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International Journal of Combinatorics
Volume 2010 (2010), Article ID 851857, 21 pages
Classification of Base Sequences
Department of Pure Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada
Received 6 February 2010; Revised 8 April 2010; Accepted 12 April 2010
Academic Editor: Masaaki Harada
Copyright © 2010 Dragomir Ž. Ðoković. 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 BS are quadruples of -sequences , with A and B of length and C and D of length n, such that the sum of their nonperiodic autocor-relation functions is a -function. The base sequence conjecture, asserting that BS exist for all n, is stronger than the famous Hadamard matrix conjecture. We introduce a new definition of equivalence for base sequences BS and construct a canonical form. By using this canonical form, we have enumerated the equivalence classes of BS for . As the number of equivalence classes grows rapidly (but not monotonically) with n, the tables in the paper cover only the cases .
Base sequences are quadruples of binary sequences, with and of length and and of length , such that the sum of their nonperiodic autocorrelation functions is a -function. In this paper we take .
Sporadic examples of base sequences have been constructed by many authors during the last 30 years; see, for example, [1–4] and the survey paper  and its references. A more systematic approach has been taken by the author in [6, 7]. The are presently known to exist for all (ibid) and for Golay numbers , where and are arbitrary nonnegative integers. However the genuine classification of is still lacking. Due to the important role that these sequences play in various combinatorial constructions such as that for -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 .
In Section 2 we recall the basic properties of base sequences of . We also recall the quad decomposition and our encoding scheme for this particular type of base sequences.
In Section 3 we enlarge the collection of standard elementary transformations of 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, , of order 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 some of the results of our computations (those for ) giving the list of representatives of the equivalence classes of . The representatives are written in the encoded form which is explained in the next section. For 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.
2. Quad Decomposition and the Encoding Scheme
We denote finite sequences of integers by capital letters. If, say, is such a sequence of length then we denote its elements by the corresponding lower case letters. Thus 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 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 .
A binary sequence is a sequence whose terms belong to . When displaying such sequences, we will often write for and for . The base sequences consist of four binary sequences , with and of length and and of length , such that Thus, for , we have
We denote by the set of such base sequences with and fixed. From now on we will consider only the case .
Let . For convenience we fix the following notation. For even (odd) we set (). We decompose the pair into quads and, if is even, the central column Similar decomposition is valid for the pair .
Recall the following basic and well-known property [3, Theorem ].
Theorem 2.1. For , the sum of the four quad entries is for the first quad of the pair and is for all other quads of and also for all quads of the pair .
Thus there are 8 possibilities for the first quad of the pair : These eight quads occur in the study of Golay sequences (see, e.g., ), and we refer to them as the Golay quads.
There are also 8 possibilities for each of the remaining quads of and all quads of : 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 and 8 are symmetric and 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:
We encode the pair by the symbol sequence where is the label of the th quad except in the case where is even and in which case is the label of the central column.
Similarly, 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).
3. The Equivalence Relation
We start by defining five types of elementary transformations of base sequences . 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  in the context of near-normal sequences. The quad notation was instrumental in the discovery of this new elementary operation.
The elementary transformations of are the following.T Negate one of the sequences .T Reverse one of the sequences .(T3) Interchange the sequences or .(T4) Replace the pair with the pair which is defined as follows: if (2.14) 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 each quad symbol 4 with the symbol 5, and vice versa. (We verify below that .) (T5) Alternate all four sequences .
In order to justify (T4) one has to verify that . For that purpose let us fix two quads and and consider their contribution to . We claim that is equal to the contribution of and to . If neither nor belongs to , then and and so . If then is the negation of . Hence if then again . Otherwise, say while , and it is easy to verify that . The pairs and also make a contribution to , but they can be treated in the same manner. Finally, if is odd then the pair also has a central column with label . In that case, if and , the contribution of and to is 0. This completes the verification.
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 (2.13) respectively (2.14) be the encoding of the pair respectively . One says that is in the canonical form if the following eleven conditions hold.(i), for even and for odd.(ii)The first symmetric quad (if any) of is 1 or 8.(iii)If is even and for then .(iv)The first skew quad (if any) of is 3 or 6.(v) for even and for odd.(vi)The first symmetric quad (if any) of is 1.(vii)The first skew quad (if any) of is 6.(viii)If is the least index such that then .(ix)If is the least index such that then .(x)If is odd and , for all , then .(xi)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 (2.13) respectively (2.14) be the encoding of respectively . By applying the first three types of elementary transformations and by Theorem 2.1 we can assume that and . By Theorem 2.1, . If is even and we apply the elementary transformation (T5). Thus we may assume that and that condition (v) for the canonical form is satisfied. To satisfy conditions (ii) and (iii), replace with (if necessary). To satisfy condition (iv), replace with (if necessary).
We now modify in order to satisfy the second part of condition (i). Note that is a BS-quad by Theorem 2.1. If there is nothing to do.
Assume that the quad is symmetric. By (ii), . From (2.8) for , we deduce that if and if . Note that if is even, then by (v). If is odd and , we switch and and apply the elementary transformation (T5). After this change we still have , , conditions (ii), (iii), and (iv) remain satisfied, and moreover .
Now assume that is skew. In view of (iv), we may assume that . Then the argument above based on (2.8) shows that , and so must be odd. After applying the elementary transformation (T5), we obtain that . Hence condition (i) is fully satisfied.
To satisfy (vi), in view of (v) we may assume that is odd and . 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 (vii), (if necessary) reverse , , or both. To satisfy (viii), (if necessary) interchange and . 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 and (if necessary). To satisfy (xi), (if necessary) replace with , with , or both.
Hence is now in the canonical form.
4. The Symmetry Group of
We will construct a group of order which acts on . Our (redundant) generating set for will consist of 12 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 aritrary member of .
To construct , we start with an elementary abelian group of order with generators , . It acts on as follows: that is, negates the th sequence of and reverses it.
Next we introduce two commuting involutory generators and . We declare that commutes with and , and commutes with and and that The group is the direct product of two isomorphic groups 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 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 (T4).
Finally, we define as the semidirect product of and the group of order 2 with generator . By definition, commutes with each and satisfies The action of on extends to by letting act as the elementary transformation (T5), 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 Golay quads, the BS-quads, and the set of central columns such that the encoding of is given by the symbol sequences On the other hand the full group has neither of these two properties.
An important feature of the action of on the BS-quads is that it preserves the symmetry type of the quads.
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 we use 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 . Let be the encoding of the pair and the encoding of the pair . The symbols (i-xi) will refer to the corresponding conditions of Definition 3.1. Observe that by (i).
We prove first preliminary claims (a-c).
For even see (v). 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 (v).
(b), that is, .
Assume first that is even. By (v), . For any the first quad of in and the one in have different symmetry types. As quad is symmetric, the equality forces to be 0. Assume now that is odd. Then for any the second quad of in and the one in have different symmetry types. Recall that and that (see (i)) and belong to . From (2.8), with , we deduce that for . We conclude that . The claim now follows from the fact that preserves while alters the symmetry type of the quad .
As an immediate consequence of (b), we point out that a quad is symmetric if and only if is, and the same is true for the quads and .
This was already proved above in the case when is odd. In general, the claim follows from (b) and the equality . Observe that each of the sets and consists of one symmetric and one skew quad and that preserves the symmetry type of quads.
Recall that . Since we have with and . Consequently, and for all ’s.
We will now prove that and . Since , the equality implies that . Thus for some .
Assume first that is symmetric. By (ii), . Then implies that . Hence, and so . If all quads , , are symmetric, then also . Otherwise let be the least index for which the quad is skew. Since and is 3 or 6 (see (iv)), we infer that . Hence and so .
Now assume that is skew. By (ii), . Then implies that . Thus and so . If all quads , , are skew, then by invoking condition (iii) we deduce that and so . Otherwise let be the least index for which the quad is symmetric. Since and is 1 or 8 (see (ii)), we infer that . Hence and so .
It remains to prove that and . We set . By (v) and claim (a) we have .
We first consider the case which occurs only for odd. Then and so . It follows that .
If some is symmetric, let be the least index such that is symmetric. Then (vi) implies that . Thus must fix the quad 1. As the stabilizer of the quad 1 in is , we infer that must also fix the quad 8. Similarly, if then (viii) implies that fixes 2 and 7. If then (ix) implies that fixes 4 and 5. These facts imply that fixes all quads in , that is, for all . It remains to show that, for odd , . If , this follows from (xi). Otherwise contains a symmetric quad and so . If then contains one of the quads 2, 4, 5, or 7. Since fixes all quads in , we infer that , and so . If , the equality follows from (x).
Finally, we consider the case . Since , . Hence fixes the quads 1 and 8.
If some is skew, then (vii) implies that fixes the quads 3 and 6. If then (viii) implies that fixes the quads 2 and 7. If then (ix) implies that fixes the quads 4 and 5. These facts imply that fixes all quads in . If is odd, then we invoke conditions (x) and (xi) to conclude that also fixes the central column of . Hence and also in this case.
5. Representatives of the Equivalence Classes
We have computed a set of representatives for the equivalence classes of base sequences for all . Due to their excessive size, we tabulate these sets only for . 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 are encoded as ,. In the tables we write 0 instead of . This convention was used in our previous papers on this and related topics.
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 (2.13) and (2.14).
In Table 2 we list the codes for the representatives of the equivalence classes of for . This table also records the values of the nonperiodic autocorrelation functions for and . For instance let us consider the first item in the list of base sequences given in Table 2. The base sequences are encoded in the first column as 0165, 6123. The first part 0165 encodes the pair , and the second part 6123 encodes the pair . The function , at the points , takes the values and 1 listed in the second column. Just below these values one finds the values of at the same points. In the third column we list likewise the values of and at the points .
Let us say that the base sequences are normal respectively near-normal if respectively for all . We denote by respectively the set of all normal respectively near-normal sequences in . Let us say also that an equivalence class is normal respectively near-normal if respectively is not void. Our canonical form has been designed so that if is normal, then its canonical representative belongs to . The analogous statement for near-normal classes is false. It is not hard to recognize which representatives in our tables are normal sequences. Let (2.13) be the encoding of the pair . Then if and only if all the quads , , belong to , and, in the case when is even, the central column symbol 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 . The base sequences 15 and 16 differ only in their first sequences, which we denote here by and respectively: Their associated polynomials are It is obvious that and are not equivalent in the above sense. On the other hand, from the factorizations and , where , and , we deduce immediately that .
This counter-example can be easily generalized. Let us define binary polynomials as polynomials associated to binary sequences. If is a polynomial of degree with , we define its dual polynomial by . Then for any positive integer we have , that is, where is the dual of the polynomial . In general we can start with any number of binary sequences, but here we take only three of them: of lengths respectively. From the associated binary polynomials , and we can form several binary polynomials of degree . The basic one is . 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.
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.
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