1. Introduction

By a binary respectively ternary sequence we mean a sequence whose terms belong to respectively . To such a sequence, we associate the polynomial . We refer to the Laurent polynomial as the norm of . Base sequences are quadruples of binary sequences, with and of length and and of length , and such that The set of such sequences will be denoted by .

In this paper, we consider only the case where or . The base sequences are normal if . We denote by the set of normal sequences of length , that is, those contained in . It is well known [1] that for normal sequences must be a sum of three squares. In particular, and are empty. Exhaustive computer searches have shown that are empty also for (see [2]) and (see [3–6]).

The base sequences are near-normal if for all . For near-normal sequences must be even or 1. We denote by the set of near-normal sequences in .

Normal sequences were introduced by Yang in [1] as a generalization of Golay sequences. Let us recall that Golay sequences are pairs of binary sequences of the same length, , and such that . We denote by the set of Golay sequences of length . It is known that they exist when where are arbitrary nonnegative integers. There exist two embeddings : the first defined by and the second by . 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 exist 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) it appears that there may be only finitely many sporadic normal sequences. For example, all 304 equivalence classes in are of Golay type. The smallest length for which the existence question of normal sequences is still unresolved is .

Table 1: Number of equivalence classes of .

Base sequences, and their special cases such as normal and near-normal sequences, play an important role in the construction of Hadamard matrices [7, 8]. For instance, the discovery of a Hadamard matrix of order 428 (see [9]) used a , constructed specially for that purpose.

Examples of normal sequences have been constructed in [1, 2, 5, 7, 10]. For various applications, it is of interest to classify the normal sequences of small length. Our main goal is to provide such classification for . The classification of near-normal sequences for and base sequences for has been carried out in our papers [5, 6, 11] and [10, 12], respectively.

We give examples of normal sequences of lengths : When displaying a binary sequence, we often write + for +1 and − for −1. We have written the sequence twice to make the quads visible (see Section 2).

If then . This has been used in our previous papers to view normal sequences as a subset of . For classification purposes it is more convenient to use the definition of as a subset of , which is closer to Yang's original definition [1].

In Section 2, we recall the basic properties of base sequences . The quad decomposition and our encoding scheme for used in our previous papers also work for , but not for arbitrary base sequences in . The quad decomposition of normal sequences is somewhat simpler than that of base sequences . We warn the reader that the encodings for the first two sequences of and are quite different.

In Section 3, we introduce the elementary transformations of . We point out that the elementary transformation (E4) is quite nonintuitive. It originated in our paper [5] where we classified near-normal sequences of small length. Subsequently, it has been extended and used to classify (see [10, 12]) the base sequences for . We use these elementary transformations to define an equivalence relation and equivalence classes in . 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 .

In Section 4, we introduce an abstract group, , of order 512 which acts naturally on all sets . Its definition depends on the parity of . The orbits of this group are just the equivalence classes of .

In Section 5, we tabulate the results of our computations giving the list of representatives of the equivalence classes of for . 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 . 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.)

2. Quad Decomposition and the Encoding Scheme

Let be an integer sequence of length . To this sequence, we associate the polynomial viewed as an element of the Laurent polynomial ring (as usual, denotes the ring of integers). The nonperiodic autocorrelation function of is defined by where for and for . Note that for all and for . The norm of is the Laurent polynomial . We have Hence, if then

The negation, , of is the sequence The reversed sequence and the alternated sequence of the sequence are defined by Observe that and for all . By we denote the concatenation of the sequences and .

Let . For convenience, we set for even (odd). We decompose the pair into quads and, if is odd, the central column . Similar decomposition is valid for the pair .

The possibilities for the quads of base sequences are described in detail in [10]. In the case of normal sequences we have 8 possibilities for the quads of : but only 4 possibilities, namely, 1, 3, 6, and 8, for the quads of . In [10], we referred to these eight quads as BS-quads. The additional eight Golay quads were also needed for the classification of base sequences . 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 are symmetric and 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:

We encode the pair by the symbol sequence when is even respectively odd. Here, is the label of the th quad for and is the label of the central column (when is odd). Similarly, we encode the pair by the symbol sequence For example, the five normal sequences displayed in the introduction are encoded as , , , , and , respectively.

3. The Equivalence Relation

We start by defining five types of elementary transformations of normal sequences (E1)Negate both sequences or one of .(E2)Reverse both sequences or one of .(E3) Interchange the sequences .(E4)Replace the pair with the pair which is defined as follows: if (2.11) is the encoding of , then the encoding of is or depending on whether is even or odd, where is the transposition (45). In other words, the encoding of is obtained from that of by replacing simultaneously each quad symbol 4 with the symbol 5, and vice versa. For the proof of the equality see [10].(E5) Alternate all four sequences .

We say that two members of 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 and let (2.10) respectively (2.11) be the encoding of the pair respectively . We say that is in the canonical form if the following twelve conditions hold.(i)For even , and for odd .(ii)The first symmetric quad (if any) of is 1.(iii)The first skew quad (if any) of is 6.(iv)If is odd and all quads of are skew, then .(v)If is odd and is the smallest index such that the consecutive quads and have the same symmetry type, then . If there is no such index and is symmetric, then .(vi) if .(vii)The first symmetric quad (if any) of is 1.(viii)The first skew quad (if any) of is 6.(ix)If is the least index such that then .(x)If is the least index such that then .(xi)If is odd and , for all , then .(xii)If is odd and , for all , 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 has at least one member having the canonical form.

Proof. Let be arbitrary and let (2.10) respectively (2.11) be the encoding of respectively . By applying the elementary transformations (E1), we can assume that . If , is in the canonical form. So, let from now on. Note that now the first quads, and , necessarily belong to and that by (2.4). In the case when is even and we apply the elementary transformation (E5). Note that (E5) preserves the quads and . Thus the conditions (i) and (vi) for the canonical form are satisfied.
The conditions (ii), (iii), and (iv) are pairwise disjoint, so at most one of them may be violated. To satisfy (ii), it suffices (if necessary) to apply to the pair the transformation (E2). To satisfy (iii) or (iv), it suffices (if necessary) to apply to the pair the transformations (E1) and (E2).
For (v), assume that and have the same symmetry type and that is the smallest such index. Also assume that , that is, .
We first consider the case where and and are symmetric. By our assumption, we have , and, by the minimality of , must be odd. We first apply (E2) to the pair and then apply (E5). The quads for remain unchanged. On the other hand, (E2) fixes because it is symmetric, while, (E5) replaces with 1 because 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 and we simply apply (E2) again.
Next, we consider the case where again while and are now skew. Thus and is even. We again apply (E2) to the pair and then apply (E5). The quads for again remain unchanged. On the other hand (E2) replaces with 6 while (E5) fixes it because is odd. Note that in this case none of the conditions (i–iv) and (vi) will be spoiled.
The remaining two cases (where ) can be treated in a similar fashion. Now assume that any two consecutive quads , have different symmetry types and that the last quad, , is symmetric. Assume also that , that is, . If then is odd and we just apply (E5). Otherwise and is even and we apply the elementary transformations (E1) and (E2) to the pair 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 . If the first symmetric quad in is 2 respectively 7, we reverse and negate respectively . If it is 8, we reverse and negate both and . Now, the first symmetric quad will be 1.
To satisfy (viii), (if necessary) reverse or , or both. To satisfy (ix), (if necessary) interchange and . 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 and and apply (E4) to preserve (x). To satisfy (xii), (if necessary) replace with or with , or both.
Hence, is now in the canonical form.

We end this section by a remark on Golay-type normal sequences. Let , with . While the Golay sequences and are always considered as equivalent (see [13]) the normal sequences and may be nonequivalent. It is easy to show that, in fact, these two normal sequences are equivalent if and only if the binary sequences and are equivalent, that is, if and only if .

The equivalence classes of Golay sequences of length ≤40 have been enumerated in [13]. This was accomplished by defining the canonical form and listing the canonical representatives of the equivalence classes. These representatives are written there in encoded form as obtained by decomposing into quads. These are Golay quads and should not be confused with the BS-quads defined in Section 2. If is one of the representatives, it is obvious that and , and it is easy to see that also . Thus. if is equivalent to we must have . Finally, one can show that the equality holds if and only if for each index . For another meaning of the latter condition see [13, Proposition 5.1]. Thus an equivalence class of Golay sequences with canonical representative provides either one or two equivalence classes of . The former case occurs if and only if for each index .

By using this criterion, it is straightforward to list the equivalence classes of of Golay type for . For instance, if there are five equivalence classes of Golay sequences. Their representatives are (see [13]) 3218, 3236, 3254, 3272, and 3315. Only the last representative violates the above condition. Hence, we have exactly equivalence classes of Golay type in .

4. The Symmetry Group of

We will construct a group of order 512 which acts on . Our (redundant) generating set for will consist of 9 involutions. Each of these generators is an elementary transformation, and we use this information to construct , that is, to impose the defining relations. We denote by an arbitrary member of .

To construct , we start with an elementary abelian group of order 64 with generators , , and , , . It acts on as follows:

Next, we introduce the involutory generator . We declare that commutes with and , and that and . The group is the direct product of two groups: of order 4 and of order 32. The action of on extends to by defining .

We add a new generator which commutes elementwise with , commutes with , , and , and satisfies . Let us denote this enlarged group by . It has the direct product decomposition where the second factor is itself a direct product of two copies of the dihedral group of order 8: The action of on extends to by letting act as the elementary transformation (E5).

Finally, we define as the semidirect product of and the group of order 2 with generator . By definition, commutes with , , and satisfies The action of on extends to by letting act as the elementary transformation (E5), that is, we have .

We point out that the definition of the subgroup is independent of and its action on has a quadwise character. By this we mean that the value of a particular quad, say , of and determine uniquely the quad of . In other words, acts on the quads and the set of central columns such that the encoding of is given by the symbol sequences On the other hand, the definition of the full group depends on the parity of , and only for odd it has the quad-wise character.

An important feature of the quad-action of is that it preserves the symmetry type of the quads. If is odd, this is also true for .

The following proposition follows immediately from the construction of and the description of its action on .

Proposition 4.1. The orbits of in are the same as the equivalence classes.

The main tool that one uses to enumerate the equivalence classes of is the following theorem.

Theorem 4.2. For each equivalence class there is a unique having the canonical form.

Proof. In view of Proposition 3.2, we just have to prove the uniqueness assertion. Let be in the canonical form. We have to prove that in fact .
By Proposition 4.1, we have for some . We can write as where and with and . Let be the encoding of the pair and the encoding of the pair . The symbols (i–xii) will refer to the corresponding conditions of Definition 3.1.
We prove first preliminary claims (a–c).
(a) and, consequently, .
For even this follows from (i). Let be odd. When we apply the generator to any , we do not change the first quad of . It follows that the quads and have the same symmetry type. The claim now follows from (i).
Clearly, we are done with the case .
If it is easy to see that we must have and . By (iv), for the central column symbols, we have . Then (2.4) for implies that for . By (xi) we must have . Hence in that case.
Thus from now on we may assume that .
(b) If is even then, .
By (i), . Note that the first quads of in and in have different symmetry types for any . As the quad is symmetric, the equality forces to be 0.
As an immediate consequence of (b), we point out that, if is even, a quad is symmetric iff is, and the same is true for the quads and .
(c) .
We first observe that and have the same symmetry type. If is even this follows from (b) since then . If is odd then under the quad action on , each of , , preserves the symmetry type of . Now the assertion (c) follows from (ii) and (iii) if and have different symmetry types, and from (v) otherwise.
We will now prove that .
Assume first that is even. Then by (i), by (b), and the equality implies that . Thus . Let be the smallest index (if any) such that the quad is skew. Then by (iii). Hence and so and follows. On the other hand, if all quads are symmetric, then all these quads are fixed by and so .
Next assume that is odd. Then by (i). Let be the smallest index (if any) such that the quads and have the same symmetry type.
We first consider the case . Since is odd, fixes the quad , and so must fix the quad 1. Thus we again have .
If is even then, by minimality of , both and are skew. By (v), we have . Since is even, fixes and so we must have . It follows that . As , the quad is skew and by (iii) we have . Since maps to its negative, we must have . Consequently, .
If is odd then both and are symmetric. By (v) we have . Since is odd, maps to its negative. Since fixes the symmetric quads, we conclude that and so . If all quads are symmetric, then they are all fixed by and so . Otherwise, let be the smallest index such that is skew. By (iii) we have , and implies that . Thus .
We now consider the case . Since is odd, fixes the quad , and so must fix the quad 6. Thus we have .
If is even then, by minimality of , both and are symmetric. By (v) we have . Since is even, fixes and so we must have . It follows that . As , the quad is symmetric and by (ii) we have . Since maps to its negative, we must have . Consequently, .
If is odd then both and are skew. By (v) we have . Since is odd, maps to its negative. Since fixes the skew quads, we conclude that and so . If all quads , , are skew, then they are all fixed by and by (iv). Now entails that and so . Otherwise let be the smallest index such that is symmetric. By (ii) we have , and implies that . Thus .
It remains to consider the case where any two consecutive quads and , , have different symmetry types. Say, the quads , , are skew for even and symmetric for odd . By (i) and (iii) we have and . Then must fix the quad 1, and so . Since , we must have and or and . In the former case, we obviously have . In the latter case, all quads , , are fixed by . Moreover, if is even also the central column is fixed by and so . On the other hand, if is odd, then the quad is symmetric and the second part of the condition (v) implies that . Hence again .
Similar proof can be used if the quads , , are symmetric for even and skew for odd . This completes the proof of the equality . The proof of the equality is the same as in [5].

5. Representatives of the Equivalence Classes

We have, computed a set of representatives for the equivalence classes of normal sequences for all . 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 , the representatives are listed in the lexicographic order of the symbol sequences (2.10) and (2.11).

In Tables 2 and 3, we list the codes for the representatives of the equivalence classes of for and , respectively. As there are 516 and 304 equivalence classes in and , respectively, we list in Table 4 only the 36 representatives of the sporadic classes of . The cases are omitted since then . We also omit 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 [13].

Table 2: Class representatives for .

Table 3: Class representatives for .

Table 4: Sporadic classes for .

Note that in the case , there are no quads and both zeros in Table 2 represent central columns.

Acknowledgments

The author is grateful to NSERC for the continuing support of his research. This paper was made possible by the facilities of the Shared Hierarchical Academic Research Computing Network (SHARCNET; www.sharcnet.ca) and Compute/Calcul Canada.