aa r X i v : . [ m a t h . C O ] A ug CLASSIFICATION OF NORMAL SEQUENCES
DRAGOMIR ˇZ. D– OKOVI ´C
Abstract.
Base sequences BS ( m, n ) are quadruples ( A ; B ; C ; D )of {± } -sequences, with A and B of length m and C and D oflength n , such that the sum of their nonperiodic autocorrelationfunctions is a δ -function. Normal sequences N S ( n ) are base se-quences ( A ; B ; C ; D ) ∈ BS ( n, n ) such that A = B . We introducea definition of equivalence for normal sequences N S ( n ), and con-struct a canonical form. By using this canonical form, we haveenumerated the equivalence classes of N S ( n ) for n ≤ Introduction
By a binary respectively ternary sequence we mean a sequence A = a , a , . . . , a m whose terms belong to {± } respectively { , ± } . Tosuch a sequence we associate the polynomial A ( z ) = a + a z + · · · + a m z m − . We refer to the Laurent polynomial N ( A ) = A ( z ) A ( z − ) asthe norm of A . Base sequences ( A ; B ; C ; D ) are quadruples of binarysequences, with A and B of length m and C and D of length n , andsuch 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 N S ( n ) the set of normal sequences of length n , i.e.,those contained in BS ( n, n ). It is well known [12] that for normalsequences 2 n must be a sum of three squares. In particular, N S (14)and
N S (30) are empty. Exhaustive computer searches have shownthat
N S ( n ) are empty also for n = 6 , , , , ,
24 (see [10]) and n = 27 , , , , , . . . ,
39 (see [2, 4]).The base sequences ( A ; B ; C ; D ) ∈ BS ( n + 1 , n ) are near-normal if b i = ( − i − a i for all i ≤ n . For near-normal sequences n must be Key words and phrases.
Base sequences, Golay sequences, normal sequences,nonperiodic autocorrelation functions, canonical form. even or 1. We denote by
N N ( n ) the set of near-normal sequences in BS ( n + 1 , n ).Normal sequences were introduced by C.H. Yang in [12] as a gener-alization 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 ) = 2 n . We denote by GS ( n ) the set of Golay sequencesof length n . It is known that they exist when n = 2 a b c where a, b, c are arbitrary nonnegative integers. There exist two embeddings GS ( n ) → N S ( n ): the first defined by ( A ; B ) → ( A ; A ; B ; B ) and thesecond 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 ofequivalence of normal sequences see section 3. However, as observed byYang, there exists normal sequences which are not of Golay type. Werefer to them as sporadic normal sequences. From the computationalresults reported in this paper (see Table 1 below) it appears that theremay be only finitely many sporadic normal sequences. E.g. all 304equivalence classes in
N S (40) are of Golay type. The smallest lengthfor which the existence question of normal sequences is still unresolvedis n = 41.Base sequences, and their special cases such as normal and near-normal sequences, play an important role in the construction of Ha-damard matrices [8, 11]. For instance, the discovery of a Hadamardmatrix of order 428 (see [9]) used a BS (71 , N S ( n ) have been constructed in [3,7, 8, 10, 12]. For various applications, it is of interest to classify thenormal sequences of small length. Our main goal is to provide suchclassification for n ≤
40. The classification of near-normal sequences
N N ( n ) for n ≤
40 and base sequences BS ( n + 1 , n ) for n ≤
30 hasbeen carried out in our papers [3, 4, 6] and [7], respectively.We give examples of normal sequences of lengths n = 1 , . . . , A = +; A = +; C = +; D = +; A = + , +; A = + , +; C = + , − ; D = + , − ; A = + , + , − ; A = + , + , − ; C = + , + , +; D = + , − , +; A = + , + , − , +; A = + , + , − , +; C = + , + , + , − ; D = + , + , + , − ; A = + , + , + , − , +; A = + , + , + , − , +; C = + , + , + , − , − ; D = + , − , + , + , − ; LASSIFICATION OF NORMAL SEQUENCES 3
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 ) ∈ N S ( n ) then ( A, +; A, − ; C ; D ) ∈ BS ( n + 1 , n ).This has been used in our previous papers to view normal sequences N S ( n ) as a subset of BS ( n +1 , n ). For classification purposes it is moreconvenient to use the definition of N S ( n ) as a subset of BS ( n, n ), whichis 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 ) usedin our previous papers also works for N S ( n ), but not for arbitrary basesequences in BS ( n, n ). The quad decomposition of normal sequences N S ( 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 ) ∈ N S ( n ) and ( A, +; A, − ; C ; D ) ∈ BS ( n + 1 , n ) are quitedifferent.In section 3 we introduce the elementary transformations of N S ( n ).We point out that the elementary transformation (E4) is quite non-intuitive. It originated in our paper [3] where we classified near-normalsequences of small length. Subsequently it has been extended and usedto classify (see [7]) the base sequences BS ( n + 1 , n ) for n ≤
30. We usethese elementary transformations to define an equivalence relation andequivalence classes in
N S ( n ). We also introduce the canonical formfor normal sequences, and by using it we were able to compute therepresentatives of the equivalence classes for n ≤ G NS , of order 512 whichacts naturally on all sets N S ( n ). Its definition depends on the parityof n . The orbits of this group are just the equivalence classes of N S ( n ).In section 5 we tabulate the results of our computations giving thelist of representatives of the equivalence classes of N S ( n ) for n ≤ N S ( n ). Note that most of the knownnormal sequences are of Golay type. The column “Gol” respectively“Spo” gives the number of equivalence classes which are of Golay typerespectively sporadic. (Blank entries are zeros.) D.ˇZ. D– OKOVI ´C
Table 1: Number of equivalence classes of
N S ( n ) n Equ Gol Spo n Equ Gol Spo1 1 1 212 1 1 223 1 1 234 1 1 245 1 1 25 4 46 26 2 27 4 4 278 7 6 1 289 3 3 29 2 210 5 4 1 3011 2 2 3112 4 4 32 516 480 3613 3 3 3314 3415 2 2 3516 52 48 4 3617 3718 1 1 3819 1 1 3920 36 34 2 40 304 3042.
Quad decomposition and the encoding scheme
Let A = a , a , . . . , a n be an integer sequence of length n . To thissequence we associate the polynomial A ( x ) = a + a x + · · · + a n x n − , viewed as an element of the Laurent polynomial ring Z [ x, x − ]. (Asusual, Z denotes the ring of integers.) The nonperiodic autocorrelationfunction N A of A is defined by: N A ( i ) = X j ∈ Z a j a i + j , i ∈ Z , where a k = 0 for k < 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 Laurentpolynomial N ( A ) = A ( x ) A ( x − ). We have N ( A ) = X i ∈ Z N A ( i ) x i . LASSIFICATION OF NORMAL SEQUENCES 5
Hence, if ( A ; B ; C ; D ) ∈ BS ( m, n ) then(2.1) N A ( i ) + N B ( i ) + N C ( i ) + N D ( i ) = 0 , i = 0 . The negation, − A , of A is the sequence − A = − a , − a , . . . , − a n . The reversed sequence A ′ and the alternated sequence A ∗ of the se-quence A are defined by A ′ = a n , a n − , . . . , a A ∗ = a , − a , a , − a , . . . , ( − n − a n . Observe that N ( − A ) = N ( A ′ ) = N ( A ) and N A ∗ ( i ) = ( − i N A ( i ) forall i ∈ Z . By A, B we denote the concatenation of the sequences A and B .Let ( A ; A ; C ; D ) ∈ N S ( n ). For convenience we set n = 2 m ( n =2 m + 1) for n even (odd). We decompose the pair ( C ; D ) into quads (cid:20) c i c n +1 − i d i d n +1 − i (cid:21) , i = 1 , , . . . , m, and, if n is odd, the central column (cid:20) c m +1 d m +1 (cid:21) . Similar decompositionis valid for the pair ( A ; A ).The possibilities for the quads of base sequences BS ( n + 1 , n ) aredescribed in detail in [7]. In the case of normal sequences we have 8possibilities for the quads of ( C ; D ):1 = (cid:20) + ++ + (cid:21) , (cid:20) + + − − (cid:21) , (cid:20) − + − + (cid:21) , (cid:20) + −− + (cid:21) , (cid:20) − ++ − (cid:21) , (cid:20) + − + − (cid:21) , (cid:20) − − + + (cid:21) , (cid:20) − −− − (cid:21) , 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 eightGolay quads were also needed for the classification of base sequences BS ( n + 1 , n ). Unless stated otherwise, the word “quad” will refer toBS-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 , , , , , , same symmetrytype if they are both symmetric or both skew. D.ˇZ. D– OKOVI ´C
There are 4 possibilities for the central column:0 = (cid:20) ++ (cid:21) , (cid:20) + − (cid:21) , (cid:20) − + (cid:21) , (cid:20) −− (cid:21) . We encode the pair ( A ; A ) by the symbol sequence(2.2) p p . . . p m respectively p p . . . p m p m +1 when n is even respectively odd. Here p i is the label of the i th quadfor 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(2.3) q q . . . q m respectively q q . . . q m q m +1 . For example, the five normal sequences displayed in the introductionare encoded as (0; 0), (1; 6), (60; 11), (16; 61) and (160; 640), respec-tively. 3.
The equivalence relation
We start by defining five types of elementary transformations of nor-mal sequences ( A ; A ; C ; D ) ∈ N S ( n ):(E1) Negate both sequences A ; A or one of C ; D .(E2) Reverse both sequences A ; A or one of C ; D .(E3) Interchange the sequences C ; D .(E4) Replace the pair ( C ; D ) with the pair ( ˜ C ; ˜ D ) which is defined asfollows: If (2.3) is the encoding of ( C ; D ), then the encoding of ( ˜ C ; ˜ D ) is τ ( q ) τ ( q ) · · · τ ( q m ) or τ ( q ) τ ( q ) · · · 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 replacingsimultaneously each quad symbol 4 with the symbol 5, and vice versa.For the proof of the equality N ˜ C + N ˜ D = N C + N D see [7].(E5) Alternate all four sequences A ; A ; C ; D .We say that two members of N S ( n ) are equivalent if one can betransformed to the other by applying a finite sequence of elementarytransformations. One can enumerate the equivalence classes by findingsuitable representatives of the classes. For that purpose we introducethe canonical form. Definition 3.1.
Let S = ( A ; A ; C ; D ) ∈ N S ( n ) and let (2.2) respec-tively (2.3) be the encoding of the pair ( A ; A ) respectively ( C ; D ). Wesay that S is in the canonical form if the following twelve conditionshold:(i) For n even p = 1, and for n > p ∈ { , } .(ii) The first symmetric quad (if any) of ( A ; A ) is 1.(iii) The first skew quad (if any) of ( A ; A ) is 6. LASSIFICATION OF NORMAL SEQUENCES 7 (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 theconsecutive quads p i and p i +1 have the same symmetry type, then p m +1 ∈ { , } . If there is no such index and p m is symmetric, then p m +1 = 0.(vi) q ∈ { , } if n > C ; D ) is 1.(viii) The first skew quad (if any) of ( C ; D ) is 6.(ix) If i is the least index such that q i ∈ { , } then q i = 2.(x) If i is the least index such that q i ∈ { , } then q i = 4.(xi) If n is odd and q i = 2, ∀ i ≤ m , then q m +1 = 2.(xii) If n is odd and q i = 1, i ≤ m , then q m +1 = 0.We can now prove that each equivalence class has a member whichis in the canonical form. The uniqueness of this member will be provedin the next section. Proposition 3.2.
Each equivalence class
E ⊆
N S ( n ) has at least onemember 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 theelementary transformations (E1), we can assume that a = c = d =+1. If n = 1, S is in the canonical form. So, let n > p and q , necessarily belong to { , } and that p = q by (2.1). In the case when n is even and p = 6 weapply the elementary transformation (E5). Note that (E5) preservesthe quads p and q . Thus the conditions (i) and (vi) for the canonicalform are satisfied.The conditions (ii),(iii) and (iv) are pairwise disjoint, and so at mostone of them may be violated. To satisfy (ii), it suffices (if necessary) toapply 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).For (v), assume that p i and p i +1 have the same symmetry type andthat i is the smallest such index. Also assume that p i +1 / ∈ { , } , i.e., p i +1 ∈ { , } .We first consider the case where p = 1 and p i and p i +1 are symmet-ric. 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 = 8with 1 because i + 1 is even. We have to make sure that previously D.ˇZ. D– OKOVI ´C established conditions are not spoiled. Only condition (iii) may beaffected. If so, we must have i = 1 and we simply apply (E2) again.Next we consider the case where again p = 1 while p i and p i +1 arenow skew. Thus p i +1 = 3 and i is even. We again apply (E2) to thepair ( A ; A ) and then apply (E5). The quads p j for j ≤ i again remainunchanged. 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 theconditions (i-iv) and (vi) will be spoiled.The remaining two cases (where p = 6) can be treated in a similarfashion. Now assume that any two consecutive quads p i , p i +1 havedifferent 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 then m is odd andwe just apply (E5). Otherwise p = 6 and m is even and we apply theelementary transformations (E1) and (E2) to the pair ( A ; A ) and thenapply (E5). After this change the conditions (i-vi) will be satisfied.To satisfy (vii), in view of (vi) we may assume that q = 6. If the firstsymmetric 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 . Nowthe 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 processwe do not violate the previously established properties.To satisfy (xi), (if necessary) switch C and D and apply (E4) topreserve (x). To satisfy (xii), (if necessary) replace C with − C ′ or D with − D ′ , or both.Hence S is now in the canonical form. (cid:3) We end this section by a remark on Golay type normal sequences. Let( A ; B ) ∈ GS ( n ), with n = 2 m >
2. While the Golay sequences ( A ; B )and ( B ; A ) are always considered as equivalent (see [1]) the normalsequences ( A ; A ; B ; B ) and ( B ; B ; A ; A ) may be non-equivalent. It iseasy to show that in fact these two normal sequences are equivalent ifand only if the binary sequences A and B ∗ are equivalent, i.e., if andonly if B ∗ ∈ { A ; − A ; A ′ ; − A ′ } .The equivalence classes of Golay sequences of length ≤
40 have beenenumerated in [1]. This was accomplished by defining the canonicalform and listing the canonical representatives of the equivalence classes.These representatives are written there in encoded form as δ δ · · · δ m obtained by decomposing ( A ; B ) into m quads. These are Golay quadsand should not be confused with the BS-quads defined in section 2. LASSIFICATION OF NORMAL SEQUENCES 9
If ( A ; B ) ∈ GS ( n ) is one of the representatives, it is obvious that B ∗ = − A and B ∗ = − A ′ , and it is easy to see that also B ∗ = A . Thus if B ∗ is equivalent to A we must have B ∗ = A ′ . Finally, one can show thatthe equality B ∗ = A ′ holds if and only if δ i ≡ i (mod 2) for each index i . For another meaning of the latter condition see [1, Proposition 5.1].Thus an equivalence class of Golay sequences GS ( n ) with canonicalrepresentative ( A ; B ) provides either one or two equivalence classes of N S ( n ). The former case occurs if and only if δ i ≡ i (mod 2) for eachindex i .By using this criterion it is straightforward to list the equivalenceclasses of N S ( n ) of Golay type for n ≤
40. For instance if n = 8 thereare five equivalence classes of Golay sequences. Their representativesare (see [1]) 3218, 3236, 3254, 3272 and 3315. Only the last represen-tative violates the above condition. Hence we have exactly 4 + 2 = 6equivalence classes of Golay type in N S (8).4.
The symmetry group of
N S ( n )We shall construct a group G NS of order 512 which acts on N S ( n ).Our (redundant) generating set for G NS will consist of 9 involutions.Each of these generators is an elementary transformation, and we usethis information to construct G NS , i.e., to impose the defining relations.We denote by S = ( A ; A ; C ; D ) an aritrary member of N S ( n ).To construct G NS , we start with an elementary abelian group E oforder 64 with generators ν, ρ , and ν i , ρ i , i ∈ { , } . It acts on N S ( n )as follows: νS = ( − A ; − A ; C ; D ) , ρS = ( A ′ ; A ′ ; C ; D ) ,ν S = ( A ; A ; − C ; D ) , ρ S = ( A ; A ; C ′ ; D ) ,ν S = ( A ; A ; C ; − D ) , ρ S = ( A ; A ; C ; D ′ ) . Next we introduce the involutory generator σ . We declare that σ commutes with ν and ρ , and that σν = ν σ and σρ = ρ σ . Thegroup H = h E, σ i is the direct product of two groups: H = h ν, ρ i oforder 4 and H = h ν , ρ , σ i of order 32. The action of E on N S ( n )extends to H by defining σS = ( A ; A ; D ; C ) . We add a new generator θ which commutes elementwise with H ,commutes with ν ρ , ν ρ and σ , and satisfies θρ = ρ θ . Let us denotethis enlarged group by ˜ H . It has the direct product decomposition˜ H = h H, θ i = H × ˜ H , where the second factor is itself direct product of two copies of thedihedral group D of order 8:˜ H = h ρ , ρ , θ i × h ν ρ , ν ρ , θσ i . The action of H on N S ( n ) extends to ˜ H by letting θ act as the ele-mentary transformation (E5).Finally, we define G NS as the semidirect product of ˜ H and the groupof order 2 with generator α . By definition, α commutes with ν, ν , ν and satisfies: αρα = ρ ( νσ ) n − ; αρ j α = ρ j ν n − j , j = 3 , αθα = θσ n − . The action of ˜ H on N S ( n ) extends to G NS by letting α act as theelementary transformation (E5), i.e., we have αS = ( A ∗ ; B ∗ ; C ∗ ; D ∗ ) . We point out that the definition of the subgroup ˜ H is independentof n and its action on N S ( n ) has a quad-wise character. By this wemean that the value of a particular quad, say p i , of S ∈ N S ( n ) and h ∈ ˜ H determine uniquely the quad p i of hS . In other words ˜ H actson the quads and the set of central columns such that the encoding of hS is given by the symbol sequences h ( p ) h ( p ) . . . and h ( q ) h ( q ) . . . . On the other hand the definition of the full group G NS depends on theparity of n , and only for n odd it has the quad-wise character.An important feature of the quad-action of ˜ H is that it preserves thesymmetry type of the quads. If n is odd, this is also true for G NS .The following proposition follows immediately from the constructionof G NS and the description of its action on N S ( n ). Proposition 4.1.
The orbits of G NS in N S ( n ) are the same as theequivalence classes. The main tool that we use to enumerate the equivalence classes of
N S ( n ) is the following theorem. Theorem 4.2.
For each equivalence class
E ⊆
N S ( n ) there is a unique S = ( A ; A ; C ; D ) ∈ E having the canonical form.Proof. In view of Proposition 3.2, we just have to prove the uniquenessassertion. Let S ( k ) = ( A ( k ) ; A ( k ) ; C ( k ) ; D ( k ) ) ∈ E , ( k = 1 , S (1) = S (2) . LASSIFICATION OF NORMAL SEQUENCES 11
By Proposition 4.1, we have gS (1) = S (2) for some g ∈ G NS . We canwrite g as g = α s h where s ∈ { , } and h = h h with h ∈ H and h ∈ ˜ H . Let p ( k )1 p ( k )2 . . . be the encoding of the pair ( A ( k ) ; A ( k ) ) and q ( k )1 q ( k )2 . . . the encoding of the pair ( C ( k ) ; D ( k ) ). The symbols (i-xii) willrefer to the corresponding conditions of Definition 3.1.We prove first preliminary claims (a-c).(a): p (1)1 = p (2)1 and, consequently, q (1)1 = q (2)1 .For n even this follows from (i). Let n be odd. When we apply thegenerator α to any S ∈ N S ( n ), we do not change the first quad of( A ; A ). It follows that the quads p (1)1 and p (2)1 = g (cid:16) p (1)1 (cid:17) = h (cid:16) p (1)1 (cid:17) 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 (1)1 = p (2)1 = 6 and q (1)1 = q (2)1 = 1. By (iv), for the central column symbols, we have p (1)2 = p (2)2 = 0. Then the equation (2.1) for i = 1 implies that q ( k )2 ∈ { , } for k = 1 ,
2. By (xi) we must have q (1)2 = q (2)2 = 1. Hence S (1) = S (2) in that case.Thus from now on we may assume that n > n is even then s = 0.By (i), p (1)1 = p (2)1 = 1. Note that the first quads of ( A ; A ) in S andin 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 (1) i is symmetric iff p (2) i is, and the same is true for the quads q (1) i and q (2) i .(c): p (1)2 = p (2)2 .We first observe that p (1)2 and p (2)2 have the same symmetry type. If n is even this follows from (b) since then g = h . If n is odd then underthe quad action on p , each of α , ν , ρ preserves the symmetry type of p . Now the assertion (c) follows from (ii) and (iii) if p (1)1 and p (1)2 havedifferent symmetry types, and from (v) otherwise.We shall now prove that A (1) = A (2) .Assume first that n is even. Then p (1)1 = p (2)1 = 1 by (i), s = 0by (b), and the equality h ( p (1)1 ) = p (2)1 implies that h (1) = 1. Thus h ∈ { , ρ } . Let i be the smallest index (if any) such that the quad p (1) i is skew. Then p (1) i = p (2) i = 6 by (iii). Hence h (6) = 6 and so h = 1 and A (1) = A (2) follows. On the other hand, if all quads p (1) i aresymmetric, then all these quads are fixed by h and so A (1) = A (2) . Next assume that n is odd. Then p (1)1 = p (1)2 ∈ { , } by (i). Let i < m be the smallest index (if any) such that the quads p (1) i and p (1) i +1 have the same symmetry type.We first consider the case p (1)1 = 1. Since n is odd α fixes the quad p , and so h must fix the quad 1. Thus we again have h ∈ { , 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 (1) i +1 = p (2) i +1 = 6. Since i is even, α fixes p i +1 and so wemust have h (6) = 6. It follows that h = 1. As i >
1, the quad p (1)2 is skew and by (iii) we have p (1)2 = p (2)2 = 6. Since α maps p to itsnegative, we must have s = 0. Consequently, A (1) = A (2) .If i is odd then both p (1) i and p (1) i +1 are symmetric. By (v) we have p (1) i +1 = p (2) i +1 = 1. Since i is odd, α maps p i +1 to its negative. Since ρ fixesthe symmetric quads, we conclude that 1 = g (1) = α s h (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 (1) j is skew. By (iii) we have p (1) j = p (2) j = 6, and 6 = p (2) j = g ( p (1) j ) = g (6) = h (6) implies that h = 1. Thus A (1) = A (2) .We now consider the case p (1)1 = 6. Since n is odd α fixes the quad p , and so h must fix the quad 6. Thus we have h ∈ { , νρ } .If i is even then, by minimality of i , both p (1) i and p (1) i +1 are symmetric.By (v) we have p (1) i +1 = p (2) i +1 = 1. Since i is even, α fixes p i +1 and so wemust have h (1) = 1. It follows that h = 1. As i >
1, the quad p (1)2 issymmetric and by (ii) we have p (1)2 = p (2)2 = 1. Since α maps p to itsnegative, we must have s = 0. Consequently, A (1) = A (2) .If i is odd then both p (1) i and p (1) i +1 are skew. By (v) we have p (1) i +1 = p (2) i +1 = 6. Since i is odd, α maps p i +1 to its negative. Since νρ fixesthe skew quads, we conclude that 6 = g (6) = α s h (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 (1) m +1 = p (2) m +1 = 0 by (iv). Now 0 = p (2) m +1 = h ( p (1) m +1 ) = h (0)entails that h = 1 and so A (1) = A (2) . Otherwise let j be the smallestindex such that p (1) j is symmetric. By (ii) we have p (1) j = p (2) j = 1, and1 = p (2) j = g ( p (1) j ) = h (1) implies that h = 1. Thus A (1) = A (2) .It remains to consider the case where any two consecutive quads p (1) i and p (1) i +1 , i < m , have different symmetry types. Say, the quads p (1) i , i ≤ m , are skew for even i and symmetric for odd i . By (i) and (iii)we have p (1)1 = p (2)1 = 1 and p (1)2 = p (2)2 = 6. Then h must fix the quad1, and so h ∈ { , ρ } . Since 6 = p (2)2 = g ( p (2)1 ) = g (6) = α s h (6), we LASSIFICATION OF NORMAL SEQUENCES 13 must have s = 0 and h = 1 or s = 1 and h = ρ . In the former casewe 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 isfixed by g and so A (1) = A (2) . On the other hand, if m is odd, then thequad p (1) m is symmetric and the second part of the condition (v) impliesthat p (1) m +1 = p (2) m +1 = 0. Hence again A (1) = A (2) .Similar proof can be used if the quads p (1) i , i ≤ m , are symmetricfor 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 thesame as in [3]. (cid:3) Representatives of the equivalence classes
We have computed a set of representatives for the equivalence classesof normal sequences
N S ( n ) for all n ≤
40. Each representative is givenin the canonical form which is made compact by using our standardencoding. The encoding is explained in detail in section 2. This com-pact notation is used primarily in order to save space, but also to avoidintroducing errors during decoding. For each n , the representativesare 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 equiv-alence classes of N S ( n ) for n ≤
15 and 16 ≤ n ≤
29, respectively. Asthere are 516 and 304 equivalence classes in
N S (32) and
N S (40) re-spectively, we list in Table 4 only the 36 representatives of the sporadicclasses of
N S (32). The cases n = 6 , , , , . . . , , , , , , , , . . . , N S ( n ) = ∅ . We also omit n = 40 because inthat case there are no sporadic classes. The Golay type equivalenceclasses of normal sequences can be easily enumerated (as explainedin section 3) by using the tables of representatives of the equivalenceclasses of Golay sequences [1]. Table 2: Class representatives for n ≤ n = 11 0 0 n = 21 6 1 n = 31 60 11 n = 41 16 61 n = 51 160 640 n = 71 1660 6122 2 6113 1623 3 6160 12624 6163 1261 n = 81 1163 6618 2 1613 6168 3 1613 64434 1638 6116 5 1661 6183 6 1686 61317 1866 6311 n = 91 16133 64140 2 16163 64150 3 61180 16640 n = 101 11863 66311 2 16166 64156 3 16613 618384 16616 61831 5 18863 63311 n = 111 611680 164231 2 616163 126232 n = 121 161383 641261 2 163868 612243 3 186338 6314224 186631 631422 n = 131 1616133 6414853 2 6116680 1286320 3 6168160 1613441 n = 151 61613163 12676761 2 61683860 12626262Note that in the case n = 1 there are no quads and both zeros inTable 2 represent central columns. LASSIFICATION OF NORMAL SEQUENCES 15
Table 3: Class representatives for 16 ≤ n ≤ n = 161 11186366 66631811 2 11186636 666311813 11631866 66186311 4 11633381 661811635 11636618 66188836 6 11638133 661836887 11661836 66116381 8 11663681 661118639 11666318 66118136 10 11668163 6611361811 11816333 66361888 12 11816663 6636111813 16131686 61686131 14 16133831 6168161315 16136168 61688386 16 16138313 6168386817 16161386 61616831 18 16163861 6161168319 16163861 64124328 20 16166138 6161831621 16166138 64127156 22 16168613 6161316823 16381331 61166813 24 16381661 6116618325 16388338 61163816 26 16388668 6116318627 16611368 61836886 28 16611638 6183611629 16618361 61833883 30 16618631 6183311331 16831313 61386868 32 16833838 6138161633 16836161 61384242 34 16836161 6138838335 16838686 61383131 36 16838863 6134431337 16861613 61316168 38 16863868 6131168639 16866131 61318313 40 16868386 6131383141 18116333 63661888 42 18116663 6366111843 18631133 63186688 44 18633388 6318116645 18636611 63188833 46 18638866 6318331147 18661163 63116618 48 18663688 6311186649 18666311 63118133 50 18668836 6311338151 18886366 63331811 52 18886636 63331181 n = 181 161633881 641242146 n = 191 1168186360 6643551210 n = 201 1166131836 6611686381 2 1166861836 66113163813 1181616633 6636161188 4 1186161633 66316161885 1186868366 6631313811 6 1188686366 66331318117 1611663138 6441827614 8 1613383113 61681613689 1613383186 6168161331 10 1616138631 616422478611 1616311386 6161866831 12 1616681386 616113683113 1616831361 6161386883 14 1616833886 616138163115 1616836113 6161388368 16 1616838638 6161383116 Table 3: (continued) n = 2017 1638133138 6116681316 18 1638133161 611668138319 1638883818 6183331633 20 1661813881 611636166621 1661863138 6183311316 22 1661863161 618331138323 1683381313 6138836868 24 1683611313 613816686825 1683831361 6138386883 26 1683833886 613838163127 1683836113 6138388368 28 1683838638 613838311629 1686613113 6131831368 30 1686613186 613183133131 1863161133 6318616688 32 1863831133 631838668833 1881616663 6336161118 34 1886161663 633161611835 1886868336 6331313881 36 1888686336 6333131881 n = 251 1616138313163 64141484851432 1616161383163 64141485841433 1616161386163 64141485851434 1616168613163 6414158585143 n = 291 161383131316830 6414148415158432 161686161313860 641515851514853 LASSIFICATION OF NORMAL SEQUENCES 17
Table 4: Sporadic classes for n = 321 1111636366331881 66661818455422772 1111663318816363 66664554118827273 1166186333886318 66412318147211764 1166186366113681 66412318586355675 1166813633883681 66143281412711676 1166813666116318 66143281853655767 1613161361683831 61686168425257478 1616168313861313 64126517658264879 1616168338613838 641262372828412610 1616168361386161 641262375656735811 1616383883163861 641221463482284312 1616386113133168 641243438467237613 1616386186866831 641228283215762314 1616613813136831 641256568467762315 1616613886863168 641271713215237616 1616616116833861 641278536517284317 1616831613868686 641234826582351218 1616831638616161 641237624343735819 1616831661383838 641237627171412620 1638163886681331 614224163147741321 1638163886681331 624114263248842322 1661166113688631 614275836852741323 1661166113688631 624185736751842324 1683161638383861 613864214216171725 1683161661616138 613864218357565626 1683383813863131 613842167171125327 1683383886136868 613816423434874628 1683616113866868 613842832121825629 1683616186133131 613883423535174330 1683838338386138 613834281657464631 1683838361613861 613834284283121232 1686168638686131 613161314247575233 1818633611886666 636344551881222234 1818666636638811 636311114455277235 1863116636816611 634126884133453736 1863116663183388 63412688142218266. Acknowledgments
The author is grateful to NSERC for the continuing support ofhis research. This work was made possible by the facilities of the
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