Classification of normal sequences

Base sequences BS(m,n) are quadruples (A;B;C;D) of {+1,-1}-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 delta-function. Normal sequences NS(n) are base sequences (A;B;C;D) in BS(n,n) such that A=B. We introduce a definition of equivalence for normal sequences NS(n), and construct a canonical form. By using this canonical form, we have enumerated the equivalence classes of NS(n) for n<= 40.


Introduction
By a binary respectively ternary sequence we mean a sequence A = a 1 , a 2 , . . . , a m whose terms belong to {±1} respectively {0, ±1}. To such a sequence we associate the polynomial A(z) = a 1 + a 2 z + · · · + a m z m−1 . We refer to the Laurent polynomial N(A) = A(z)A(z −1 ) as the norm of A. Base sequences (A; B; C; D) are quadruples of binary sequences, with A and B of length m and C and D of length n, and such that (1.1) N(A) + N(B) + N(C) + N(D) = 2(m + n).
The set of such sequences will be denoted by BS(m, n).
In this paper we consider only the case where m = n or m = n + 1. The base sequences (A; B; C; D) ∈ BS(n, n) are normal if A = B. We denote by NS(n) the set of normal sequences of length n, i.e., those contained in BS(n, n). It is well known [12] that for normal sequences 2n must be a sum of three squares. In particular, NS(14) and NS(30) are empty. Exhaustive computer searches have shown that NS(n) are empty also for n = 6, 17, 21, 22, 23, 24 (see [10]) and n = 27, 28, 31, 33, 34, . . . , 39 (see [2,4]).
The base sequences (A; B; C; D) ∈ BS(n + 1, n) are near-normal if b i = (−1) i−1 a i for all i ≤ n. For near-normal sequences n must be D.Ž. D -OKOVIĆ even or 1. We denote by NN(n) the set of near-normal sequences in BS(n + 1, n). Normal sequences were introduced by C.H. Yang in [12] as a generalization of Golay sequences. Let us recall that Golay sequences (A; B) are pairs of binary sequences of the same length, n, and such that N(A) + N(B) = 2n. We denote by GS(n) the set of Golay sequences of length n. It is known that they exist when n = 2 a 10 b 26 c where a, b, c are arbitrary nonnegative integers. There exist two embeddings GS(n) → NS(n): the first defined by (A; B) → (A; A; B; B) and the second by (A; B) → (B; B; A; A). We say that these normal sequences (and those equivalent to them) are of Golay type. For the definition of equivalence of normal sequences see section 3. However, as observed by Yang, there exists normal sequences which are not of Golay type. We refer to them as sporadic normal sequences. From the computational results reported in this paper (see Table 1 below) it appears that there may be only finitely many sporadic normal sequences. E.g. all 304 equivalence classes in NS(40) are of Golay type. The smallest length for which the existence question of normal sequences is still unresolved is n = 41.
Base sequences, and their special cases such as normal and nearnormal sequences, play an important role in the construction of Hadamard matrices [8,11]. For instance, the discovery of a Hadamard matrix of order 428 (see [9]) used a BS(71, 36), constructed specially for that purpose.
Examples of normal sequences NS(n) have been constructed in [3,7,8,10,12]. For various applications, it is of interest to classify the normal sequences of small length. Our main goal is to provide such classification for n ≤ 40. The classification of near-normal sequences NN(n) for n ≤ 40 and base sequences BS(n + 1, n) for n ≤ 30 has been carried out in our papers [3,4,6] and [7], respectively.
We give examples of normal sequences of lengths n = 1, . . . , 5: When displaying a binary sequence, we often write + for +1 and − for −1. We have written the sequence A twice to make the quads visible (see the next section). If (A; A; C; D) ∈ NS(n) then (A, +; A, −; C; D) ∈ BS(n + 1, n). This has been used in our previous papers to view normal sequences NS(n) as a subset of BS(n+1, n). For classification purposes it is more convenient to use the definition of NS(n) as a subset of BS(n, n), which is closer to Yang's original definition [12].
In section 2 we recall the basic properties of base sequences BS(m, n). The quad decomposition and our encoding scheme for BS(n+1, n) used in our previous papers also works for NS(n), but not for arbitrary base sequences in BS(n, n). The quad decomposition of normal sequences NS(n) is somewhat simpler than that of base sequences BS(n + 1, n). We warn the reader that the encodings for the first two sequences of (A; A; C; D) ∈ NS(n) and (A, +; A, −; C; D) ∈ BS(n + 1, n) are quite different.
In section 3 we introduce the elementary transformations of NS(n). We point out that the elementary transformation (E4) is quite nonintuitive. It originated in our paper [3] where we classified near-normal sequences of small length. Subsequently it has been extended and used to classify (see [7]) the base sequences BS(n + 1, n) for n ≤ 30. We use these elementary transformations to define an equivalence relation and equivalence classes in NS(n). We also introduce the canonical form for normal sequences, and by using it we were able to compute the representatives of the equivalence classes for n ≤ 40.
In section 4 we introduce an abstract group, G NS , of order 512 which acts naturally on all sets NS(n). Its definition depends on the parity of n. The orbits of this group are just the equivalence classes of NS(n).
In section 5 we tabulate the results of our computations giving the list of representatives of the equivalence classes of NS(n) for n ≤ 40. The representatives are written in the encoded form which is explained in the next section.
The summary is given in Table 1. The column "Equ" gives the number of equivalence classes in NS(n). Note that most of the known normal sequences are of Golay type. The column "Gol" respectively "Spo" gives the number of equivalence classes which are of Golay type respectively sporadic. (Blank entries are zeros.) Let A = a 1 , a 2 , . . . , a n be an integer sequence of length n. To this sequence we associate the polynomial A(x) = a 1 + a 2 x + · · · + a n x n−1 , viewed as an element of the Laurent polynomial ring Z[x, x −1 ]. (As usual, Z denotes the ring of integers.) The nonperiodic autocorrelation function N A of A is defined by: where a k = 0 for k < 1 and for k > n. Note that N A (−i) = N A (i) for all i ∈ Z and N A (i) = 0 for i ≥ n. The norm of A is the Laurent polynomial N(A) = A(x)A(x −1 ). We have Hence, if (A; B; C; D) ∈ BS(m, n) then The negation, −A, of A is the sequence The reversed sequence A ′ and the alternated sequence A * of the sequence A are defined by A ′ = a n , a n−1 , . . . , a 1 A * = a 1 , −a 2 , a 3 , −a 4 , . . . , (−1) n−1 a n .
By A, B we denote the concatenation of the sequences A and B.
Let (A; A; C; D) ∈ NS(n). For convenience we set n = 2m (n = 2m + 1) for n even (odd). We decompose the pair (C; D) into quads and, if n is odd, the central column c m+1 d m+1 . Similar decomposition is valid for the pair (A; A).
The possibilities for the quads of base sequences BS(n + 1, n) are described in detail in [7]. In the case of normal sequences we have 8 possibilities for the quads of (C; D): but only 4 possibilities , namely 1,3,6 and 8, for the quads of (A; A). In [7] we referred to these eight quads as BS-quads. The additional eight Golay quads were also needed for the classification of base sequences BS(n + 1, n). Unless stated otherwise, the word "quad" will refer to BS-quads. We say that a quad is symmetric if its two columns are the same, and otherwise we say that it is skew. The quads 1, 2, 7, 8 are symmetric and 3, 4, 5, 6 are skew. We say that two quads have the same symmetry type if they are both symmetric or both skew.

D.Ž. D -OKOVIĆ
There are 4 possibilities for the central column: We encode the pair (A; A) by the symbol sequence when n is even respectively odd. Here p i is the label of the ith quad for i ≤ m and p m+1 is the label of the central column (when n is odd). Similarly, we encode the pair (C; D) by the symbol sequence For example, the five normal sequences displayed in the introduction are encoded as (0; 0), (1; 6), (60; 11), (16; 61) and (160; 640), respectively.

The equivalence relation
We 3) is the encoding of (C; D), then the encoding of (C;D) is τ (q 1 )τ (q 2 ) · · · τ (q m ) or τ (q 1 )τ (q 2 ) · · · τ (q m )q m+1 depending on whether n is even or odd, where τ is the transposition (45). In other words, the encoding of (C;D) is obtained from that of (C; D) by replacing simultaneously each quad symbol 4 with the symbol 5, and vice versa. For the proof of the equality NC + ND = N C + N D see [7].
(E5) Alternate all four sequences A; A; C; D.
We say that two members of NS(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. (iv) If n is odd and all quads of (A; A) are skew, then p m+1 = 0.
(v) If n is odd and i < m is the smallest index such that the consecutive quads p i and p i+1 have the same symmetry type, then p m+1 ∈ {1, 6}. If there is no such index and p m is symmetric, then 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 ⊆ NS(n) has at least one member having the canonical form.
Proof. Let S = (A; A; C; D) ∈ E be arbitrary and let (2.2) respectively (2.3) be the encoding of (A; A) respectively (C; D). By applying the elementary transformations (E1), we can assume that a 1 = c 1 = d 1 = +1. If n = 1, S is in the canonical form. So, let n > 1 from now on. Note that now the first quads, p 1 and q 1 , necessarily belong to {1, 6} and that p 1 = q 1 by (2.1). In the case when n is even and p 1 = 6 we apply the elementary transformation (E5). Note that (E5) preserves the quads p 1 and q 1 . Thus the conditions (i) and (vi) for the canonical form are satisfied.
The conditions (ii),(iii) and (iv) are pairwise disjoint, and so at most one of them may be violated. To satisfy (ii), it suffices (if necessary) to apply to the pair (A; A) the transformation (E2). To satisfy (iii) or (iv), it suffices (if necessary) to apply to the pair (A; A) the transformations (E1) and (E2).
We first consider the case where p 1 = 1 and p i and p i+1 are symmetric. By our assumption we have p i+1 = 8 and, by the minimality of i, i must be odd. We first apply (E2) to the pair (A; A) and then apply (E5). The quads p j for j ≤ i remain unchanged. On the other hand (E2) fixes p i+1 because it is symmetric, while (E5) replaces p i+1 = 8 with 1 because i + 1 is even. We have to make sure that previously established conditions are not spoiled. Only condition (iii) may be affected. If so, we must have i = 1 and we simply apply (E2) again.
Next we consider the case where again p 1 = 1 while p i and p i+1 are now skew. Thus p i+1 = 3 and i is even. We again apply (E2) to the pair (A; A) and then apply (E5). The quads p j for j ≤ i again remain unchanged. On the other hand (E2) replaces p i+1 = 3 with 6, while (E5) fixes it because i + 1 is odd. Note that in this case none of the conditions (i-iv) and (vi) will be spoiled.
The remaining two cases (where p 1 = 6) can be treated in a similar fashion. Now assume that any two consecutive quads p i , p i+1 have different symmetry types and that the last quad, p m , is symmetric. Assume also that p m+1 = 0, i.e., p m+1 = 3. If p 1 = 1 then m is odd and we just apply (E5). Otherwise p 1 = 6 and m is even and we apply the elementary transformations (E1) and (E2) to the pair (A; A) and then apply (E5). After this change the conditions (i-vi) will be satisfied.
To satisfy (vii), in view of (vi) we may assume that q 1 = 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 (viii), (if necessary) reverse C or D, or both. To satisfy (ix), (if necessary) interchange C and D. To satisfy (x), (if necessary) apply the elementary transformation (E4). Note that in this process we do not violate the previously established properties.
To satisfy (xi), (if necessary) switch C and D and apply (E4) to preserve (x). To satisfy (xii), (if necessary) replace C with −C ′ or D with −D ′ , or both.
Hence S is now in the canonical form.

The symmetry group of NS(n)
We shall construct a group G NS of order 512 which acts on NS(n). Our (redundant) generating set for G NS will consist of 9 involutions. Each of these generators is an elementary transformation, and we use this information to construct G NS , i.e., to impose the defining relations. We denote by S = (A; A; C; D) an aritrary member of NS(n).
To construct G NS , we start with an elementary abelian group E of order 64 with generators ν, ρ, and ν i , ρ i , i ∈ {3, 4}. It acts on NS(n) as follows: Next we introduce the involutory generator σ. We declare that σ commutes with ν and ρ, and that σν 3 = ν 4 σ and σρ 3 = ρ 4 σ. The group H = E, σ is the direct product of two groups: H 1 = ν, ρ of order 4 and H 2 = ν 3 , ρ 3 , σ of order 32. The action of E on NS(n) extends to H by defining σS = (A; A; D; C).
We add a new generator θ which commutes elementwise with H 1 , commutes with ν 3 ρ 3 , ν 4 ρ 4 and σ, and satisfies θρ 3 = ρ 4 θ. Let us denote this enlarged group byH. It has the direct product decompositioñ where the second factor is itself direct product of two copies of the dihedral group D 8 of order 8: The action of H on NS(n) extends toH by letting θ act as the elementary transformation (E5). Finally, we define G NS as the semidirect product ofH and the group of order 2 with generator α. By definition, α commutes with ν, ν 3 , ν 4 and satisfies: The action ofH on NS(n) extends to G NS by letting α act as the elementary transformation (E5), i.e., we have αS = (A * ; B * ; C * ; D * ).
We point out that the definition of the subgroupH is independent of n and its action on NS(n) has a quad-wise character. By this we mean that the value of a particular quad, say p i , of S ∈ NS(n) and h ∈H determine uniquely the quad p i of hS. In other wordsH acts on the quads and the set of central columns such that the encoding of hS is given by the symbol sequences h(p 1 )h(p 2 ) . . . and h(q 1 )h(q 2 ) . . . .
On the other hand the definition of the full group G NS depends on the parity of n, and only for n odd it has the quad-wise character.
An important feature of the quad-action ofH is that it preserves the symmetry type of the quads. If n is odd, this is also true for G NS .
The following proposition follows immediately from the construction of G NS and the description of its action on NS(n). The main tool that we use to enumerate the equivalence classes of NS(n) is the following theorem. Proof. In view of Proposition 3.2, we just have to prove the uniqueness assertion. Let S (k) = (A (k) ; A (k) ; C (k) ; D (k) ) ∈ E, (k = 1, 2) be in the canonical form. We have to prove that in fact S (1) = S (2) . By Proposition 4.1, we have gS (1) = S (2) for some g ∈ G NS . We can write g as g = α s h where s ∈ {0, 1} and h = h 1 h 2 with h 1 ∈ H 1 and h 2 ∈H 2 . Let p . . the encoding of the pair (C (k) ; D (k) ). The symbols (i-xii) will refer to the corresponding conditions of Definition 3.1.
We prove first preliminary claims (a-c). (a): p 1 and, consequently, q 1 . For n even this follows from (i). Let n be odd. When we apply the generator α to any S ∈ NS(n), we do not change the first quad of (A; A). It follows that the quads p have the same symmetry type. The claim now follows from (i).
Clearly, we are done with the case n = 2. If n = 3 it is easy to see that we must have p  (2) in that case.
Thus from now on we may assume that n > 3. (b): If n is even then s = 0. By (i), p 1 = 1. Note that the first quads of (A; A) in S and in αS have different symmetry types for any S ∈ E. As the quad h(1) is symmetric, the equality α s hS (1) = S (2) forces s to be 0.
As an immediate consequence of (b), we point out that, if n is even, a quad p  We shall now prove that A (1) = A (2) . Assume first that n is even. Then p    have the same symmetry type.
We first consider the case p (1) 1 = 1. Since n is odd α fixes the quad p 1 , and so h 1 must fix the quad 1. Thus we again have h 1 ∈ {1, ρ}.
If i is even then, by minimality of i, both p (1) i and p (1) i+1 are skew. By (v) we have p Since i is even, α fixes p i+1 and so we must have h 1 (6) = 6. It follows that h 1 = 1. As i > 1, the quad p (2) 2 = 6. Since α maps p 2 to its negative, we must have s = 0. Consequently, A (1) = A (2) .
If i is odd then both p Since i is odd, α maps p i+1 to its negative. Since ρ fixes the symmetric quads, we conclude that 1 = g(1) = α s h 1 (1) = α s (1) and so s = 0. If all quads p (1) i are symmetric, then they are all fixed by g and so A (1) = A (2) . Otherwise let j be the smallest index such that p We now consider the case p (1) 1 = 6. Since n is odd α fixes the quad p 1 , and so h 1 must fix the quad 6. Thus we have h 1 ∈ {1, νρ}.
If i is even then, by minimality of i, both p (2) i+1 = 6. Since i is odd, α maps p i+1 to its negative. Since νρ fixes the skew quads, we conclude that 6 = g(6) = α s h 1 (6) = α s (6) and so s = 0. If all quads p (1) i , i ≤ m, are skew, then they are all fixed by g and p entails that h 1 = 1 and so A (1) = A (2) . Otherwise let j be the smallest index such that p (1) j is symmetric. By (ii) we have p (2) . It remains to consider the case where any two consecutive quads p 1 ) = g(6) = α s h 1 (6), we must have s = 0 and h 1 = 1 or s = 1 and h 1 = ρ. In the former case we obviously have A (1) = A (2) . In the latter case all quads p (1) i , i ≤ m, are fixed by g. Moreover, if m is even also the central column p m+1 is fixed by g and so A (1) = A (2) . On the other hand, if m is odd, then the quad p (1) m is symmetric and the second part of the condition (v) implies that p (2) . Similar proof can be used if the quads p (1) i , i ≤ m, are symmetric for even i and skew for odd i. This completes the proof of the equality A (1) = A (2) . The proof of the equality (C (1) ; D (1) ) = (C (2) ; D (2) ) is the same as in [3].

Representatives of the equivalence classes
We have computed a set of representatives for the equivalence classes of normal sequences NS(n) for all n ≤ 40. 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. 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 (2.2) and (2.3).
In Table 2 and 3 we list the codes for the representatives of the equivalence classes of NS(n) for n ≤ 15 and 16 ≤ n ≤ 29, respectively. As there are 516 and 304 equivalence classes in NS(32) and NS(40) respectively, we list in Table 4 only the 36 representatives of the sporadic classes of NS(32). The cases n = 6, 14, 17, 21, . . . , 24, 27, 28, 30, 31, 33, 34, . . . , 39 are omitted since then NS(n) = ∅. We also omit n = 40 because in that case there are no sporadic classes. The Golay type equivalence classes of normal sequences can be easily enumerated (as explained in section 3) by using the tables of representatives of the equivalence classes of Golay sequences [1].

Acknowledgments
The author is grateful to NSERC for the continuing support of his research. This work was made possible by the facilities of the