
0182668
0182883
0183277
0183553
0183644
0183734
0183757
0183774
0186652
0186847
0611364
0611546
0611654
0611764
0611874
0612457
0612536
0612556
0612645
0612752
0612753
0612844
0613554
0613753
0614274
0614571
0614638
0614681
CD
6415223
6445160
6161242
6412511
6412710
6184643
6431511
6141272
6142172
6124831
6116413
6121742
6145161
6142411
6141440
6121641
6118273
6431311
6122863
6141522
6121763
6182231
6121641
6142281
6142240
6141232
6121642
6131422
6131222
1662871
1626832
1286740
1642733
1638220
1662171
1287762
1286422
1286441
1624282
1661181
1661271
1628243
1664122
1662142
1286372
1286321
1623233
141
144
147
150
153
156
159
162
165
168
171
174
177
180
183
186
189
192
195
198
201
204
207
210
213
216
219
222
225
228
231
234
237
240
243
246
249
252
255
258
261
264
267
270
273
276
279
AB
0173362
0173521
0173612
0173744
0176164
0176424
0176541
0176557
0176641
0176834
0176847
0177663
0178262
0178346
0178365
0178683
0182357
0182657
0182664
0182761
0183241
0183521
0183557
0183657
0183744
0183767
0183776
0186774
0187626
0611453
0611554
0611686
0611853
0612286
0612517
0612547
0612585
0612646
0612752
612758
0612853
0613647
0614236
0614475
0614571
0614642
0614773
CD
6416122
6162781
6164880
6162760
6182672
6146511
6124682
6145313
6142780
6127172
6121632
6431413
6116271
6142420
6431310
6112422
6126411
6141222
6124263
6127773
6184683
6124681
6127130
6122481
6142411
6124233
6124142
6132312
6132281
1648653
1627841
1286320
1626233
1663480
1642753
1624820
1623820
1624730
1624670
1624250
1626142
1628323
1664450
1628213
1286671
1286341
1643450
International Journal of Combinatorics
13
Table 7: Continued.
280
283
286
289
292
295
298
301
304
307
310
313
316
319
322
325
328
331
334
337
340
343
346
349
352
355
358
361
364
367
370
373
376
379
382
385
388
391
394
397
400
403
406
409
412
415
418
AB
0614774
0614884
0615147
0615627
0615783
0615874
0616237
0616472
0616534
0617426
0617556
0617625
0617655
0617786
0617844
0617853
0618516
0618557
0618714
0618825
0618883
0631458
0634135
0635132
0635152
0635441
0636178
0638171
0638221
0638281
0638384
0638457
0638744
0641163
0641278
0641451
0641517
0641867
0643536
0643814
0643843
0644145
0644813
0645387
0645871
0646387
0648176
CD
1643270
1645130
1268452
1612342
1228631
1612341
1264280
1614551
1261732
1268423
1612741
1262433
1613422
1612741
1616450
1614222
1262443
1613450
1612662
1263172
1613451
1682421
1686220
1644270
1624740
1646161
1641321
1661181
1645681
1644141
1661242
1624420
1632281
1286372
1624260
1286372
1632482
1621370
1286881
1638221
1646150
1286240
1634121
1263821
1261623
1261233
1614430
281
284
287
290
293
296
299
302
305
308
311
314
317
320
323
326
329
332
335
338
341
344
347
350
353
356
359
362
365
368
371
374
377
380
383
386
389
392
395
398
401
404
407
410
413
416
419
AB
0614784
0615118
0615471
0615741
0615784
0615883
0616281
0616481
0616841
0617526
0617572
0617626
0617682
0617824
0617845
0617874
0618517
0618616
0618748
0618847
0631147
0631557
0634187
0635137
0635311
0635857
0636882
0638177
0638237
0638285
0638427
0638487
0638747
0641163
0641361
0641451
0641712
0643113
0643545
0643836
0644124
0644381
0644823
0645651
0646123
0648121
0648225
CD
1624250
1268632
1276443
1263482
1271660
1262313
1228632
1262343
1611822
1616431
1614651
1262343
1613640
1264620
1266212
1271662
1262433
1613441
1228631
1612361
1686422
1681411
1682321
1638220
1645860
1621341
1631443
1286420
1642620
1643440
1638221
1626132
1641272
1286671
1624842
1286671
1632783
1648422
1661421
1661222
1624570
1626122
1634141
1226343
1261373
1264760
1263482
282
285
288
291
294
297
300
303
306
309
312
315
318
321
324
327
330
333
336
339
342
345
348
351
354
357
360
363
366
369
372
375
378
381
384
387
390
393
396
399
402
405
408
411
414
417
420
AB
0614873
0615146
0615487
0615743
0615785
0616234
0616358
0616481
0617413
0617556
0617586
0617626
0617786
0617824
0617853
0617884
0618527
0618625
0618824
0618874
0631352
0631778
0635124
0635138
0635381
0636178
0638171
0638188
0638248
0638372
0638428
0638522
0641147
0641264
0641361
0641457
0641754
0643412
0643611
0643842
0644134
0644813
0645172
0645857
0646138
0648175
0648272
CD
1643211
1268462
1268111
1616242
1614350
1264760
1267113
1614232
1618861
1271642
1271660
1614232
1271642
1616231
1612461
1262450
1612441
1613441
1264142
1613241
1686222
1681411
1644340
1626240
1626710
1634410
1644131
1286620
1642470
1624821
1638421
1633822
1638422
1643430
1643833
1622453
1621662
1662713
1624573
1661271
1627611
1621263
1616441
1261243
1261622
1262282
1612662
14
International Journal of Combinatorics
Table 7: Continued.
421
424
427
430
433
436
439
442
445
448
451
454
457
460
463
466
469
472
475
AB
0648275
0648376
0648472
0648587
0655416
0658125
0658147
0658364
0661274
0663875
0685236
0685637
0685827
0686142
0686215
0686357
0686424
0687525
0688763
CD
1614460
1616231
1264141
1226380
1126471
1164341
1162451
1162461
1127641
1163241
1128263
1126370
1162742
1163422
1163322
1163222
1162731
1162631
1164561
422
425
428
431
434
437
440
443
446
449
452
455
458
461
464
467
470
473
AB
0648276
0648412
0648481
0648635
0655416
0658135
0658173
0658463
0663582
0663885
0685424
0685724
0685846
0686154
0686253
0686374
0686451
0687561
CD
1262282
1264523
1264161
1226481
1162182
1166450
1127621
1127762
1164522
1163422
1128623
1162323
1162762
1122671
1126452
1164650
1126341
1126361
423
426
429
432
435
438
441
444
447
450
453
456
459
462
465
468
471
474
AB
0648376
0648426
0648586
0648863
0655417
0658146
0658275
0658487
0663614
0682412
0685536
0685753
0685863
0686213
0686273
0686413
0687515
0687645
CD
1264620
1263323
1226480
1226453
1126371
1162461
1126760
1127631
1163441
1186822
1162740
1126341
1126432
1162471
1126760
1164560
1162451
1126760
when n is even respectively odd. Here qi is the label of the ith quad for i ≤ m, and qm1 is the
label of the central column when n is odd.
3. The Equivalence Relation
We start by defining five types of elementary transformations of base sequences BSn 1, n.
These elementary transformations include the standard ones, as described in 3, 6. But we
also introduce one additional elementary transformation, see item T4, which made its
first appearance in 10 in the context of near-normal sequences. The quad notation was
instrumental in the discovery of this new elementary operation.
The elementary transformations of A; B; C; D ∈ BSn 1, n are the following.
T1 Negate one of the sequences A; B; C; D.
T2 Reverse one of the sequences A; B; C; D.
T3 Interchange the sequences A; B or C; D.
D
which is defined as follows: if 2.14
T4 Replace the pair C; D with the pair C;
D
is τq1 τq2 · · · τqm qm1
is the encoding of C; D, then the encoding of C;
or τq1 τq2 · · · τqm depending on whether n is even or odd, where τ is the
D
is obtained from that
transposition 45. In other words, the encoding of C;
of C; D by replacing each quad symbol 4 with the symbol 5, and vice versa. We
verify below that NC ND NC ND .
T5 Alternate all four sequences A; B; C; D.
In order to justify T4 one has to verify that NC ND NC ND . For that purpose
let us fix two quads qk and qki and consider their contribution δi to NC i ND i. We claim
International Journal of Combinatorics
15
that δi is equal to the contribution δi of τqk and τqki to NC i ND i. If neither qk nor
qki belongs to {4, 5}, then τqk qk and τqki qki and so δi δi . If q ∈ {4, 5}, then τq
is the negation of q. Hence if {qk , qki } ⊆ {4, 5}, then again δi δi . Otherwise, say qk ∈ {4, 5}
while qki /
∈ {4, 5}, and it is easy to verify that δi 0 δi . The pairs qk and qn1−k−i also make
a contribution to NC i ND i, but they can be treated in the same manner. Finally, if n is
odd then the pair C; D also has a central column with label qm1 . In that case, if k m 1 − i
and qk ∈ {4, 5}, the contribution of qk and qm1 to NC i ND i is 0. This completes the
verification.
We say that two members of BSn 1, n are equivalent if one can be transformed to
the other by applying a finite sequence of elementary transformations. One can enumerate
the equivalence classes by finding suitable representatives of the classes. For that purpose we
introduce the canonical form.
Definition 3.1. Let S A; B; C; D ∈ BSn 1, n and let 2.13 respectively 2.14 be the
encoding of the pair A; B respectively C; D. One says that S is in the canonical form if the
following eleven conditions hold.
i p1 3 , p2 ∈ {6, 8} for n even and p2 ∈ {1, 6} for n odd.
ii The first symmetric quad if any of A; B is 1 or 8.
iii If n is even and pi ∈ {3, 4, 5, 6} for 2 ≤ i ≤ m, then pm1 ∈ {0, 3}.
iv The first skew quad if any of A; B is 3 or 6.
v q1 1 for n even and q1 ∈ {1, 6} for n odd.
vi The first symmetric quad if any of C; D is 1.
vii The first skew quad if any of C; D is 6.
viii If i is the least index such that qi ∈ {2, 7}, then qi 2.
ix If i is the least index such that qi ∈ {4, 5}, then qi 4.
x If n is odd and qi ∈ {1, 3, 6, 8}, for all i ≤ m, then qm1 /
2.
xi If n is odd and qi ∈ {3, 4, 5, 6}, for all i ≤ m, then qm1 0.
We can now prove that each equivalence class has a member which is in the canonical
form. The uniqueness of this member will be proved in the next section.
Proposition 3.2. Each equivalence class E ⊆ BSn 1, n has at least one member having the
canonical form.
Proof. Let S A; B; C; D ∈ E be arbitrary and let 2.13 respectively 2.14 be the encoding
of A; B respectively C; D. By applying the first three types of elementary transformations
and by Theorem 2.1 we can assume that p1 3 and c1 d1 1. By Theorem 2.1, q1 ∈ {1, 6}.
If n is even and q1 6 we apply the elementary transformation T5. Thus we may assume
that p1 3 , and that condition v for the canonical form is satisfied. To satisfy conditions
ii and iii, replace B with −B if necessary. To satisfy condition iv, replace A with A if
necessary.
We now modify S in order to satisfy the second part of condition i. Note that p2 is a
BS-quad by Theorem 2.1. If p2 6 there is nothing to do.
Assume that the quad p2 is symmetric. By ii, p2 ∈ {1, 8}. From 2.8 for i n − 1, we
deduce that p2 8 if q1 1 and p2 1 if q1 6. Note that if n is even, then q1 1 by v. If
16
International Journal of Combinatorics
n is odd and q1 1, we switch A and B and apply the elementary transformation T5. After
this change we still have p1 3 , q1 1, conditions ii, iii, and iv remain satisfied, and
moreover p2 6.
Now assume that p2 is skew. In view of iv, we may assume that p2 3. Then the
argument above based on 2.8 shows that q1 6, and so n must be odd. After applying the
elementary transformation T5, we obtain that p2 1. Hence condition i is fully satisfied.
To satisfy vi, in view of v we may assume that n is odd and q1 6. If the first
symmetric quad in C; D is 2 respectively 7, we reverse and negate C respectively D. If it is
8, we reverse and negate both C and D. Now the first symmetric quad will be 1.
To satisfy vii, if necessary reverse C, D, or both. To satisfy viii, if necessary
interchange C and D. Note that in this process we do not violate the previously established
properties. To satisfy ix, if necessary apply the elementary transformation T4. To satisfy
x, switch C and D if necessary. To satisfy xi, if necessary replace C with −C , D with
−D , or both.
Hence S is now in the canonical form.
4. The Symmetry Group of BSn 1, n
We will construct a group GBS of order 212 which acts on BSn 1, n. Our redundant
generating set for GBS will consist of 12 involutions. Each of these generators is an elementary
transformation, and we use this information to construct GBS , that is, to impose the defining
relations. We denote by S A; B; C; D an aritrary member of BSn 1, n.
To construct GBS , we start with an elementary abelian group E of order 28 with
generators νi , ρi , i ∈ {1, 2, 3, 4}. It acts on BSn 1, n as follows:
ν1 S −A; B; C; D,
ν2 S A; −B; C; D,
ν3 S A; B; −C; D,
ν4 S A; B; C; −D,
ρ1 S A ; B; C; D ,
ρ2 S A; B ; C; D ,
ρ3 S A; B; C ; D ,
ρ4 S A; B; C; D ,
4.1
that is, νi negates the ith sequence of S and ρi reverses it.
Next we introduce two commuting involutory generators σ1 and σ2 . We declare that
σ1 commutes with ν3 , ν4 , ρ3 , and ρ4 , and σ2 commutes with ν1 , ν2 , ρ1 , and ρ2 and that
σ1 ν1 ν2 σ1 ,
σ1 ρ1 ρ2 σ1 ,
σ2 ν3 ν4 σ2 ,
σ2 ρ3 ρ4 σ2 .
4.2
The group H E, σ1 , σ2 is the direct product of two isomorphic groups of order 32:
H1 ν1 , ρ1 , σ1 ,
H2 ν3 , ρ3 , σ2 .
4.3
The action of E on BSn 1, n extends to H by defining
σ1 S B; A; C; D,
σ2 S A; B; D; C.
4.4
International Journal of Combinatorics
17
We add a new generator θ which commutes elementwise with H1 , commutes with
It has the
ν3 ρ3 , ν4 ρ4 , and σ2 , and satisfies θρ3 ρ4 θ. Let us denote this enlarged group by H.
direct product decomposition
2 ,
H, θ H1 × H
H
4.5
where the second factor is itself direct product of two copies of the dihedral group D8 of order
8:
2 ρ3 , ρ4 , θ × ν3 ρ3 , ν4 ρ4 , θσ2 .
H
4.6
by letting θ act as the elementary transformation
The action of H on BSn 1, n extends to H
T4.
and the group of order 2 with
Finally, we define GBS as the semidirect product of H
generator α. By definition, α commutes with each νi and satisfies
αρi α ρi νin ,
αρj α ρj νjn−1 ,
i 1, 2;
j 3, 4;
4.7
αθα θσ2n−1 .
on BSn1, n extends to GBS by letting α act as the elementary transformation
The action of H
T5, that is, we have
αS A∗ ; B∗ ; C∗ ; D∗ .
4.8
is independent of n, and its action
We point out that the definition of the subgroup H
on BSn 1, n has a quadwise character. By this we mean that the value of a particular quad,
determine uniquely the quad pi of hS. In other words H
say pi , of S ∈ BSn 1, n and h ∈ H
acts on the Golay quads, the BS-quads, and the set of central columns such that the encoding
of hS is given by the symbol sequences
h p1 h p2 · · · h pm1 ,
h q1 h q2 · · · .
4.9
On the other hand the full group GBS has neither of these two properties.
on the BS-quads is that it preserves the
An important feature of the action of H
symmetry type of the quads.
The following proposition follows immediately from the construction of GBS and the
description of its action on BSn 1, n.
Proposition 4.1. The orbits of GBS in BSn 1, n are the same as the equivalence classes.
The main tool that we use to enumerate the equivalence classes of BSn 1, n is the
following theorem.
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International Journal of Combinatorics
Theorem 4.2. For each equivalence class E ⊆ BSn 1, n there is a unique S A; B; C; D ∈ E
having the canonical form.
Proof. In view of Proposition 3.2, we just have to prove the uniqueness assertion. Let
Sk Ak ; Bk ; Ck ; Dk ∈ E,
k 1, 2
4.10
be in the canonical form. We have to prove that in fact S1 S2 .
By Proposition 4.1, we have gS1 S2 for some g ∈ GBS . We can write g as g αs h
Let pk pk · · · pk be the encoding of the pair Ak ; Bk where s ∈ {0, 1} and h ∈ H.
2
1
m1
k k
and q1 q2 · · · the encoding of the pair Ck ; Dk . The symbols i-xi will refer to the
1
2
corresponding conditions of Definition 3.1. Observe that p1 p1 3 by i.
We prove first preliminary claims a-c.
1
2
a q1 q1 .
For n even see v. Let n be odd. When we apply the generator α to any S ∈ BSn1, n,
1
2
we do not change the first quad of C; D. It follows that the quads q1 and q1 have the same
symmetry type. The claim now follows from v.
that is, s 0.
b g ∈ H,
1
2
Assume first that n is even. By v, q1 q1 1. For any S ∈ E the first quad of C; D
in S and the one in αS have different symmetry types. As quad h1 is symmetric, the equality
αs hS1 S2 forces s to be 0. Assume now that n is odd. Then for any S ∈ E the second quad
1
2
of A; B in S and the one in αS have different symmetry types. Recall that q1 q1 ∈ {1, 6}
1
2
and that see i p2 and p2 belong to {1, 6}. From 2.8, with i n − 1, we deduce that
k
k
1
2
p2 /
q1 for k 1, 2. We conclude that p2 p2 . The claim now follows from the fact that h
preserves while α alters the symmetry type of the quad p2 .
1
As an immediate consequence of b, we point out that a quad pi is symmetric if and
2
1
2
only if pi is, and the same is true for the quads qi and qi .
1
2
c p2 p2 .
This was already proved above in the case when n is odd. In general, the claim follows
1
2
from b and the equality hp2 p2 . Observe that each of the sets {6, 8} and {1, 6} consists
of one symmetric and one skew quad and that h preserves the symmetry type of quads.
2 . Since s 0, we have g h h1 h2 with h1 ∈ H1 and h2 ∈ H
2 .
H1 × H
Recall that H
1
2
1
2
Consequently, h1 pi pi and h2 qi qi for all i’s.
1
2
We will now prove that A1 A2 and B1 B2 . Since p1 p1 3 , the equality
1
2
h1 p1 p1 implies that h1 3 3 . Thus h1 ρ1e ν2 ρ2 f for some e, f ∈ {0, 1}.
1
1
1
2
1
Assume first that p2 is symmetric. By ii, p2 ∈ {1, 8}. Then h1 p2 p2 p2
1
implies that f 0. Hence, h1 ρ1e and so B1 B2 . If all quads pi , i / 1, are symmetric,
1
2
1
1
then also A h1 A A . Otherwise let i be the least index for which the quad pi is
1
skew. Since B1 B2 and pi is 3 or 6 see iv, we infer that e 0. Hence h1 1 and so
A1 A2 .
International Journal of Combinatorics
1
19
1
1
2
1
Now assume that p2 is skew. By ii, p2 6. Then h1 p2 p2 p2 implies that
1
1, are skew, then by invoking
e 0. Thus h1 ν2 ρ2 f and so A1 A2 . If all quads pi , i /
condition iii we deduce that f 0 and so B1 B2 . Otherwise let i be the least index for
1
1
which the quad pi is symmetric. Since A1 A2 and pi is 1 or 8 see ii, we infer that
f 0. Hence h1 1 and so B1 B2 .
1
It remains to prove that C1 C2 and D1 D2 . We set Q {qi : 1 ≤ i ≤ m}. By
1
2
v and claim a we have q1 q1 ∈ {1, 6}.
1
2
We first consider the case q1 q1 6 which occurs only for n odd. Then h2 6 6
and so h2 ∈ ν3 ρ3 , ν4 ρ4 , θ, σ2 . It follows that h2 3 3.
1
If some q ∈ Q is symmetric, let i be the least index such that qi is symmetric. Then
1
2
vi implies that qi qi 1. Thus h2 must fix the quad 1. As the stabilizer of the quad 1
in ν3 ρ3 , ν4 ρ4 , θ, σ2 is θ, σ2 , we infer that h2 must also fix the quad 8. Similarly, if 2 ∈ Q,
then viii implies that h2 fixes 2 and 7. If 4 ∈ Q, then ix implies that h2 fixes 4 and 5. These
1
2
facts imply that h2 fixes all quads in Q, that is, qi qi for all i ≤ m. It remains to show
1
2
that, for odd n, qm1 qm1 . If Q ⊆ {3, 4, 5, 6}, this follows from xi. Otherwise Q contains a
⊆{1, 3, 6, 8} then Q contains one of the quads 2, 4, 5,
symmetric quad and so h2 ∈ θ, σ2 . If Q/
1
2
or 7. Since h2 fixes all quads in Q, we infer that h2 ∈ θ, and so qm1 qm1 . If Q ⊆ {1, 3, 6, 8},
1
2
the equality qm1 qm1 follows from x.
1
2
1
2
Finally, we consider the case q1 q1 1. Since h2 q1 q1 , h2 ∈ ρ3 , ρ4 , θ, σ2 .
Hence h2 fixes the quads 1 and 8.
If some q ∈ Q is skew, then vii implies that h2 fixes the quads 3 and 6. If 2 ∈ Q, then
viii implies that h2 fixes the quads 2 and 7. If 4 ∈ Q, then ix implies that h2 fixes the quads
4 and 5. These facts imply that h2 fixes all quads in Q. If n is odd, then we invoke conditions
x and xi to conclude that h2 also fixes the central column of C1 ; D1 . Hence C1 C2
and D1 D2 also in this case.
5. Representatives of the Equivalence Classes
We have computed a set of representatives for the equivalence classes of base sequences
BSn 1, n for all n ≤ 30. Due to their excessive size, we tabulate these sets only for n ≤ 13.
Each representative is given in the canonical form which is made compact by using our
standard encoding. The encoding is explained in detail in Section 2.
As an example, the base sequences
A , , , , −, −, , −, ,
B , , , −, , , , −, −,
C , , −, −, , −, −, ,
5.1
D , , , , −, , −, ,
are encoded as 3 6142, 1675. In the tables we write 0 instead of 3 . This convention was used
in our previous papers on this and related topics.
20
International Journal of Combinatorics
This compact notation is used primarily in order to save space, but also to avoid
introducing errors during decoding. For each n, the representatives are listed in the
lexicographic order of the symbol sequences 2.13 and 2.14.
In Table 2 we list the codes for the representatives of the equivalence classes of BSn 1, n for n ≤ 8. This table also records the values NX k of the nonperiodic autocorrelation
functions for X ∈ {A, B, C, D} and k ≥ 0. For instance let us consider the first item in the list of
base sequences BS8, 7 given in Table 2. The base sequences are encoded in the first column
as 0165, 6123. The first part 0165 encodes the pair A; B, and the second part 6123 encodes
the pair C; D. The function NA , at the points 0, 1, . . . , 7, takes the values 8, −1, 2, −1, 0, 1, 2,
and 1 listed in the second column. Just below these values one finds the values of NB at
the same points. In the third column we list likewise the values of NC and ND at the points
i 0, 1, . . . , 6.
Tables 3, 4, 5, 6, and 7 contain only the list of codes for the representatives of the
equivalence classes of BSn 1, n for 9 ≤ n ≤ 13.
Let us say that the base sequences S A; B; C; D ∈ BSn1, n are normal respectively
near-normal if bi ai respectively bi −1i−1 ai for all i ≤ n. We denote by NSn respectively
NNn the set of all normal respectively near-normal sequences in BSn 1, n. Let us say
also that an equivalence class E ⊆ BSn 1, n is normal respectively near-normal if E ∩ NSn
respectively E ∩ NNn is not void. Our canonical form has been designed so that if E is
normal, then its canonical representative S belongs to NSn. The analogous statement for
near-normal classes is false. It is not hard to recognize which representatives S in our tables
are normal sequences. Let 2.13 be the encoding of the pair A; B. Then S ∈ NSn if and
only if all the quads pi , i / 1, belong to {1, 3, 6, 8}, and, in the case when n 2m is even, the
central column symbol pm1 is 0 or 3.
It is an interesting question to find the necessary and sufficient conditions for two
binary sequences to have the same norm. The group of order four generated by the negation
and reversal operations acts on binary sequences. We say that two binary sequences are
equivalent if they belong to the same orbit. Note that the equivalent binary sequences have
the same norm. However the converse is false. Here is a counter-example which occurs in
Table 2 for the case n 8. The base sequences 15 and 16 differ only in their first sequences,
which we denote here by U and V, respectively:
U − − − − − ,
V − − − − .
5.2
Their associated polynomials are
Ux 1 x − x2 − x3 − x4 x5 − x6 − x7 x8 ,
V x 1 x − x2 x3 x4 − x5 − x6 − x7 x8 .
5.3
It is obvious that U and V are not equivalent in the above sense. On the other hand, from the
factorizations Ux pxqx and V x pxrx, where px 1x−x2 , qx 1−x3 −x6 ,
and rx 1 x3 − x6 −x6 qx−1 , we deduce immediately that NU x NV x.
This counter-example can be easily generalized. Let us define binary polynomials as
polynomials associated to binary sequences. If fx is a polynomial of degree d with f0 /
0,
we define its dual polynomial f ∗ by f ∗ x xd fx−1 . Then for any positive integer k we
International Journal of Combinatorics
21
have f ∗ xk fxk ∗ , that is, f ∗ xk g ∗ x where g ∗ is the dual of the polynomial fxk .
In general we can start with any number of binary sequences, but here we take only three
of them: A; B; C of lengths m, n, k, respectively. From the associated binary polynomials
Ax, Bx, and Cx we can form several binary polynomials of degree mnk − 1. The basic
one is AxBxm Cxmn . The others are obtained from this one by replacing one or more of
the three factors by their duals. It is immediate that the binary sequences corresponding to
these binary polynomials all have the same norm. In general many of these sequences will
not be equivalent. However, note that if we replace all three factors with their duals, we will
obtain a binary sequence equivalent to the basic one.
Acknowledgments
The author has pleasure to thank the three anonymous referees for their valuable comments.
The author is grateful to NSERC for the continuing support of his research. This work was
made possible by the facilities of the Shared Hierarchical Academic Research Computing
Network SHARCNET: http://www.sharcnet.ca and Compute/Calcul Canada.
References
1 H. Kharaghani and C. Koukouvinos, “Complementary, base and Turyn sequences,” in Handbook of
Combinatorial Designs, C. J. Colbourn and J. H. Dinitz, Eds., pp. 317–321, CRC Press, Boca Raton, Fla,
USA, 2nd edition, 2007.
2 S. Kounias and K. Sotirakoglou, “Construction of orthogonal sequences,” in Proceedings of the 14th
Greek Statistics Conference, pp. 229–236, Skiathos, Greece, 2001.
3 C. Koukouvinos, S. Kounias, and K. Sotirakoglou, “On base and Turyn sequences,” Mathematics of
Computation, vol. 55, no. 192, pp. 825–837, 1990.
4 C. H. Yang, “On composition of four-symbol δ-codes and Hadamard matrices,” Proceedings of the
American Mathematical Society, vol. 107, no. 3, pp. 763–776, 1989.
5 J. Seberry and M. Yamada, “Hadamard matrices, sequences, and block designs,” in Contemporary
Design Theory, Wiley-Intersci. Ser. Discrete Math. Optim., pp. 431–560, Wiley, New York, NY, USA,
1992.
6 D. Ž. −D oković, “Aperiodic complementary quadruples of binary sequences,” Journal of Combinatorial
Mathematics and Combinatorial Computing, vol. 27, pp. 3–31, 1998, Correction: ibid 30, p. 254, 1999.
7 D. Ž. −D oković, “On the base sequence conjecture,” Discrete Mathematics, vol. 310, pp. 1956–1964, 2010.
8 C. Koukouvinos, S. Kounias, J. Seberry, C. H. Yang, and J. Yang, “Multiplication of sequences with
zero autocorrelation,” The Australasian Journal of Combinatorics, vol. 10, pp. 5–15, 1994.
9 D. Ž. −D oković, “Equivalence classes and representatives of Golay sequences,” Discrete Mathematics,
vol. 189, no. 1–3, pp. 79–93, 1998.
10 D. Ž. −D oković, “Classification of near-normal sequences,” Discrete Mathematics, Algorithms and
Applications, vol. 1, no. 3, pp. 389–399, 2009.