New Infinite Families of Perfect Quaternion Sequences And

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New Inﬁnite Families of Perfect Quaternion Sequences and Williamson Sequences Curtis Bright, Ilias Kotsireas Member, IEEE, and Vijay Ganesh

Abstract—We present new constructions for perfect and odd construction a decade prior as an aircraft engineer at the Sperry perfect sequences over the quaternion group Q8. In particular, we Gyroscope Company. Moreover, Frank was granted a patent show for the first time that perfect and odd perfect quaternion t for a communication system based on his sequences [15]. sequences exist in all lengths 2 for t ≥ 0. In doing so we 2 disprove the quaternionic form of Mow’s conjecture that the Frank’s construction generates perfect sequences of length n longest perfect Q8-sequence that can be constructed from an over the complex nth roots of unity. Given the theoretical orthogonal array construction is of length 64. Furthermore, we elegance of perfect sequences and their importance to many use a connection to combinatorial design theory to prove the fields of engineering it would be extremely interesting and existence of a new infinite class of Williamson sequences, showing t useful if a construction could be found that produces perfect that Williamson sequences of length 2 n exist for all t ≥ 0 when 2 Williamson sequences of odd length n exist. Our constructions sequences longer than n using nth roots of unity. However, explain the abundance of Williamson sequences in lengths that over 60 years of effort has failed to find such a construction and are multiples of a large power of two. it has been conjectured by Mow [16] that such a construction Index Terms—Perfect sequences, quaternions, Williamson se- does not exist. In light of this, several researchers have searched quences, odd perfect sequences, periodic autocorrelation, odd for perfect sequences over other alphabets such as the group periodic autocorrelation, array orthogonality property of quaternions [17]. Quaternions are generalizations of the complex numbers that include the additional numbers j and k I.INTRODUCTION that satisfy the relationships Sequences that have zero correlation with themselves after i2 = j2 = k2 = −1, ij = k, jk = i, ki = j. a nontrivial cyclic shift are known as perfect [1]. Such sequences have a long history [2] and an amazing wealth Note that these relationships imply that quaternions are of applications, for example, appearing in the 3GPP LTE generally noncommutative. standard [3]. Perfect sequences and their generalizations have Communication systems have been described that are based been applied to spread spectrum multiple access systems [4], on the quaternions [18], [19] and standard mathematical radar systems [5], fast start-up equalization [6], channel techniques like the Fourier transform have been generalized estimation and synchronization [7], peak-to-average power to the quaternions for usage in signal processing [20] and ratio reduction [8], image watermarking [9], and constructing image processing [21]. Perfect sequences over the quaternions complementary sets [10]. They also tend to have small aperiodic were first studied by Kuznetsov [22]. Kuznetsov and Hall [23] correlation [11] and small ambiguity function sidelobes [12]. showed that Mow’s conjecture cannot be directly extended One of the first researchers to study perfect sequences to these sequences by constructing a perfect sequence over was Heimiller [13] who in 1961 gave a construction for a quaternion alphabet with 24 elements and whose length perfect sequences using matrices with array orthogonality. His is over 5 billion. In 2012, Barrera Acevedo and Hall [24] construction generated perfect sequences of length p2 over constructed perfect sequences over the alphabet {±1, ±i, j} arXiv:1905.00267v2 [cs.IT] 25 Nov 2020 the complex pth roots of unity for any prime p. Shortly after in all lengths of the form q + 1 where q ≡ 1 (mod 4) is a Heimiller’s paper was published, Frank and Zadoff published a prime power. Most recently, Blake [25] has run extensive response [14] pointing out that Frank had discovered the same searches for perfect sequences over the basic quaternion alphabet Q8 := {±1, ±i, ±j, ±k} (see Section VI for more Manuscript received May 3, 2019; revised July 21, 2020; accepted July 22, 2020. Date of publication August 13, 2020; date of current version November on previous work). The longest perfect Q8-sequence generated 20, 2020. (Corresponding author: Curtis Bright.) from an orthogonal array construction that Blake found had Curtis Bright was with the Department of Electrical and Computer length 64. Intriguingly, this length is exactly the square of the Engineering at the University of Waterloo, Waterloo, ON N2L 3G1, Canada. He is now with the School of Computer Science at the University of Windsor, alphabet size of 8. This led Blake to conjecture a quaternionic Windsor, ON N9B 3P4, Canada (e-mail: [email protected]). form of Mow’s conjecture that the quadratic Frank–Heimiller Ilias Kotsireas is with the Department of Physics and Computer Science bound applies to perfect sequences over Q8 generated from an at Wilfrid Laurier University, Waterloo, ON N2L 3C5, Canada (e-mail: [email protected]). orthogonal array construction. Vijay Ganesh is with the Department of Electrical and Computer Engineering In this paper we show that Blake’s conjecture is false. This at the University of Waterloo, Waterloo, ON N2L 3G1, Canada (e-mail: is accomplished via a new construction for perfect quaternion [email protected]). Communicated by K.-U. Schmidt, Associate Editor for Sequences. sequences that can be used to construct arbitrarily long perfect Digital Object Identifier 10.1109/TIT.2020.3016510 Q8-sequences (see Section V) and odd perfect Q8-sequences

(see Section IV). In particular, we construct the first infinite present Barrera Acevedo and Dietrich’s equivalence between family of odd perfect quaternion sequences and show for the perfect quaternion sequences and Williamson sequences. first time that perfect and odd perfect quaternion sequences of length 2t exist for all t. This is starkly different from what A. Complementary sequences and perfect sequences happens if one restricts the alphabet to only include the purely real or complex elements of Q8. It is known that the longest The aperiodic crosscorrelation of two sequences A = t perfect {±1}-sequence of length 2 is of length four [26] [a0, . . . , an−1] and B = [b0, . . . , bn−1] of length n is given by t and the longest perfect {±1, ±i}-sequence of length 2 is n−t−1 of length sixteen [27]. Additionally, the longest odd perfect X ∗ CA,B(t) := arb {±1} r+t -sequence has length two [28]. r=0 Recently Barrera Acevedo and Dietrich [29], [30] discov- ∗ ered a correspondence between perfect sequences over the where z denotes the conjugate of z [35]. For a general quater- ∗ quaternions and Williamson sequences from combinatorial nionic number we have (a + bi + cj + dk) := a − bi − cj − dk design theory [31] (see Section II for the definition of where a, b, c, d are real. The aperiodic autocorrelation of A is Williamson sequences). Using this relationship we construct given by CA(t) := CA,A(t). More generally, we also define the new Williamson sequences in even lengths, including all powers periodic and odd periodic (or negaperiodic) crosscorrelations of two. Prior to our construction Williamson sequences of by t length 2 were only known to exist for t ≤ 6 [32]. See ∗ RA,B(t) := CA,B(t) + CB,A(n − t) and Section III for our Williamson sequence construction and ˆ ∗ Section IV for a construction for a variant of Williamson RA,B(t) := CA,B(t) − CB,A(n − t) for 0 ≤ t < n. sequences that our Williamson sequence construction relies on. The periodic and odd periodic autocorrelations of A are given In 1944, when Williamson introduced the sequences that by RA(t) := RA,A(t) and RÂ(t) := RÂ,A(t) respectively. now bear his name [31] he showed that the existence of These functions may be extended to all integers t via the Williamson sequences of odd length n implied the existence expressions of Williamson sequences of length 2n. In 1970, twenty-six years later, Turyn [27] generalized Williamson’s result by n−1 X ∗ showing that the existence of Williamson sequences of odd RA,B(t) := arbr+t mod n length n implied the existence of Williamson sequences of r=0 lengths 2tn for t ≤ 4. Nearly fifty years after Turyn’s result, and Barrera Acevedo and Dietrich [29] improved this to t ≤ 6 in n−1 2019. In this paper we complete this process of generalizing ˆ X b(r+t)/nc ∗ RA,B(t) := (−1) arbr+t mod n. Williamson’s doubling result by showing the result in fact holds r=0 for all t ≥ 0. In other words, the existence of Williamson sequences of odd length n implies the existence of Williamson A set S of sequences of length n is called complementary P 1 t if CA(t) = 0 for all 1 ≤ t < n. Similarly, S is called sequences of length 2 n for all t and this provides a large new A∈S P infinite class of Williamson sequences. periodic complementary if A∈S RA(t) = 0 and odd periodic P ˆ Exhaustive searches for Williamson sequences [33], [34] complementary (or negacomplementary) if A∈S RA(t) = 0 have shown that they exist in all lengths n < 65 except for for all 1 ≤ t < n. A single sequence A is called perfect if ˆ n = 35, 47, 53, and 59. They are generally very abundant in RA(t) = 0 for all 1 ≤ t < n and odd perfect if RA(t) = 0 the even lengths (particularly in the lengths that are divisible by for all 1 ≤ t < n. a large power of two) but not in the odd lengths. For example, Note that the periodic correlation values RA(t) of a over 50,000 inequivalent sets of Williamson sequences exist sequence are preserved under the cyclic shift operator in length 64 but fewer than 100 sets of Williamson sequences [a0, . . . , an−2, an−1] 7→ [an−1, a0, . . . , an−2] and the negape- ˆ exist in all odd lengths up to 65. Previously this dichotomy was riodic correlation values RA(t) of a sequence are preserved unexplained but the constructions that we provide in this paper under the negacyclic shift operator [a0, . . . , an−2, an−1] 7→ produce approximately 75% of the Williamson sequences that [−an−1, a0, . . . , an−2] (see [36], [37]). Therefore applying a exist in all even lengths n ≤ 70 (see Section VII). cyclic shift to any sequence in a set of periodic complementary sequences or applying a negacyclic shift to any sequence in a set of negacomplementary sequences does not disturb the property II.PRELIMINARIES of the set being periodic complementary or negacomplementary. In this section we provide the preliminaries necessary to Let (−1) ∗ X denote the sequence whose rth entry is r explain our construction for perfect quaternion sequences, odd (−1) xr, i.e., the alternating negation operation. A set of perfect quaternion sequences, and Williamson sequences. First periodic complementary sequences of odd length n can be we define the concept of sequence perfection in terms of converted into a set of negacomplementary sequences and vice the amount of correlation that a sequence has with cyclically versa by applying the alternating negation operation (see [38]). shifted copies of itself. Following this, we discuss the array orthogonality property used in several constructions for perfect 1Technically S should be defined to be a multiset but we follow standard sequences. We then define Williamson sequences and finally convention and refer to it as a set. 3

Lemma 1. If n is odd then (A, B, C, D) are periodic comple- (A, B, C, D) nega Williamson if they are negacomplementary, mentary sequences of length n if and only if ((−1) ∗ A, (−1) ∗ i.e., RÂ(t) + RˆB(t) + RˆC (t) + RˆD(t) = 0 for all 1 ≤ t < n. B, (−1) ∗ C, (−1) ∗ D) are negacomplementary sequences. As is conventional, we require by definition that Williamson sequences are symmetric. If the symmetry condition is replaced Example 1. (+--, +--, +--, +++) is a set of complementary with the weaker property that the sequences are instead sequences of length 3 and (++-, ++-, ++-, +-+) is the set amicable (i.e., R (t) = R (t) for all t and X,Y ∈ of negacomplementary sequences generated from it using X,Y Y,X {A, B, C, D}) then the sequences are known as Williamson- Lemma 1. Note that we follow the convention of writing 1s type. For nega Williamson sequences we do not require them to by + and −1s by -. be symmetric. Instead, our work has discovered the importance of palindromic and antipalindromic nega Williamson sequences; B. Matrices with array orthogonality see Section III for details. The sequences in a set S are periodically uncorrelated if Example 3. (++, ++, +-, +-) are Williamson sequences of any two distinct sequences A, B in S satisfy RA,B(t) = 0 for length 2. Similarly, (+-, +-, +-, +-) are antipalindromic nega all 0 ≤ t < n. If a set of periodic complementary sequences Williamson sequences and (++, ++, ++, ++) are palindromic S = {S1,...,Sm} (with each sequence of length n a multiple nega Williamson sequences. (++-+, ++-+, ++-+, ++-+) of m) are periodically uncorrelated then the n × m matrix are Williamson sequences of length 4. Similarly, whose columns are given by S1, ... , Sm is said to have array (+-+-, +-+-, ++--, ++--) are antipalindromic nega orthogonality. Williamson sequences and (+--+, +--+, ++++, ++++) are Matrices with array orthogonality are important because they palindromic nega Williamson sequences. are used in many constructions for perfect sequences such as in the Frank–Heimiller and Mow constructions. In particular, the D. The Barrera Acevedo–Dietrich correspondence sequence formed by concatenating the rows of a matrix with array orthogonality is perfect. Additionally, matrices with array Let Q8 := {±1, ±i, ±j, ±k} be the quaternion group and orthogonality are themselves perfect arrays. Perfect arrays Q+ := Q8 ∪ qQ8 where q := (1 + i + j + k)/2 (note that Q+ are often studied as a way of generalizing the concept of is a set of sixteen quaternions that is not a group). Barrera perfection from sequences to matrices (e.g., see [39], [40]). Acevedo and Dietrich [29, Theorem 2.4] show that there is an They are defined to be n × m matrices A = (ar,s) that satisfy equivalence between Williamson-type sequences and perfect sequences over Q+. n−1 m−1 X X ∗ ar,sar+t mod n,s+t0 mod n = 0 Theorem 1. There is a one-to-one correspondence between r=0 s=0 sets of Williamson-type sequences of length n and perfect for all (t, t0) 6= (0, 0) with 0 ≤ t < n, 0 ≤ t0 < m. sequences of length n over Q+. Example 2. The columns of the matrix This correspondence is made explicit through the following mapping between the rth entries of a set of Williamson-type + + sequences (ar, br, cr, dr) and the rth entry of the corresponding + - perfect sequence sr: have array orthogonality and therefore this is a perfect array. ar - + + + + + + + +++- It generates the perfect sequence [ ]. br - - + - - + + - cr - - - + - - + + C. Williamson and nega Williamson sequences dr - + - - - + - + s 1 i j k q qi qj qk First, we describe the symmetry properties that are used in r the definition of Williamson sequences and will be important Additionally, we have the rule that if (ar, br, cr, dr) maps to sr in our construction for Williamson sequences. A sequence A of then (−ar, −br, −cr, −dr) maps to −sr. length n is called symmetric if at = an−t for all 1 ≤ t < n and Because of this theorem a construction for Williamson is palindromic if at = an−t−1 for all 0 ≤ t < n. For example, sequences of length n also produces symmetric perfect [x, y, z, y] is symmetric and [y, z, y] is palindromic. Note that Q+-sequences. In this paper we state our construction in a sequence is symmetric if and only if the subsequence formed terms of Williamson sequences with the understanding that it by removing its first element is palindromic. Additionally, equivalently produces perfect Q+-sequences. In Section V we we call a sequence antipalindromic if at = −an−t−1 for all also show how our construction can be used to produce perfect 0 ≤ t < (n − 1)/2 and antisymmetric if at = −an−t for Q8-sequences in many lengths including all powers of two. all 1 ≤ t < n/2. If [X; Y ] denotes sequence concatenation If the entries of a set of Williamson sequences (A, B, C, D) ˜ ˜ and X denotes the reversal of X then [X; X] is palindromic of length n satisfy arbrcrdr = 1 for all 0 ≤ r < n then the and [X; −X˜] is antipalindromic. Barrera Acevedo–Dietrich correspondence produces perfect A quadruple of symmetric {±1}-sequences (A, B, C, D) Q8-sequences. For this reason we say that a quadruple of are known as Williamson if they are periodic complementary, sequences has the Q8-property if the entries of its sequences i.e., if RA(t) + RB(t) + RC (t) + RD(t) = 0 for all 1 ≤ satisfy arbrcrdr = 1 for all 0 ≤ r < n. In his original t < n. Additionally, we call a quadruple of {±1}-sequences paper [31] Williamson proved that the entries of all Williamson 4

sequences in odd lengths n satisfy arbrcrdr = −a0b0c0d0 for 6) If X is symmetric then d(X) is symmetric. 1 ≤ r < n. As a consequence, no Williamson sequence of 7) If X is antipalindromic and of even length then n(X) odd length n > 1 can have the Q8-property. However, many is palindromic. Williamson sequences in even lengths have the Q8-property. 8) If X is symmetric and Y is palindromic then X x Y is Exhaustive searches [34] have found Williamson sequences symmetric. 5 with the Q8-property in all even lengths n ≤ 2 except for 6, Our main construction for Williamson sequences is given 12, and 28. by the following theorem. Barrera Acevedo and Dietrich also use the periodic product Theorem 3. If (A, B, C, D) are Williamson sequences of construction of Lüke [41] that generates perfect sequences 0 0 0 0 from shorter perfect sequences. Let X × Y be the sequence even length n and (A ,B ,C ,D ) are antipalindromic nega Williamson sequences of length n then whose rth entry is xr mod nyr mod m for 0 ≤ r < nm (where X has length n and Y has length m). (d(A) x n(A0), d(B) x n(B0), d(C) x n(C0), d(D) x n(D0)) Theorem 2. Suppose X and Y have coprime lengths n and m. are Williamson sequences of length 4n. If X is a perfect Q8-sequence and Y is a perfect Q+-sequence This theorem can be used to construct a large number of new then X × Y is a perfect Q+-sequence. Furthermore, if X and Y are symmetric then X × Y is symmetric. Williamson sequences, assuming that sequences that satisfy the preconditions are known. For example, if N Williamson Theorems 1 and 2 immediately yield the following corollary. sequences of length n are known and M antipalindromic nega t Williamson sequences of length n are known this theorem Corollary 1. If a perfect symmetric Q8-sequence of length 2 exists and Williamson sequences of odd length n exist then immediately implies that at least NM Williamson sequences Williamson sequences of length 2tn exist. of length 4n exist. We now prove Theorem 3. Barrera Acevedo and Dietrich also provide examples of Proof. Let X := d(X) x n(X0) for X ∈ {A, B, C, D}. symmetric perfect Q8-sequences for all t ≤ 6. In this paper new we extend their result by showing that perfect symmetric By construction it is clear that Xnew is a {±1}-sequence of Q -sequences of length 2t exist for all t ≥ 0 and therefore length 4n. Additionally, Xnew is symmetric by property 8 since 8 0 show Williamson sequences exist in all lengths of the form 2tn d(X) is symmetric by property 6 and n(X ) is palindromic whenever Williamson sequences exist in odd length n. by property 7. It remains to show that

Example 4. Using the Barrera Acevedo–Dietrich correspon- RAnew (t)+RBnew (t)+RCnew (t)+RDnew (t) = 0 for 1 ≤ t < 4n. dence the Williamson sequences (++-+, ++-+, ++-+, ++-+) Note that by properties 4 and 5 we have RX (t) is produce the perfect Q -sequence [--+-], the Williamson new 8 ( sequences (++--+, -+--+, -++++, -++++) produce the per- Rd(X)(t/2) + Rn(X0)(t/2) if t is even, fect Q+-sequence [q-jj-] and the Williamson sequences t−1 t+1 Rd(X),n(X0) 2 + Rn(X0),d(X) 2 if t is odd. (++--+--+, ++--+--+, +++-+-++, +++-+-++) produce R (t) = 0 t t the perfect Q8-sequence [--j+-+j-]. (We denote i, j, k, By property 3 we have Xnew when is odd. When and q by i, j, k, and q.) is even by properties 1 and 2 we have

ˆ 0 RXnew (t) = 2RX (t/2) + 2RX (t/2). III.CONSTRUCTIONSFOR WILLIAMSONSEQUENCES When t = 2n we have RX (t/2) = n and RˆX0 (t/2) = −n Our main construction is based on the following simple 0 (because X and X both have length n) so RXnew (t) = 0 in sequence operations. this case. Otherwise we have 1) The doubling of X, denoted by d(X), i.e., d(X) := X X RX (t/2) = 0 and RˆX0 (t/2) = 0 [x0, . . . , xn−1, x0, . . . , xn−1]. 2) The negadoubling of X, denoted by n(X), i.e., n(X) := X=A,B,C,D X=A,B,C,D [x0, . . . , xn−1, −x0,..., −xn−1]. for even t 6= 2n with 1 ≤ t < 4n since (A, B, C, D) 3) The interleaving of X and Y , denoted by X x Y , i.e., are Williamson sequences and (A0,B0,C0,D0) are nega P X x Y := [x0, y0, x1, y1, . . . , xn−1, yn−1]. Williamson sequences. It follows that X=A,B,C,D RXnew (t) = We will use the following properties of these operations in 0, as required. our construction. For completeness, proofs of these properties Example 5. Using the set of Williamson sequences are given in the appendix. In each property X and Y are (++-+, ++-+, ++-+, ++-+) and the set of antipalindromic arbitrary sequences of the same length and t is an arbitrary nega Williamson sequences (+-+-, +-+-, ++--, ++--) in integer. Theorem 3 produces the set of Williamson sequences 1) R (t) = 2R (t). d(X) X (+++--++-+-++--++, +++--++-+-++--++, 2) R (t) = 2Rˆ (t). n(X) X ++++--+-+-+--+++, ++++--+-+-+--+++). 3) Rd(X),n(Y )(t) = 0 and Rn(Y ),d(X)(t) = 0. 4) RXxY (2t) = RX (t) + RY (t). Note that the assumption that n is even is essential to 5) RXxY (2t + 1) = RX,Y (t) + RY,X (t + 1). the theorem. If n is odd the constructed sequences will not 5

be symmetric. Additionally, antipalindromic nega Williamson IV. NEGA WILLIAMSONSEQUENCESANDODDPERFECT sequences do not exist in odd lengths n > 1, as we now show. SEQUENCES

Lemma 2. Antipalindromic nega Williamson sequences do not Although antipalindromic nega Williamson sequences do not exist in odd lengths except for n = 1. exist in odd lengths larger than 1 we now present constructions Proof. Let (A, B, C, D) be a hypothetical set of antipalin- showing that they exist in many even lengths. First, note that dromic nega Williamson sequences of odd length n. Note that when the length is even there is a correspondence between a sequence X of odd length n is antipalindromic if and only if palindromic and antipalindromic nega Williamson sequences. In X0 := (−1)∗X is antipalindromic. By Lemma 1, it follows that other words, to use Theorem 3 it is sufficient to find palindromic (A0,B0,C0,D0) are antipalindromic periodic complementary nega Williamson sequences. sequences. If sum(X) denotes the row sum of X it is well- Lemma 3. There is a one-to-one correspondence between known (e.g., via the Wiener–Khinchin theorem or sequence palindromic and antipalindromic nega Williamson sequences compression [42]) that in even lengths. 0 2 0 2 0 2 0 2 sum(A ) + sum(B ) + sum(C ) + sum(D ) = 4n. Proof. Applying n/2 negacyclic shifts to each sequence in a set However, if X0 is antipalindromic we have sum(X0) = ±1 of palindromic nega Williamson sequences of even length n and the above sum of squared row sums must be equal to four, yields a set of antipalindromic nega Williamson sequences. implying that n = 1. Similarly, a set of palindromic nega Williamson sequences can be produced from a set of antipalindromic nega Williamson Although the construction in Theorem 3 requires sequences sequences by applying the negacyclic shift operator n/2 times. of even lengths there is a variant of the construction that uses sequences of odd lengths. The proof is similar and uses the following additional properties. Example 7. (+--++-, +-+-+-, ++-+--, +++---) are an- 9) If X is antisymmetric and of odd length then n(X) is tipalindromic nega Williamson sequences that may be con- symmetric. verted into the palindromic nega Williamson sequences 10) If X is palindromic then d(X) is palindromic. (--++--, +-++-+, -++++-, ++++++) using the translation Theorem 4. Let n be odd and let (A, B, C, D) and in Lemma 3. 0 0 0 0 (A ,B ,C ,D ) be quadruples of sequences of length n. We now provide a construction that shows that infinitely If (A, B, C, D) is the result of applying (n − 1)/2 cyclic many palindromic nega Williamson sequences exist. Recall that shifts to each member in a set of Williamson sequences and X˜ denotes the reverse of the sequence X and [X; Y ] denotes 0 0 0 0 (A ,B ,C ,D ) is the result of applying (n + 1)/2 negacyclic the concatenation of X and Y . Note that if X and Y have shifts to each member in a set of palindromic nega Williamson ∗ length n then C[X;Y ](t) = CX (t) + CY (t) + CY,X (n − t) sequences then and C[X;Y ](2n − t) = CX,Y (n − t) for 0 ≤ t ≤ n. First we (n(A0) x d(A), n(B0) x d(B), n(C0) x d(C), n(D0) x d(D)) prove a simple lemma. are Williamson sequences of length 4n. Lemma 4. Applying the negadoubling operator to the sequences in a complementary set produces a negacomplementary Proof. Since a symmetric sequence of odd length n becomes set. palindromic after (n − 1)/2 cyclic shifts, (A, B, C, D) are palindromic. Since a palindromic sequence of odd length n Proof. Note that if X is a sequence of length n then ˆ becomes antisymmetric after (n + 1)/2 negacyclic shifts, Rn(X)(t) = 2CX (t) for 0 ≤ t ≤ n; by definition we have 0 0 0 0 ˆ ∗ (A ,B ,C ,D ) are antisymmetric. Rn(X)(t) = Cn(X)(t) − Cn(X)(2n − t) and when 0 ≤ t ≤ n 0 ∗ Let Xnew := n(X ) x d(X) for X ∈ {A, B, C, D}. By we have Cn(X)(t) = CX (t) + C−X (t) + C−X,X (n − t) = ∗ construction it is clear that Xnew is a {±1}-sequence of 2CX (t) − CX (n − t) and Cn(X)(2n − t) = CX,−X (n − t) = ˆ ∗ length 4n. Additionally, Xnew is symmetric by property 8 since −CX (n − t). Thus Rn(X) = 2CX (t) − CX (n − t) + CX (n − 0 ∗ n(X ) is symmetric by property 9 and d(X) is palindromic t) = 2CX (t). by property 10. The remainder of the proof now proceeds as Suppose S is a complementary set of sequences of P ˆ in the proof of Theorem 3. length n. Using the above property we have A∈S Rn(A)(t) = 2 P C (t) for all 0 ≤ t ≤ n. Since S is complementary Example 6. Note (++--+, -+--+, -++++, -++++) are A∈S A this implies P Rˆ (t) = 0 for all 1 ≤ t ≤ n. Using the Williamson sequences and (+---+, ++-++, +---+, +++++) A∈S n(A) ˆ ˆ ∗ are palindromic nega Williamson sequences. Then us- symmetry RX (t) = RX (−t) shows that this also holds for ing (A, B, C, D) = (-+++-, -+-+-, ++-++, ++-++) and all n < t < 2n. (A0,B0,C0,D0) = (++-+-, +--++, ++-+-, ---++) in The- orem 4 produces the following Williamson sequences of Example 8. Using the set of four complementary se- length 20: quences (+++, +--, +-+, ++-) of length three with Lemma 4 produces the set of four negacomplementary sequences (+-++-+++-----+++-++-, +--+--+++---+++--+--, (+++---, +---++, +-+-+-, ++---+) of length six. ++++--++-+-+-++--+++, -+-+--+++++++++--+-+). 6

Theorem 5. If A and B are complementary {±1}-sequences Golay sequences, originally defined by Golay [44], are (i.e., (A, B) is a Golay pair) then known to exist in lengths 2, 10, and 26 [45]. Since Turyn has shown that Golay pairs in lengths n and m can be composed ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ([A; B; B; A], [B; A; A; B], [−B; A; A; −B], [−A; B; B; −A]) to form Golay pairs in length nm [46] they also exist in all a b c and lengths of the form 2 5 13 with a ≥ b + c. Thus, Theorem 5 implies that palindromic nega Williamson sequences with the a b c ([A; B; B˜; A˜; A; B; B˜; A˜], Q8-property exist in all lengths of the form 2 5 13 with [A; B; −B˜; −A˜; −A; −B; B˜; A˜], a ≥ b + c + 2. In particular, taking b = c = 0 gives that palindromic nega Williamson sequences with the Q -property ˜ ˜ ˜ ˜ 8 [A; −B; B; −A; −A; B; −B; A], exist in all lengths that are powers of two (since palindromic [A; −B; −B˜; A˜; A; −B; −B˜; A˜]) nega Williamson sequences with the Q8-property of length 2 exist, see Example 3). These are not the only lengths in are sets of palindromic nega Williamson sequences with the which palindromic nega Williamson sequences exist, however. Q8-property. Palindromic nega Williamson sequences in many other lengths Proof. The fact that the sequences are palindromic is immediate may be constructed using Lüke’s product construction with from the manner in which they were constructed. Note that odd perfect sequences. (A,˜ B˜) is a Golay pair since (A, B) is a Golay pair. Thus First, note that in odd lengths there is an equivalence be- (A, B, B,˜ A˜) is a complementary set and since tween Williamson sequences and palindromic nega Williamson sequences. + + + + + + - - Lemma 5. There is a one-to-one correspondence between   + - + - palindromic nega Williamson sequences and Williamson se- + - - + quences in odd lengths. is an orthogonal matrix the sequences Proof. Let (A, B, C, D) be a set of Williamson sequences of odd length n. By Lemma 1, ((−1) ∗ A, (−1) ∗ B, (−1) ∗ ([A; B; B˜; A˜], [A; B; −B˜; −A˜], (∗) C, (−1) ∗ D) will be a set of nega Williamson sequences. [A; −B; B˜; −A˜], [A; −B; −B˜; A˜]) The sequences in this set will be antisymmetric since if X is symmetric and of odd length then (−1) ∗ X is antisymmetric. form a complementary set of sequences by [43, Thm. 7]. Since Applying (n − 1)/2 negacyclic shifts to each sequence in this they are complementary they are also negacomplementary and set produces a set of palindromic nega Williamson sequences. applying 2n negacyclic shifts (where A and B are of length n) Similarly, arbitrary palindromic nega Williamson sequences of to the second and third sequences and negating the fourth odd length can be transformed into Williamson sequences by shows the first set in the theorem is negacomplementary. applying the inverse of the above transformations. Since (∗) are complementary, by Lemma 4 applying the negadoubling operator to these sequences will produce negacomplementary sequences. In other words, Example 10. The set of Williamson sequences (-++--++, ---++--, -+----+, -+----+) generates the set of palin- ˜ ˜ ˜ ˜ ([A; B; B; A; −A; −B; −B; −A], dromic nega Williamson sequences (++---++, --+-+--, [A; B; −B˜; −A˜; −A; −B; B˜; A˜], +-----+, +-----+) and vice versa using the transformation [A; −B; B˜; −A˜; −A; B; −B˜; A˜], in Lemma 5. [A; −B; −B˜; A˜; −A; B; B˜; −A˜]) Since Williamson sequences are known to exist for all lengths n < 35 this implies that palindromic nega Williamson are negacomplementary. Applying 4n negacyclic shifts (where sequences exist for all odd lengths up to 33. A and B are of length n) to the first and last sequences shows Next, we note that odd perfect sequences in even lengths the second set in the theorem is negacomplementary. can be composed with odd perfect sequences in odd lengths to Lastly, we show that the produced sequences X, Y , U, generate longer odd perfect sequences. Let X ×ˆ Y be the se- and V have the Q -property. In the first set we have x = a , 8 r r quence whose rth entry is (−1)br/nc+br/mcx y y = b , u = −y , and v = −x for all 0 ≤ r < n r mod n r mod m r n−r+1 r r r r for 0 ≤ r < nm (where X has length n and Y has length m). and in the second set we have xr = yr = ur = vr = ar for all 0 ≤ r < n. In each case xryrurvr = 1 for all 0 ≤ r < n Lemma 6. Suppose X and Y have coprime lengths n and m, and more generally we have xryrurvr = 1 for all 0 ≤ r < 4n one of which is even. If X is odd perfect and Y is odd perfect in the first set (and 0 ≤ r < 8n in the second set). then X ×ˆ Y is odd perfect. Furthermore, if X and Y are palindromic then X ×ˆ Y is antipalindromic. Example 9. Using the Golay pair (++, +-) the first set of palindromic nega Williamson sequences generated by Theorem 5 is Proof. The fact that X ×ˆ Y is odd perfect follows from ˆ ˆ ˆ (+++--+++, -++++++-, +-++++-+, --+--+--); the sec- RX×ˆ Y (t) = RX (t)RY (t) as given in [28, Eq. 9]. Sup- ond set is (+++--++++++--+++, +++-+------+-+++, pose that 0 ≤ r < nm is arbitrary. Since X and Y ++-+-+----+-+-++, ++-++-++++-++-++). are palindromic we have xnm−r−1 mod nynm−r−1 mod m = 7

xn−r−1 mod nym−r−1 mod m = xr mod nyr mod m. Thus the In particular, palindromic odd perfect sequences exist in all (nm − r − 1)th entry of X ×ˆ Y is lengths that are a power of two since Theorem 6 implies they a b c b(nm−r−1)/nc+b(nm−r−1)/mc exist in all lengths of the form 2 5 13 with a ≥ b + c + 3 (−1) xr mod nyr mod m and they exist in the lengths 2 and 4 as shown by the n+m+b−(r+1)/nc+b−(r+1)/mc = (−1) xr mod nyr mod m. examples [++] and [+ii+]. Using the fact that palindromic nega Williamson sequences exist in all odd lengths up to 33 Using the fact that b−(r + 1)/sc = −(br/sc + 1) for positive we used the Barrera Acevedo–Dietrich correspondence to integers r, s and that n + m is odd this becomes construct palindromic odd perfect sequences in all odd lengths 1+br/nc+br/mc (−1) xr mod nyr mod m up to 33. (This relies on the fact that a set of palindromic real sequences {A, B, C, D} are necessarily “nega-amicable” in which is the negative of the rth entry of X ×ˆ Y as required. that RˆX,Y (t) = RˆY,X (t) for all t and X,Y ∈ {A, B, C, D}.) Example 11. The palindromic odd perfect sequences X = Furthermore, using the fact that palindromic odd perfect [++] and Y = [+q+] used with Lemma 6 produces the Q8-sequences exist in all lengths that are powers of two we antipalindromic odd perfect sequence [+q-+Q-]. (We use Q used Lemma 6 to construct palindromic odd perfect sequences to denote −q.) in all even lengths up to 68 (see the appendix for an explicit list). The Barrera Acevedo–Dietrich correspondence applied to these We now show that infinitely many palindromic odd perfect sequences gives palindromic nega Williamson sequences in all Q8-sequences exist using a variant of Theorem 5 and the even lengths up to 68. It is conceivable that palindromic odd Barrera Acevedo–Dietrich correspondence. perfect sequences and palindromic nega Williamson sequences Theorem 6. If (A, B) is a Golay pair then actually exist in all even lengths. P := [−A; jB; kB˜; iA˜; iA; kB; jB˜; −A˜] V. PERFECTQUATERNIONSEQUENCES is a palindromic odd perfect Q8-sequence. We now use our constructions for Williamson sequences Proof. This follows by a direct but tedious calculation of and palindromic nega Williamson sequences to show that Rˆ (t); we give an example of how this may be done. Let P Williamson sequences and perfect Q -sequences exist in all P = [P 0; P˜0] have length 8n. We find for 0 ≤ t ≤ 4n that 8 lengths 2t with t ≥ 0. ˆ ∗ RP (t) = C[P 0;P˜0](t) − C[P 0;P˜0](8n − t) Theorem 7. Symmetric perfect sequences over Q8 exist for ∗ ∗ t = CP 0 (t) + CP˜0 (t) + CP˜0,P 0 (4n − t) − CP 0,P˜0 (4n − t) . all lengths 2 . ∗ t Suppose that 1 ≤ t ≤ n. Then CP 0 (t) = φ(t) + ψ(n − t) Proof. First, note that Golay sequences exist in all lengths 2 . where By Theorem 5 and Lemma 3 it follows that antipalindromic nega Williamson sequences with the Q -property exist in φ(t) := C (t) + C (t) + C (t) + C (t) 8 −A jB kB˜ iA˜ all lengths 2t for t ≥ 2 (they also exist for smaller t, see ψ(t) := CjB,−A(t) + CkB,jB˜ (t) + CiA,k˜ B˜ (t). Example 3). Theorem 3 then implies that if Williamson sequences with the Q -property exist in length 2t they also Since multiplying a sequence by a constant does not change 8 exist in length 2t+2 for all t ≥ 1. Additionally, it is known that its autocorrelation values and since (A, B, A,˜ B˜) are comple- Williamson sequences with the Q -property exist in lengths 2 mentary we find that φ(t) = 0. Furthermore, using the fact 8 and 4 (see below). By induction, Williamson sequences with C (t) = xy∗C (t) and C (t) = C (t) (since A xA,yB A,B A,˜ B˜ B,A the Q -property exist in all lengths 2t for t ≥ 0. By the and B have real entries) we find 8 Barrera Acevedo–Dietrich correspondence symmetric perfect t ψ(t) = −jCB,A(t) + iCB,B˜ (t) + jCB,A(t) = iCB,B˜ (t). Q8-sequences exist in all lengths 2 as well. ∗ Additionally, we have C 0 (t) = C 0 (t) , C 0 0 (4n − t) = P˜ P P ,P˜ Example 13. We use the base Golay pair (+, +), the C (n − t) = C (n − t), and C 0 0 (4n − t) = −A,−A˜ A,A˜ P˜ ,P base Williamson sequences (+, +, +, +), (+-, +-, ++, ++), and C (n − t) = C (n − t). Then iA,iA˜ A,A˜ the base nega Williamson sequences (++, ++, ++, ++). Ad- ∗ ditionally, we use Golay’s interleaving doubling construc- RˆP (t) = (iC ˜ (n − t)) + iC ˜ (n − t) B,B B,B tion [44] to generate larger Golay pairs via the mapping + C (n − t)∗ − C (n − t)∗ = 0. A,A˜ A,A˜ (A, B) 7→ (A x B,A x −B). From Theorem 4 we generate the Williamson sequences (++-+, ++-+, ++-+, ++-+), from Similarly one can show RˆP (t) = 0 for n < t ≤ 4n from which the symmetry Rˆ (t) = Rˆ (−t)∗ implies Rˆ (t) = 0 Theorem 5 and Lemma 3 we generate the antipalindromic X X P ++++ ++++ -++- -++- for all 1 ≤ t < 8n. nega Williamson sequences ( , , , ), and from Theorem 3 we generate the Williamson sequences Example 12. Using the Golay pair (++, +-) with The- (+-+++++-, +-+++++-, +--+++--, +--+++--). Continu- orem 6 produces the palindromic odd perfect sequence ing in this way and using the Barrera Acevedo–Dietrich t [--jJKkiiiikKJj--]. (We denote −i by I, −j by J, and correspondence produces perfect Q8-sequences of lengths 2 −k by K.) for all t ≥ 0. We denote −i by I, −j by J, and −k by K and 8

∗ explicitly give the sequences produced with this construction We have RˆB0,B00 (t) = CB0,B00 (t) − CB00,B0 (n/8 − t) by deﬁ- for t ≤ 7: nition. Suppose that 0 ≤ t ≤ n/32 so that CB00,B0 (n/8 − t) = C (n/32 − t) = −C (n/32 − t) and C 0 00 (t) = - -j --+- -+j---j+ -+-J+j-----j+J-+ kE,−kE˜ E,E˜ B ,B [ ], [ ], [ ], [ ], [ ] φ(t) + ψ(n/32 − t)∗ where [-i++jJ-K-k-jj-+I-I+-jj-k-K-Jj++i], φ(t) := CiD,kE(t)+CjE,˜ −D˜ (t)+CD,−jE(t)+C−kE,˜ −iD˜ (t),

[-+++-iJI+kjK-J-J-j-j-kjK+iJI--+- ψ(t) := C−D,iD˜ (t) + C−jE,jE˜ (t) + C−iD,D˜ (t). --+--IJi+Kjk-j-j-J-J-Kjk+IJi-+++], Note that CE,˜ D˜ (t) = CD,E(t) since D and E contain real [-iii+i+Ij+J+--K+-Jkj-JjJjk-K+KIK entries. Then -kIk+k-Kjjjj-Jkj--K+--J-ji+I+IiI φ(t) = jCD,E(t) − jCE,˜ D˜ (t) + jCD,E(t) − jCE,˜ D˜ (t) = 0, -IiI+I+ij-J--+K--jkJ-jjjjK-k+kIk ψ(t) = −iCD,D˜ (t) − CE,E˜ (t) + iCD,D˜ (t) = −CE,E˜ (t). -KIK+K-kjJjJ-jkJ-+K--+J+jI+i+iii] ∗ Finally, RM ,M (t) = 2RˆB0,B00 (t) = 2 −C ˜ (n/32 − t) − The perfect sequences generated by Theorem 7 for t ≥ 7 are 1 3 E,E (−C (n/32 − t))∗ = 0 for 0 ≤ t ≤ n/32. A similar counterexamples to the quaternionic form of Mow’s conjecture E,E˜ calculation shows Rˆ 0 00 (t) = 0 for n/32 < t < n/8 from presented by Blake [25] because they can be generated using B ,B which it follows that R (t) = 0 for all t. an orthogonal matrix construction, as we now show. M1,M3 Theorem 8. Let n ≥ 32 and let M be the (n/4) × 4 matrix Example 14. For n = 16 we use the perfect sequence found containing the entries of a perfect sequence P generated in Example 13 and the matrix it generates has the array by Theorem 7 using the second set from Theorem 5. Write orthogonality property as well. Otherwise we give the matrices the entries of P in M from left to right and top to bottom constructed using Theorem 8 for n = 32, 64, and 128. The transpose of the matrices are displayed to save space. (i.e., Mi,j = P4i+j). Then the columns of M have the array orthogonality property. -+-+ -j-j-j-j --+---+---+---+- +j-J IJ-kij+K IKj+-jKiikJ-+JkI Proof. Let M0, M1, M2, M3 denote the columns of M as       3 ---- +--++--+ +Jj--jJ++Jj--jJ+ sequences. We must show that P R (t) = 0 for all 1 ≤       r=0 Mr J-j+ K+jik-JI IkJ+-JkiiKj-+jKI t < n/4 and that RMr ,Ms (t) = 0 for all t and 0 ≤ r < s < 4. -+j---j+-+j---j+-+j---j+-+j---j+ By construction P = (M0 x M2) x (M1 x M3) is a perfect sequence and therefore IIKKJj-+--jjkKIiiikkjJ+-++JJKkiI   IKJ+-jkiikj-+JKIIKJ+-jkiikj-+JKI R(M0xM2)x(M1xM3)(t) = 0 for all 1 ≤ t < n. IikKJJ++-+JjkkiiiIKkjj--+-jJKKII

By property 4 in Section III this implies RM0xM2 (t) + P3 VI.PREVIOUS WORK RM1xM3 (t) = 0 for all 1 ≤ t < n/2 and r=0 RMr (t) = 0 for all 1 ≤ t < n/4. Despite an enormous amount of work the conjecture that the Since P was generated by applying the Barrera Acevedo– longest perfect sequences over the complex nth roots of unity Dietrich correspondence to the sequences generated in Theo- have length n2 remains open, though some special cases of this rem 3 we have P = d(A) x n(B) for some perfect symmetric conjecture have been resolved. In the case n = 2, Turyn [26] sequence A and antipalindromic sequence B of even length n/4. showed that any perfect binary sequence of length longer than Let X0 denote the sequence formed by the even entries of X four must have length 4m2 for odd m that is not a prime and let X00 denote the sequence formed by the odd entries power. Despite this progress the general conjecture even for of X. Then we have n = 2 remains open [47]. In the case n = 4, the conjecture states that the longest perfect sequence over {±1, ±i} has 0 0 00 00 M0 = d(A ),M1 = n(B ),M2 = d(A ),M3 = n(B ). length 16 and Turyn [27] showed that the length of any longer k By property 3 of Section III this representation yields perfect sequence cannot be of the form 2p for any prime p and integer k. Most recently, Ma and Ng [48] showed many RM ,M (t) = RM ,M (t) = RM ,M (t) = RM ,M (t) = 0 0 1 0 3 1 2 2 3 restrictions on the length of perfect sequences over the pth for all t and only the crosscorrelation of the pairs (M0,M2) 0 00 roots of unity for prime p. In particular, they showed that no and (M1,M3) are left to consider. Since A = A x A is k+2 k+3 perfect and was generated by applying Theorem 3 we have perfect sequences of length 2p or p exist for k ≥ 0. 0 00 Several other constructions for perfect sequences over that A is of the form d(C1) for some C1 and A is of the complex roots of unity have been found since the construction form n(C2) for some C2. Property 3 of Section III then yields of Frank and Heimiller. In 1972, Chu described a method [49] RM ,M (t) = 2RA0,A00 (t) = 2Rd(C ),n(C )(t) = 0 for all t. 0 2 1 2 of producing perfect sequences of any length n. Shortly after Lastly, we must show RM1,M3 (t) = 0. Suppose the antipalindromic B was generated from Theorem 5 using the Golay Chu’s paper was published, Frank published a response [50] pair (D,E). An analysis of the Barrera Acevedo–Dietrich pointing out that Zadoff had discovered the same construction correspondence shows that we have and had been granted a patent for a communication system based on his construction [51]. Other variants of Zadoff and B0 = [iD; jE˜; D; −kE˜],B00 = [kE; −D˜; −jE; −iD˜]. Chu’s sequences have been described by Alltop [52] and Lewis 9

and Kretschmer [53]. In 1983, Milewski [54] found a new constructions for sequences that are optimal, i.e., ones that construction for perfect sequences of length n2t+1 over nt+1th meet the bound exactly. roots of unity for all n, t ≥ 1. In 2004, Liu and Fan [55] found Williamson sequences have been quite well-studied since a new construction for perfect sequences of length n over nth introduced by Williamson 75 years ago [31]. They are often roots of unity when n is a multiple of four. In 2014, Blake presented using matrix notation and known as “Williamson and Tirkel [56] gave a construction for perfect sequences of matrices” since one of their traditional applications has been to length 4mnt+1 over 2mntth roots of unity for m, n, t ≥ 1. construct Hadamard matrices. In this formulation Williamson Blake has run extensive searches for perfect sequences over matrices are defined to be circulant (i.e., each row is the cyclic complex roots of unity and quaternions [25] and has found a shift of the previous row) and Williamson sequences are simply number of perfect sequences over the nth roots of unity that can- the first rows of Williamson matrices. Baumert and Hall [68] not be generated using matrices with array orthogonality (like performed an exhaustive search for Williamson sequences of those from the Frank–Heimiller and Mow constructions). The odd length n ≤ 23 and presented a doubling construction for a sequences that he found are counterexamples to the conjecture generalization of Williamson matrices that are symmetric but that Mow’s unified construction produces all perfect sequences not circulant. Turyn [69] constructed Williamson sequences over nth roots of unity [16] but are shorter than n2 and thus are in all lengths (q + 1)/2 where q ≡ 1 (mod 4) is a prime not counterexamples to Mow’s original conjecture [57] (also power, Whiteman [70] constructed Williamson sequences in all generalized by Lüke, Schotten, and Hadinejad-Mahram [28] lengths q(q+1)/2 where q ≡ 1 (mod 4) is a prime power, and to odd perfect sequences). Spence [71] constructed Williamson sequences in all lengths t Apparently no infinite family of odd perfect Q8-sequences q (q + 1)/2 where q ≡ 1 (mod 4) is a prime power and t ≥ 0. has been previously constructed.2 It is known that no infinite Computer searches have determined that Williamson se- family of odd perfect sequences can exist over the alpha- quences in odd lengths are particularly rare. Following Baumert bet {±1}—Lüke, Schotten, and Hadinejad-Mahram show that and Hall’s exhaustive search, Baumert [72] found a new set the longest odd perfect {±1}-sequence has length two and of Williamson sequences in length 29, Sawade [73] found they conjecture that the longest odd perfect sequence over the eight new sets in lengths 25 and 27, Yamada [74] found one alphabet {±1, ±i} has length four [28]. However, Lüke has con- new set in length 37, Koukouvinos and Kounias [75] found structed almost perfect and odd perfect {±1, ±i}-sequences A four new sets in length 33, Ðokovic´ [76]–[78] found six new (with RA(t) = 0 or RÂ(t) = 0 for all 1 ≤ t < n except sets in the lengths 25, 31, 33, 37, and 39, van Vliet found t = n/2) in all lengths q + 1 where q ≡ 1 (mod 4) is a prime one new set in length 51 (unpublished but appears in [33]), power [58]. Holzmann, Kharaghani, and Tayfeh-Rezaie [33] found one Some work has also been done on perfect quaternion arrays. new set in length 43, and Bright, Kotsireas, and Ganesh [34] In 2013, Barrera Acevedo and Jolly [59] constructed perfect found one new set in length 63. Ðokovic´ [77] also found that Q8-arrays of size 2 × (p + 1)/2 for primes p ≡ 1 (mod 4). no Williamson sequences exist in length 35 and Holzmann, Furthermore, they extended a construction of Arasu and de Kharaghani, and Tayfeh-Rezaie [33] found that no Williamson Launey [60] to produce perfect Q8-arrays of size 2p×p(p+1)/2 sequences exist in lengths 47, 53, and 59. for primes p ≡ 1 (mod 4). Additionally, Blake [25] found a In even lengths Williamson sequences are much more t t construction for perfect Q8-arrays of size 2 × 2 with 2 ≤ common, as first shown by an exhaustive search up to length 18 t < 7. by Kotsireas and Koukouvinos [79]. Non-exhaustive searches There does not seem to be much prior work on nega were performed up to length 34 by Bright et al. [80], and up Williamson sequences, though Xia, Xia, Seberry, and Wu [38] to length 42 by Zulkoski et al. [81]. Bright [82] completed an study them under the name “4-suitable negacyclic matrices”. exhaustive search up to length 44 and Bright, Kotsireas, and They give constructions for them in terms of Golay pairs, Ganesh [34] completed an exhaustive search in the even lengths Williamson sequences, and base sequences. However, we up to 70. Barrera Acevedo and Dietrich’s correspondence [29] could find no prior constructions that specifically generated can be used to generate Williamson sequences in many lengths palindromic or antipalindromic nega Williamson sequences. including 70. Lüke presents a method [61] of constructing pairs of negacomplementary binary sequences, also studied under the name VII.CONCLUSION “associated pairs” by Ito [62] and “negaperiodic Golay pairs” We have shown that perfect and odd perfect Q -sequences ´ 8 by Balonin and Ðokovic [63]. Lüke and Schotten [64] also exist in all lengths that are a power of two. Barrera Acevedo give a construction for odd perfect almost binary sequences and Dietrich [30] summarize the knowledge (as of 2018) of that Wen, Hu, and Jin [65] use to construct negacomplementary perfect Q -sequences as follows: binary sequences. Jin et al. [66] give necessary conditions for 8 the existence of negacomplementary sequences and a doubling Currently there exists only one infinite family of construction for negacomplementary sequences. Yang, Tang, perfect sequences over the quaternions (of magnitude and Zhou [67] show that a {±1}-sequence A of length n one). . . ˆ satisfies max1≤t

the second known infinite family of perfect Q8-sequences. [4] N. Suehiro and M. Hatori, “Modulatable orthogonal sequences and their application to SSMA systems,” IEEE Transactions on Information Theory, Additionally, our construction for odd perfect Q8-sequences vol. 34, no. 1, pp. 93–100, 1988. is the first known infinite family of odd perfect Q8-sequences. [5] F. F. Kretschmer and K. Gerlach, “Low sidelobe radar waveforms Because our perfect sequences can be constructed using derived from orthogonal matrices,” IEEE Transactions on Aerospace and matrices with array orthogonality (as shown in Theorem 8) Electronic Systems, vol. 27, no. 1, pp. 92–102, 1991. [6] S. Qureshi, “Fast start-up equalization with periodic training sequences,” they disprove Blake’s conjecture [25, Conjecture 8.2.1] that the IEEE Transactions on Information Theory, vol. 23, no. 5, pp. 553–563, longest perfect Q8-sequences generated from an orthogonal 1977. array construction have length 64. Theorem 8 also implies the [7] G. Gong, F. Huo, and Y. Yang, “Large zero autocorrelation zones of Golay t sequences and their applications,” IEEE Transactions on Communications, existence of perfect Q8-arrays of sizes 2 × 4 for all t ≥ 2 vol. 61, no. 9, pp. 3967–3979, 2013. and the construction of Barrera Acevedo and Jolly [59] then [8] Y. Rahmatallah and S. Mohan, “Peak-to-average power ratio reduction t implies the existence of perfect Q8-arrays of size 2 p × 4p in OFDM systems: A survey and taxonomy,” IEEE Communications when p = 2t+2 − 1 is prime. Surveys & Tutorials, vol. 15, no. 4, pp. 1567–1592. [9] A. Z. Tirkel, C. F. Osborne, and T. E. Hall, “Image and watermark Furthermore, we generalize Williamson’s doubling construc- registration,” Signal processing, vol. 66, no. 3, pp. 373–383, 1998. tion [31] from 1944, showing that the existence of Williamson [10] B. M. Popovic,´ “Complementary sets based on sequences with ideal sequences of odd length n implies not only the existence periodic autocorrelation,” Electronics Letters, vol. 26, no. 18, pp. 1428– 1430, 1990. of Williamson sequences of length 2n but also implies the [11] N. Zhang and S. W. Golomb, “Polyphase sequence with low autocorre- existence of Williamson sequences of length 2tn for all t ≥ 1. lations,” IEEE Transactions on Information Theory, vol. 39, no. 3, pp. We have also shown the importance of nega Williamson 1085–1089, 1993. [12] B. M. Popovic,´ “Generalized chirp-like polyphase sequences with sequences, a class of sequences defined by Xia et al. [38] optimum correlation properties,” IEEE Transactions on Information in 2006. In particular, we have demonstrated the importance Theory, vol. 38, no. 4, pp. 1406–1409, 1992. of palindromic nega Williamson sequences. [13] R. Heimiller, “Phase shift pulse codes with good periodic correlation properties,” IRE Transactions on Information Theory, vol. 7, no. 4, pp. Lastly, our constructions provide an explanation for the 254–257, 1961. abundance of Williamson sequences in lengths that are divisible [14] R. L. Frank and S. A. Zadoff, “Phase shift pulse codes with good periodic by a large power of two. Prior to this work it was noticed that correlation properties (corresp.),” IRE Transactions on Information Theory, vol. 8, no. 6, pp. 381–382, 1962. Williamson sequences are much more abundant in these lengths. [15] R. L. Frank, “Phase coded communication system,” Jul. 30 1963, US For example, fewer than 100 sets of Williamson sequences are Patent 3,099,795. known to exist in the odd lengths, but an exhaustive computer [16] W. H. 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Rajan, and V. Shashidhar, “Full-diversity, high-rate space-time block codes from division algebras,” IEEE Transactions on multiples of four. Information Theory, vol. 49, no. 10, pp. 2596–2616, 2003. It would also be interesting to find a construction that works [20] T. Bulow and G. Sommer, “Hypercomplex signals—a novel extension in the even lengths that are not multiples of four. Williamson’s of the analytic signal to the multidimensional case,” IEEE Transactions on signal processing, vol. 49, no. 11, pp. 2844–2852, 2001. doubling result can be used for lengths 2n but requires [21] T. A. Ell and S. J. Sangwine, “Hypercomplex Fourier transforms of color that Williamson sequences of odd length n exist. Barrera images,” IEEE Transactions on Image Processing, vol. 16, no. 1, pp. Acevedo and Dietrich’s correspondence can be used in certain 22–35, 2007. [22] O. 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APPENDIX Note that x2n−r mod n = xn−r mod n = xr mod n. Thus A. Proofs the (2n − r)th entry of d(X) is equal to the rth entry, as required. We now give proofs of the properties from Section III. In 7) If X is antipalindromic and of even length then n(X) each case we let X = [x , . . . , x ] and Y = [y , . . . , y ] 0 n−1 0 n−1 is palindromic. be arbitrary sequences of length n. Let Y be the ﬁrst half of X (i.e., X = [Y ; −Y˜ ]) so that 1) Rd(X)(t) = 2RX (t). n(X) = [Y ; −Y˜ ; −Y ; Y˜ ] is a palindrome. The rth entry of d(X) is xr mod n. Then 8) If X is symmetric and Y is palindromic then X x Y is 2n−1 symmetric. X ∗ Rd(X)(t) = xr mod nxr+t mod n = 2RX (t). First, we show the even entries of X x Y satisfy the r=0 symmetric property. The (2r)th entry of X x Y is xr ˆ and the (2n − 2r)th entry of X x Y is xn−r (for r 6= 0). 2) Rn(X)(t) = 2RX (t). br/nc Since X is symmetric xn−r = xr showing the (2r)th The rth entry of n(X) is (−1) xr mod n. Then and (2n − 2r)th entries are equal. Rn(X)(t) is Second, we show the odd entries of X x Y satisfy the 2n−1 X br/nc b(r+t)/nc ∗ symmetric property. The (2r +1)th entry of X xY is yr (−1) xr mod n(−1) x r+t mod n and the (2n−2r−1)the entry of X xY is yn−r−1. Since r=0 Y is palindromic y = y showing the (2r + 1)th n−1 n−r−1 r X and (2n − 2r − 1)th entries are equal. = x (−1)b(r+t)/ncx∗ r r+t mod n 9) If X is antisymmetric and of odd length then n(X) is r=0 n−1 symmetric. X + (−x )(−1)b(r+n+t)/ncx∗ Let Y be the ﬁrst half of the palindromic part r r+t mod n ˜ r=0 of X (i.e., X = [x0; Y ; −Y ]) so that n(X) = [x0; Y ; −Y˜ ; −x0; −Y ; Y˜ ] is symmetric. = 2RˆX (t). 10) If X is palindromic then d(X) is palindromic. 3) Rd(X),n(Y )(t) = 0 and Rn(Y ),d(X)(t) = 0. Note that x2n−r−1 mod n = xn−r−1 mod n = xr mod n. The rth entry of d(X) is xr mod n and the rth entry of Thus the (2n − r − 1)th entry of d(X) is equal to the br/nc n(Y ) is (−1) yr mod n. Then Rd(X),n(Y )(t) is rth entry, as required. 2n−1 X b(r+t)/nc ∗ xr mod n(−1) yr+t mod n B. List of odd perfect quaternion sequences r=0 We now give palindromic odd perfect Q+-sequences in all n−1 X lengths n < 70 except for 35, 47, 53, 59, 65, and 67. The = x (−1)b(r+t)/ncy∗ r r+t mod n sequences in odd lengths were constructed using Lemma 5 r=0 n−1 with a previously known set of Williamson sequences of X b(r+n+t)/nc ∗ length n. The sequences in even lengths were constructed + xr(−1) yr+t mod n r=0 using Lemma 6, Theorem 6, and Lemma 3 and are new to the best of our knowledge, though perfect quaternion sequences = Rˆ (t) − Rˆ (t) = 0. X,Y X,Y may be constructed in these lengths using the results of Barrera The second property follows because RX,Y (t) = Acevedo and Dietrich [29]. ∗ RY,X (−t) for all X, Y , and t. The sequences are denoted by Pn where n is the length of the 4) RXxY (2t) = RX (t) + RY (t). sequence. The symbols + and - denote 1 and −1, capitalization The (2r)th entry of X xY is xr and the (2r +1)th entry denotes negation of an entry, and an overlined entry denotes is yr. Then RXxY (2t) is left multiplication by q, i.e., the symbol I denotes the entry 2n−1 2n−1 −qi. X ∗ X ∗ P1 = [+] xr/2xr/2+t mod n + y(r−1)/2y(r−1)/2+t mod n P2 = [++] r=0 r=0 r even r odd P3 = [Iq-I] P4 = [+ii+] = RX (t) + RY (t). P5 = [J+q++J] P = [iq-IIq-i] 5) RXxY (2t + 1) = RX,Y (t) + RY,X (t + 1). 6 P7 = [j-Jq-J-j] The (2r)th entry of X xY is xr and the (2r +1)th entry P8 = [-jkiikj-] is yr. Then RX Y (2t + 1) is x P9 = [KjI+q++IjK] 2n−1 2n−1 P10 = [J-q-+JJ+q--J] X ∗ X ∗ P = [i+K-Jq-J-K+i] xr/2y + y(r−1)/2x 11 r/2+t mod n (r+1)/2+t mod n P = [+-i+KIIK+i-+] r=0 r=0 12 q q q q r even r odd P13 = [KiJKjiq+ijKJiK] = RX,Y (t) + RY,X (t + 1). P14 = [J-Jq+j-jj-jq+J-J] P15 = [Jj-JJ++q-++JJ-jJ] 6) If X is symmetric then d(X) is symmetric. P16 = [--jJKkiiiikKJj--] 13

P17 = [+iK++jjJq+Jjj++Ki+] P68 = [IIKiijJkqKjjki+k+I+IJI-jkKq+jKK-+j-++-j+-K P18 = [KJi+q+-ijKKji-q++iJK] Kjq+Kkj-IJI+I+k+ikjjqKkJjiiKII] P19 = [kKj-JJK++q-++KJJ-jKk] P69 = [-----j-JJj--+j-+-JJJj-j+Jj-jj-Jj+-q+-+jJ- P20 = [k-q+iKj+qkiJJiqk+jKiq+-k] jj-jJ+j-jJJJ-+-j+--jJJ-j-----] P21 = [++IIi++i+-q+-+i++iII++] P22 = [I+K+jq-J+k+ii+k+Jq-j+K+I] P23 = [j+k-i--KJiIq-IiJK--i-k+j] ACKNOWLEDGMENT P24 = [+qJkIq+kjqk-JqIiiqIJ-qkjkq+IkqJ+] The authors would like to thank the editors and reviewers P25 = [kJJIikk+j-+iq+i+-j+kkiIJJk] P26 = [KIjKjIq-ijkjiKKijkjiq-IjKjIK] for their careful proofreading of our work. P27 = [IikJII-j+K-JKq-KJ-K+j-IIJkiI] P28 = [k-jqkk+jKijq-KIjjIKq-jiKj+kqkj-k] P29 = [+iIIKkjJkIIKKKq+KKKIIkJjkKIIi+] Curtis Bright received a PhD in computer science from the University of P30 = [jj-jj++q+-+Jj+jJJj+jJ+-q+++jj-jj] Waterloo in 2017. He was a research intern at Maplesoft (2017, 2018) and a P31 = [-II+i-I---IiIiiq-iiIiI---I-i+II-] postdoctoral fellow at the University of Waterloo (2017–2019) and Carleton P32 = [---+jJjjkkKkIiiiiiiIkKkkjjJj+---] University (2020). He joined the faculty of the School of Computer Science at the University of Windsor in 2020 and starts as a tenure-track professor in 2021. P33 = [J+IjKKkjkk+Iik-iq+i-kiI+kkjkKKjI+J] +Ik++JJJ+jJj+-ki++ik-+jJj+JJJ++kI+ He is currently the lead developer of the MathCheck project for combining P34 = [ q q ] satisfiability solvers with computer algebra systems to solve mathematical P36 = [JJIiqk+iKJKJ-Iq-++kKKk++q-I-JKJKi+qkiIJJ] conjectures. His research interests include automated reasoning search algo- P37 = [jikIIJI+KkJ-JJ+-K+q++K-+JJ-JkK+IJIIkij] rithms, computer-assisted proofs, experimental mathematics, formal verification P38 = [KKj+jJK--q-+-kJJ+JKkkKJ+JJk-+q---KJj+jKK] or counterexample generation of conjectures, and discrete mathematics. He P39 = [Jkk-J+I+KkJ+++kjKJjq-jJKjk+++JkK+I+J-kkJ] proved that Williamson sequences exist in all lengths that are powers of two as a postdoctoral fellow at the University of Waterloo under the supervision of P40 = [kKqI+j-KqKII+-q+JIKiqjjjjjqjiKIJq+-+IIqKK-j+qIKk] iKkKikIIKKikiKiiIikk+kkiIiiKikiKKIIkiKkK professor Vijay Ganesh and as a research affiliate of Wilfrid Laurier University P41 = [ q under the supervision of professor Ilias Kotsireas. i] P42 = [+-iIi--i++q--+I-+iii++++iii+-I+-q-++i--iIi -+] P43 = [j-IJJjIKKK+j+k-+KijK-q+-KjiK+-k+j+KKKIjJJ Ilias Kotsireas (Member, IEEE) serves as a professor of Computer Science I-j] at Wilfrid Laurier University (Waterloo, Ontario, Canada) and Director of the P = [-+kik+JIj+I-iK+kkj-jiiiij-jkk+Ki-I+jIJ+k CARGO Lab. He has over 190 refereed journal and conference publications, 44 q q q q chapters in books, edited books and special issues of journals, in the research ik+-] areas of Computational Algebra, Metaheuristics, High-Performance Computing, P45 = [kkkKkIiKIkikKKIKIiKKiiq+iiKKiIKIKKkikIKiI Dynamical Systems and Combinatorial Design Theory. He serves on the kKkkk] Editorial Board of 8 international journals. He serves as the Managing P46 = [J+k+I--kjiIq+iiJk+-i+K+jj+K+i-+kJiiq+Iijk- Editor of 2 Springer journals and as Editor in Chief of a Springer journal -I+k+J] and a Birkhauser book series. He has organized a very large number of international conferences in Europe, North America and Asia, often serving as P48 = [+qKJJqIKiq+IiqIKJqj-+qK+jqJkkq+iiq+kkqJj+qK+-qjJKqIiI q+iKqIJJqK+] a Program Committee Chair or General Chair. His research work has gathered approximately 1200 citations. His research is and has been funded by NSERC, P49 = [kkiIIkIIiIIIkiKkikkIIKkKq+KkKIIkkikKikIII the European Union and the National Natural Science Foundation of China iIIkIIikk] (NSFC). He has received funding for conference organization from Maplesoft, P50 = [kjjIiKK+j+-iq+I--j-KkiijJkkJjiikK-j--Iq+i- the Fields Institute, and several Wilfrid Laurier University offices. He served as +j+KKiIjjk] Chair of the ACM Special Interest Group on Symbolic Computation (SIGSAM) P51 = [JJ-j+-+-j++j+J+++j-jJ--+jq+j+--Jj-j+++J+j on a 4-year term: July 2013 to July 2017. He is a co-chair of the Applications ++j-+-+j-JJ] of Computer Algebra Working Group (ACA WG), a group of 45 internationally P = [JIJjkiq-+KkJ-jkiKjjIqK+JKK-KK-KKJ+qKIjjKikj renowned researchers that oversee the organization of the ACA conference 52 series. He has delivered more than 10 invited/plenary talks at conferences -JkK+q-ikjJIJ] around the world. P54 = [iikjiI-J-K-jkq-Kj+K+J+IIjKiIIiKjII+J+K+jK q-kj-K-J-Iijkii] P55 = [JJJ++jJ++-+-++jj-j+-J+--j+jq+j+j--+J-+j-j j++-+-++Jj++JJJ] Vijay Ganesh is an associate professor at the University of Waterloo. Prior P = [kK++JjikIiIj-+IIKJ++j-KKKiJJJJiKKK-j++JK to joining Waterloo in 2012, he was a research scientist at MIT (2007–2012) 56 q q q q q q and completed his PhD in computer science from Stanford University in 2007. II+-jqIiIkijJq++Kk] Vijay’s primary area of research is the theory and practice of automated P57 = [Ii++I+-++-iiI---+i+I-i--II-Iq+I-II--i-I+i reasoning aimed at software engineering, formal methods, security, and +---Iii-++-+I++iI] mathematics. In this context he has led the development of many SAT/SMT P58 = [+IiIKKJJkiiKKkq-KKkiIkjJkKiii++iiiKkJjkIi solvers, most notably, STP, the Z3 string solver, MapleSAT, and MathCheck. kKKq-kKKiikJJKKIiI+] He has also proved several decidability and complexity results in the context of first-order theories. He has won over 25 research awards, best paper awards, P60 = [Kj+kk-+qKi+jkiJJkK+JKi+q+IIjJIkJJkIJjIIq++i KJ+KkJJikj+iK+-kk+jK distinctions, and medals for his research to-date. He recently won an ACM q ] ISSTA Impact Paper Award 2019 (Best Paper in 10 Years Award at ISSTA P61 = [KkKKIiKiKIIkkKiIIKIIKkiKKiiiIKq+KIiiiKKik conference), an ACM Test of Time Award at CCS 2016 (Best Paper in 10 KIIKIIiKkkIIKiKiIKKkK] Years Award at CCS), the Early Researcher Award (ERA) given by the Ontario P62 = [+II-I-I++-IIiiiq+IiIIi--+i-i-iI--Ii-i-i+- Government in 2016, Outstanding Paper Award at ACSAC 2016, an IBM -iIIiIq+iiiII-++I-I-II+] Research Faculty Award in 2015, two Google Research Faculty Awards in 2013 and 2011, a Ten-Year Most Influential paper citation at DATE 2008, and 10 P63 = [-iiiIi+jjjJ+kiKJ-Kk-ik-Ij---+iIq-Ii+---jI -ki-kK-JKik+Jjjj+iIiii-] best paper awards/honors of different kinds at conferences like CAV, IJCAI, ---+--+-jJjjjJJJKKKkkkKkiIiiIiiiiiiIiiIi CADE, ISSTA, SAT, SPLC, and CCS. His solvers STP and MapleSAT have P64 = [ won numerous awards at the highly competitive international SMT and SAT kKkkkKKKJJJjjjJj-+--+---] solver competitions. In 2013 he was invited to the first Heidelberg Laureate P66 = [J-ijKkKjkK-IiK+iq+I+kii-kkJKKKJi+JJ+iJKKK Forum, a gathering where a select group of young researchers from around Jkk-iik+Iq+i+KiI-KkjKkKji-J] the world met with Turing, Fields and Abel Laureates.