Perfect Sequences over the Quaternions and Relative Difference Sets Santiago Barrera-Acevedo June 9, 2017 Santiago Barrera-Acevedo Pefect Sequences over Quaternions and RDS
Definitions Autocorrelation of a sequence An ordered n -tuple S = ( s 0 , . . . , s n − 1 ) of elements from a set A ⊂ C is called a finite sequence . The set A is called an alphabet and the number n is called the length of the sequence. We define, for all t ∈ { 0 , . . . , n − 1 } , the t - autocorrelation value of S as n − 1 � s l s ∗ AC S ( t ) = l + t l =0 where s ∗ l + t is the complex conjugation of s l + t , and the indices l and l + t are taken modulo n . Santiago Barrera-Acevedo Pefect Sequences over Quaternions and RDS
Definitions Perfect sequences The autocorrelation sequence of S is defined as AC S = (AC S (0) , . . . , AC S ( n − 1)) , with AC S (0) being the peak-value and all other values being off-peak values . The sequence S has constant off-peak autocorrelation if all its off-peak autocorrelation values are equal. In particular, S is perfect if all its off-peak autocorrelation values are zero. The sequences S 1 = (1 , 1 , 1 , − 1) and S 2 = (1 , 1 , i, 1 , 1 , − 1 , i, − 1) over the binary and quaternary alphabet, respectively, are perfect since we have AC S 1 = (4 , 0 , 0 , 0) and AC S 2 = (8 , 0 , 0 , 0 , 0 , 0 , 0 , 0) . Santiago Barrera-Acevedo Pefect Sequences over Quaternions and RDS
Definitions Applications Sequences with “good” autocorrelation properties, such as being perfect, have important applications in information technology, for example, in digital watermarking, frequency hopping patterns for radar or sonar communications and signal correlation (synchronisation of signals). In this work we focus exclusively on the mathematical aspects of sequences with good autocorrelation. Santiago Barrera-Acevedo Pefect Sequences over Quaternions and RDS
Definitions It is very difficult to construct perfect sequences over 2 nd-, 4 th-, and in general over n -th roots of unity. † It is conjectured that perfect sequences over n -th roots of unity do not exist for lengths greater that n 2 , Ma and Ng [7]. 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 [5], who defined perfect sequences over the quaternion algebra. † This problem is related to the construction of (generalised) circulant Hadamard matrices over n -th roots of unity. Santiago Barrera-Acevedo Pefect Sequences over Quaternions and RDS
Definitions Quaternions H The quaternion algebra H is a 4-dimensional real vector space with R -basis { 1 , i, j, k } and non-commutative multiplication defined by i 2 = j 2 = k 2 = − 1 and ij = k. It follows from these relations that jk = i, ki = j, ji = − k, kj = − i, and ik = − j. The R -linear complex conjugation on H is denoted h �→ h ∗ , and uniquely defined by 1 ∗ = 1 , i ∗ = − i, j ∗ = − j, and k ∗ = − k. The norm of a quaternion q , denoted by || q || , is defined by || q || = qq ∗ . Santiago Barrera-Acevedo Pefect Sequences over Quaternions and RDS
Definitions Note that the basic quaternions Q 8 = {± 1 , ± i, ± j, ± k } form a group under multiplication, the quaternion group of order 8. The multiplicative group consisting of all elements {± 1 , ± i, ± j, ± k, ( ± 1 ± i ± j ± k ) / 2 } (where signs may be taken in any combination) is the so-called binary tetrahedral group and has size 24. By abuse of notation we call it the quaternion group Q 24 . In the following, we often decompose Q 24 into the cosets Q 24 = Q 8 ∪ qQ 8 ∪ q ∗ Q 8 q = 1+ i + j + k 1+ i + j + k 1+ i + j + k where q q . 2 2 2 Santiago Barrera-Acevedo Pefect Sequences over Quaternions and RDS
Definitions Let S = ( s 0 , . . . , s n − 1 ) be a sequence of length n over an arbitrary quaternion alphabet. We define, for all t ∈ { 0 , . . . , n − 1 } , the left and right t - autocorrelation values of S as n − 1 n − 1 AC L � s ∗ AC R � s l s ∗ S ( t ) = l s l + t and S ( t ) = l + t l =0 l =0 Left and right AC values of S = ( j, j, − 1 , − k, i, − j ) AC L || AC L AC R || AC R t S || S || S S 0 6 36 6 36 1 0 0 2 j + 2 k 8 2 − 1 + 3 i − j − k 12 − 1 + i + j − k 4 3 0 0 0 0 4 − 1 − 3 i + j + k 12 − 1 − i − j + k 4 5 0 0 − 2 j − 2 k 8 Santiago Barrera-Acevedo Pefect Sequences over Quaternions and RDS
Definitions Perfect Sequences over Quaternions A sequence S = ( s 0 , . . . , s n − 1 ) of length n over an arbitrary quaternion alphabet is called left ( right ) perfect when all left (right) off-peak t -autocorrelation values are equal to zero, for t ∈ { 1 , . . . , n − 1 } . S = ( i, j, − k, j, i, 1 , k, − 1 , k, 1) AC L S = (10 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0) AC R S = (10 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0) Theorem (Kuznetsov [5]) Let S be a sequence over an arbitrary quaternion alphabet. Then the sequence S is right perfect if and only if it is left perfect. Santiago Barrera-Acevedo Pefect Sequences over Quaternions and RDS
Definitions Motivation Kuznetsov and Hall [6] showed a construction of a perfect sequence of length 5 , 354 , 228 , 880 over Q 24 . At this point two main questions were stated: Are there perfect sequences of unbounded lengths over Q 24 ? If so, is it possible to restrict the alphabet size to a small one, say the basic quaternions Q 8 = { 1 , − 1 , i, − i, j, − j, k, − k } ? Theorem (Barrera Acevedo and Hall [4]) There exists a family of perfect sequences over Q 8 of length n = p a + 1 ≡ 2 mod 4 , where p is prime and a ∈ N . Santiago Barrera-Acevedo Pefect Sequences over Quaternions and RDS
Symmetry of perfect sequences over the quaternions Symmetry type 1 A sequence S = ( s 0 , . . . , s n − 1 ) has symmetry type 1 if s r = s n − r for r = 1 , . . . , n − 1 . 1 1 Length 8: (1 1 , 1 , i, − 1 , 1 1 , − 1 , i, 1) 1 j Length 10: (1 1 , i, − 1 , − i,j j, − i, − 1 , i ) Length 11: (1 , k, − j, − i, − 1 ,q q q, − 1 , − i, − j, k, 1) Length 16: (1 1 1 , i, − 1 , i, j, k, − j, − i − i − i, − j, k, j, i, − 1 , i ) Santiago Barrera-Acevedo Pefect Sequences over Quaternions and RDS
Symmetry of perfect sequences over the quaternions Symmetry type 2 A sequence S = ( s 0 , . . . , s n − 1 ) has symmetry type 2 † if n is even 2 = ( − 1) r s r for all r = 0 , . . . , n and s r + n 2 − 1 . Length 8: (1 , 1 , i, − 1 , 1 , − 1 , − i, − 1) Length 8: (1 , 1 , i, − 1 , 1 , − 1 , − 1 , − 1) Length 16: (1 , − 1 , 1 , − i, − 1 , i, 1 , 1 , 1 , 1 , 1 , i, − 1 , − i, 1 , − 1) Length 16: (1 , i, j, − k, 1 , − k, − j, i, 1 , − i, j, k, 1 , k, − j, − i ) Length 32: (1 , − 1 , 1 , − i, i, − j, 1 , − k, 1 , k, − 1 , j, i, i, − 1 , 1 , Length 32 :::1 , 1 , 1 , i, i, j, 1 , k, 1 , − k, − 1 , − j, i, − i, − 1 , − 1) † A sequence can have symmetry type 1 and 2. Santiago Barrera-Acevedo Pefect Sequences over Quaternions and RDS
Symmetry of perfect sequences over the quaternions Symmetry type 3 A sequence S = ( s 0 , . . . , s n − 1 ) has symmetry type 3 if n is 2 = ( − 1) r s 2 r + e for r = 0 , . . . , n divisible by 4 and s 2 r + e + n 2 − 1 and e = 0 , 1 . Length 16: (1 , i, − j, j, 1 , − i, − k, − k, 1 , i, − j, − j, 1 , − i, − k, − k ) Length 16: (1 , i, − j, j, 1 , − i, − k, − k, 1 , 1 , − 1 , − 1 , 1 , − 1 , − 1 , − 1) Length 48: (1 , − qk, − j, j, − q, − i, − k, qj, 1 , i, − qi, − j, 1 , qk, k, k, − q, i, − j, − qi, 1 , − i, qj, − k, 1 , − qk, j, − j, − q, − i, k, − qj, 1 , i, qi, j, 1 , qk, − k, − k, − q, i, j, qi, 1 , − i, − qj, k ) Santiago Barrera-Acevedo Pefect Sequences over Quaternions and RDS
Relative difference sets An ( m, n, l, λ ) - relative difference set (RDS) R in a group G of order mn , relative to a (forbidden) subgroup N of order n , is a l -subset of G with the property that the list of quotients r 1 r − 1 2 with distinct r 1 , r 2 ∈ R contains each element in G \ N exactly λ times and does not contain the elements of N . We also call R an ( m, n, l, λ ) -RDS or simply RDS. For example R = { 1 , i, j, k } is a (4 , 2 , 4 , 2) -RDS in Q 8 with forbidden subgroup N = { 1 , − 1 } . 1 i − 1 = − i i 1 − 1 = i j 1 − 1 = j k 1 − 1 = k 1 j − 1 = − j ij − 1 = − k ji − 1 = k ki − 1 = − j 1 k − 1 = − k ik − 1 = j jk − 1 = − i kj − 1 = i Santiago Barrera-Acevedo Pefect Sequences over Quaternions and RDS
Relative difference sets Group ring : If G is a multiplicatively written group and K is a ring with 1, then the group ring �� � K [ G ] = g ∈ G a g g | a g ∈ K and only finite a g � = 0 is the free K -module with basis G , equipped with the multiplication � � � g ∈ G a g g h ∈ G b h h = g,h ∈ G a g b h gh. We identify the multiplicative identities 1 G , 1 K , and 1 K [ G ] , and denote them all by 1. Santiago Barrera-Acevedo Pefect Sequences over Quaternions and RDS
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