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Perfect Sequences Over the Quaternions and (4N, 2, 4N, 2N)-Relative Difference Sets in Cn × Q8

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Perfect Sequences Over the Quaternions and (4N, 2, 4N, 2N)-Relative Difference Sets in Cn × Q8 Perfect Sequences over the Quaternions and (4n; 2; 4n; 2n)-Relative Difference Sets in Cn × Q8 Santiago Barrera Acevedo1? and Heiko Dietrich2? 1 Monash University, School of Mathematical Sciences, Clayton 3800 VIC, Australia [email protected] 2 Monash University, School of Mathematical Sciences, Clayton 3800 VIC, Australia [email protected] Abstract. Perfect sequences over general quaternions were introduced in 2009 by Kuznetsov. The existence of perfect sequences of increasing lengths over the basic quaternions Q8 = {±1; ±i; ±j; ±kg was established in 2012 by Barrera Acevedo and Hall. The aim of this paper is to prove a 1–1 correspondence be- tween perfect sequences of length n over Q8 [ qQ8 with q = (1 + i + j + k)=2, and (4n; 2; 4n; 2n)-relative difference sets in Cn × Q8 with forbidden subgroup a C2; here Cm is a cyclic group of order m. We show that if n = p + 1 for a prime p and integer a ≥ 0 with n ≡ 2 mod 4, then there exists a (4n; 2; 4n; 2n)- relative different set in Cn × Q8 with forbidden subgroup C2. Lastly, we show that every perfect sequence of length n over Q8 [ qQ8 yields a Hadamard matrix of order 4n (and a quaternionic Hadamard matrix of order n over Q8 [ qQ8). The final publication is available at Springer via http://link.springer.com/article/10.1007/s12095-017-0224-y 1 Introduction The periodic autocorrelation of a sequence is a measure for how much the sequence differs from its cyclic shifts. If the autocorrelation values for all nontrivial cyclic shifts are 0, then the sequence is perfect. It is well known that sequences with good autocor- relation properties, such as being perfect, have important applications in information technology, for example, in digital watermarking, frequency hopping patterns for radar, or sonar communications. However, it is very difficult to construct perfect sequences over 2nd-, 4th-, and in general over n-th roots of unity. In fact, it is conjectured that perfect sequences over n-th roots of unity do not exist for lengths greater that n2 [14]. Perfect sequences are closely related to relative difference sets and Hadamard matrices. Relative difference sets are important objects in combinatorial design theory and finite geometry, see [15, 16]. They have been studied extensively in abelian groups (see for example [5, 6] and their references). A Hadamard matrix H of order n is a an n × n matrix with entries in {±1g and HH| = nIn. Hadamard matrices have significant ap- plications, for example, in spectroscopy, object recognition, or coding of digital signals and encryption, see [9] for more details. Hadamard matrices exist for many orders, and it is the famous Hadamard Conjecture which claims that there is a Hadamard matrix ? This research was partly funded by ARC DECRA projects DE140100088 and DE140101201. of order 4n for all positive integers n. Starting 2001, Arasu, de Launey, and Ma [1, 3] established and studied a connection between perfect arrays of size m × n over 4th- roots of unity and (2nm; 2; 2nm; nm)-relative difference sets in Cn × Cm × C4 with forbidden subgroup of size 2. Such relative difference sets are now related to Hadamard matrices: Jungnickel [11] proved that every (n; 2; n; n=2)-relative difference set with forbidden subgroup of size 2 yields a Hadamard matrix of order n. We note that for every n divisible by 4, Ito [10] constructed a group and conjectured that it contains an (n; 2; n; n=2)-relative difference set. This conjecture has been verified for all n ≤ 42, see [18], and if it is true, then so is the Hadamard Conjecture. Due to the importance of perfect sequences and the difficulty to construct them over n-th roots of unity, there has been some focus on other classes of sequences with good autocorrelation. One of these classes has been introduced by Kuznetsov [13], who de- fined perfect sequences over the quaternion algebra. In 2012, Barrera Acevedo and Hall [4] constructed the first infinite family of perfect sequences of increasing lengths over the basic quaternions Q8 = {±1; ±i; ±j; ±kg. The aim of the present paper is to study such quaternionic sequences further and to exploit and establish relations with relative difference sets and Hadamard matrices. In the following let q = (1 + i + j + k)=2. Motivated by the result of Arasu et al., our main result is a 1–1 correspondence between perfect sequences of length n over the alphabet Q8 [ qQ8 and (4n; 2; 4n; 2n)-relative difference sets in Cn × Q8 with forbidden subgroup C2 ≤ Q8. We discuss these relative difference sets in Section 3, and then prove the 1–1 correspondence in Section 4. In Section 5, we comment on the well known connections between relative difference sets and Hadamard matrices. We show that every perfect sequence of length n over the alphabet Q8 [ qQ8 yields a quaternionic Hadamard matrix of order n and a Hadamard matrix of order 4n. We conclude in Section 6 with a discussion of possible future work. 2 Notation We denote by IR and C the field of real and complex numbers, respectively. For a posi- tive integer n we denote by Cn the cyclic group of order n. The quaternion algebra IH is the 4-dimensional real algebra with IR-basis f1; i; j; kg and non-commutative mul- tiplication defined by i2 = j2 = k2 = −1 and ij = k. It follows readily from these relations that jk = i, ki = j, ji = −k, kj = −i, and ik = −j. The IR-linear com- plex conjugation on IH is denoted h 7! h∗, and uniquely defined by 1∗ = 1, i∗ = −i, ∗ ∗ j = −j, and k = −k. Note that the basic quaternions Q8 = {±1; ±i; ±j; ±kg form a group under multiplication, the quaternion group of order 8. The multiplica- tive group consisting of all elements {±1; ±i; ±j; ±k; (±1±i±j ±k)=2g (where signs may be taken in any combination) is the so-called binary tetrahedral group, it has size 24 and it is the unit group of the so-called Hurwitz quaternions. By abuse of notation, in this paper, we call it the quaternion group Q24. Note that every element in Q24 has norm 1. If G is a multiplicatively written group and K is a ring with 1, then the group ring K[G] is the free K-module with basis G, equipped with the multiplication X X X agg bhh = agbhgh: g2G h2G g;h2G We identify the multiplicative identities 1G, 1K , and 1K[G], and denote them all by 1. Consequently, a positive integer n can be identified with the sum of n copies of 1K in K, and with the sum of n copies of 1G in K[G]. The following notation is also standard in the literature of relative difference sets, cf. Jungnickel and Pott [12]. Definition 1. Let K[G] be as before, let H ⊆ G be a subset and A 2 K[G]. P P a) We identify H with h2H h 2 K[G], in particular, G = g2G g 2 K[G]. P (−1) P −1 b) If A = g2G agg, then we define A = g2G agg 2 K[G]. Notation 1. Throughout this paper, n is a positive integer and Gn is the group n 4 4 2 2 −1 −1 Gn = hw; x; y j w = x = y = 1; x = y ; yxy = x ; wx = xw; wy = ywi: 2 We define H = hx; yi, Cn = hwi, and N = hx i as subgroups of Gn, so that Gn = Cn × H, and Cn, H, and N are isomorphic to Cn, Q8, and C2, respectively. 3 Relative difference sets An (m; n; k; λ)-relative difference set R in a group G of order mn, relative to a for- bidden normal subgroup N of order n, is a k-subset of G with the property that the list −1 of quotients r1r2 with distinct r1; r2 2 R contains each element in G n N exactly λ times and does not contain the elements of N. We also call R an (m; n; k; λ)-RDS, 2 2 3 3 or simply RDS. For example, R = fx; y; wy; xy; x ; wx ; wx ; wxy g ⊂ G2 is an 2 (8; 2; 8; 4)-RDS in G2 with forbidden subgroup N = hx i. Note that R ⊆ G is an (m; n; k; λ)-RDS if and only if in the real group ring IR[G] RR(−1) = k + λ(G − N): P Recall that every subset R ⊆ Gn is identified with r2R r 2 IR[Gn], and that the d u v latter is a sum of elements of the form w x y 2 Gn, that is, R is identified with a polynomial expression in the generators fx; y; wg of Gn. It is useful to introduce the following characteristic functions. We comment on it in the subsequent remark. Definition 2. For a subset R of Gn, we define a) χR : Gn ! f0; 1g by χR(g) = 1 if g 2 R, and χR(g) = 0 otherwise, b) ξR : Gn ! {−1; 1g by ξR(g) = −1 if g 2 R, and ξR(g) = 1 otherwise, P d u u d u v c) P (w; x; y) = d u v χ (w x y )w x y 2 IR[G ], R w x y 2Gn R n P d u u d u v d) T (w; x; y) = d u v ξ (w x y )w x y 2 IR[G ]. R w x y 2Gn R n The main advantage of this notation is that it allows us to describe evaluations, that is, certain substitutions for the generators x and y, for example, we have X P (w; i; j) = χ (wdxuyu)wdiujv 2 IH[C ]; (1) R d u v R n w x y 2Gn which is an element in the quaternion group ring of Cn = hwi.
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