Tan, Ming Ming
Construction of Relative Difference Sets and Hadamard Groups Bernhard Schmidt, Ming Ming Tan Nanyang Technological University, Singapore
July 5, 2013
Outline
Introduction Hadamard Matrices (2m, 2, 2m, m) Relative Difference Sets
Constructions of (2m, 2, 2m, m) Relative Difference Sets Example of Construction by Quaternary Golay Pairs
Table
Hadamard Matrices Definition A square matrix H of order m with entries ±1 is an Hadamard matrix if it satisfies HH T = mI.
Hadamard Matrices Definition A square matrix H of order m with entries ±1 is an Hadamard matrix if it satisfies HH T = mI.
Example
1 1 1 1 1 1 −1 −1 1 −1 1 −1 1 −1 −1 1
Hadamard Matrices Definition A square matrix H of order m with entries ±1 is an Hadamard matrix if it satisfies HH T = mI.
Example
1 1 1 1 1 1 −1 −1 1 −1 1 −1 1 −1 −1 1
Hadamard Conjecture There exists an Hadamard matrix of order 4t for every positive integer t.
Hadamard Matrices Definition A square matrix H of order m with entries ±1 is an Hadamard matrix if it satisfies HH T = mI.
Example
1 1 1 1 1 1 −1 −1 1 −1 1 −1 1 −1 −1 1
Hadamard Conjecture There exists an Hadamard matrix of order 4t for every positive integer t. Smallest open case: 668
Hadamard Matrices
(2m, 2, 2m, m) Relative Difference Sets
Hadamard Groups
Cocyclic Hadamard Matrices
Hadamard Matrices
(2m, 2, 2m, m) Relative Difference Sets
Hadamard Groups
Cocyclic Hadamard Matrices
Hadamard Matrices
(2m, 2, 2m, m) Relative Difference Sets
Hadamard Groups
Cocyclic Hadamard Matrices
Hadamard Matrices
(2m, 2, 2m, m) Relative Difference Sets
Hadamard Groups
Ito’s Conjecture
Cocyclic Hadamard Matrices
Hadamard Matrices
Hadamard Groups
(2m, 2, 2m, m) Relative Difference Sets
Cocyclic Hadamard Matrices
Asymptotic Existence (Launey, Kharaghani) 10+t q k 2 (q−1) , q is odd natural number. t ≥ 8 log210
j
Hadamard Matrices
(2m, 2, 2m, m) Relative Difference Sets
Hadamard Groups
Cocyclic Hadamard Matrices
(2m, 2, 2m, m) Relative Difference Sets Definition Let G be a group of order 4m. Let N be a normal subgroup of G of order 2. Let R be a 2m-subset of G. Then R is a normal (2m, 2, 2m, m) relative difference set in G relative to N if I
every g ∈ G \ N has exactly m representations g = r1 r2−1 with r1 , r2 ∈ R
I
no non-identity element in N has such a representation.
N is called the forbidden subgroup.
Example of (2m, 2, 2m, m) Relative Difference Sets Quaternion group: Q8 = hx, y|x4 = y 4 = 1, x2 = y 2 , y −1 xy = x−1 i
Example of (2m, 2, 2m, m) Relative Difference Sets Quaternion group: Q8 = hx, y|x4 = y 4 = 1, x2 = y 2 , y −1 xy = x−1 i N = hy 2 i
Example of (2m, 2, 2m, m) Relative Difference Sets Quaternion group: Q8 = hx, y|x4 = y 4 = 1, x2 = y 2 , y −1 xy = x−1 i N = hy 2 i R = {1, x, y, xy} is a (4, 2, 4, 2) relative difference set of Q8 relative to N
Example of (2m, 2, 2m, m) Relative Difference Sets Quaternion group: Q8 = hx, y|x4 = y 4 = 1, x2 = y 2 , y −1 xy = x−1 i N = hy 2 i R = {1, x, y, xy} is a (4, 2, 4, 2) relative difference set of Q8 relative to N 1
x
y
xy
1
−
x3
x2 y x3 y
x
x
−
x3 y
y
y
y
xy
−
x3
xy xy x2 y
x
−
Example of (2m, 2, 2m, m) Relative Difference Sets Quaternion group: Q8 = hx, y|x4 = y 4 = 1, x2 = y 2 , y −1 xy = x−1 i N = hy 2 i R = {1, x, y, xy} is a (4, 2, 4, 2) relative difference set of Q8 relative to N 1
x
y
xy
1
−
x3
x2 y x3 y
x
x
−
x3 y
y
y
y
xy
−
x3
xy xy x2 y
x
−
Example of (2m, 2, 2m, m) Relative Difference Sets Quaternion group: Q8 = hx, y|x4 = y 4 = 1, x2 = y 2 , y −1 xy = x−1 i N = hy 2 i R = {1, x, y, xy} is a (4, 2, 4, 2) relative difference set of Q8 relative to N
R = 1 + x + y + xy RR(−1) = (1 + x + y + xy)(1 + x−1 + y −1 + (xy)−1 )
Example of (2m, 2, 2m, m) Relative Difference Sets Quaternion group: Q8 = hx, y|x4 = y 4 = 1, x2 = y 2 , y −1 xy = x−1 i N = hy 2 i R = {1, x, y, xy} is a (4, 2, 4, 2) relative difference set of Q8 relative to N
R = 1 + x + y + xy RR(−1) = (1 + x + y + xy)(1 + x−1 + y −1 + (xy)−1 ) = 4 + 2(Q8 − N )
Example of (2m, 2, 2m, m) Relative Difference Sets Quaternion group: Q8 = hx, y|x4 = y 4 = 1, x2 = y 2 , y −1 xy = x−1 i N = hy 2 i R = {1, x, y, xy} is a (4, 2, 4, 2) relative difference set of Q8 relative to N
R = 1 + x + y + xy RR(−1) = (1 + x + y + xy)(1 + x−1 + y −1 + (xy)−1 ) = 4 + 2(Q8 − N ) RR(−1) = 2m + m(G − N )
Hadamard Groups Definition A group of order 4m which contains a (2m, 2, 2m, m) relative difference set is called an Hadamard group.
Hadamard Groups Definition A group of order 4m which contains a (2m, 2, 2m, m) relative difference set is called an Hadamard group.
Example (Ito) The generalized quaternion group Q8t = hx, y|x4t = y 4 = 1, x2t = y 2 , y −1 xy = x−1 i is an Hadamard group for all t such that 2t − 1 or 4t − 1 is a prime power.
Hadamard Groups Definition A group of order 4m which contains a (2m, 2, 2m, m) relative difference set is called an Hadamard group.
Example (Ito) The generalized quaternion group Q8t = hx, y|x4t = y 4 = 1, x2t = y 2 , y −1 xy = x−1 i is an Hadamard group for all t such that 2t − 1 or 4t − 1 is a prime power.
Ito’s Conjecture Q8t is an Hadamard group for all t > 0 Smallest open case: t = 47
Partial Semidirect Product of Z4 and Abelian Groups
Q8t = hx, y|x4t = 1, y 4 = 1, y 2 = x2t , y −1 xy = x−1 i
Partial Semidirect Product of Z4 and Abelian Groups
Q8t = hx, y|x4t = 1, y 4 = 1, y 2 = x2t , y −1 xy = x−1 i Q(H, g) = hH, y|y 4 = 1, y 2 = g, y −1 hy = h−1
H is abelian group of order 2m. g is an element of order 2 in H.
∀h ∈ Hi
(2m, 2, 2m, m) RDS in Q(H, g)
Q(H, g) = hH, y|y 4 = 1, y 2 = g, yhy −1 = h−1 R = A + By
where A, B ∈ H
∀h ∈ Hi
(2m, 2, 2m, m) RDS in Q(H, g)
Q(H, g) = hH, y|y 4 = 1, y 2 = g, yhy −1 = h−1 R = A + By
Example Q8 = Q(Z4 , x2 ) N = hy 2 i
where A, B ∈ H
∀h ∈ Hi
(2m, 2, 2m, m) RDS in Q(H, g)
Q(H, g) = hH, y|y 4 = 1, y 2 = g, yhy −1 = h−1 R = A + By
where A, B ∈ H
Example Q8 = Q(Z4 , x2 ) N = hy 2 i R = 1 + x + y + xy = (1 + x) + (1 + x)y.
∀h ∈ Hi
(2m, 2, 2m, m) RDS in Q(H, g) Q(H, g) = hH, y|y 4 = 1, y 2 = g, yhy −1 = h−1 R = A + By
where A, B ∈ H
AA(−1) + BB (−1) = 2m + m(H − N )
Example Q8 = Q(Z4 , x2 ) N = hy 2 i R = 1 + x + y + xy = (1 + x) + (1 + x)y.
∀h ∈ Hi
Recursive Construction Methods for (2m, 2, 2m, m) Relative Difference Sets R = A + By Q(H, g)
Recursive Construction Methods for (2m, 2, 2m, m) Relative Difference Sets R = A + By Q(H, g)
R 0 = A0 + B 0 y Q(H 0 , g)
Recursive Construction Methods for (2m, 2, 2m, m) Relative Difference Sets R 0 = A0 + B 0 y Q(H 0 , g)
R = A + By Q(H, g)
I
Golay pairs (binary and quaternary)
I
Williamson matrices
I
Building sets
Golay Pairs Definition Let a = (a0 , . . . , am−1 ), Write f=
m−1 X
ai xi ,
b = (b0 , . . . , bm−1 ).
g=
i=0
m−1 X
bi xi .
i=0
If ai , bi ∈ {±1}, and f (x)f (x−1 ) + g(x)g(x−1 ) = 2m, then (a, b) is a Golay pair of length m.
Golay Pairs Definition Let a = (a0 , . . . , am−1 ), Write f=
m−1 X
ai xi ,
b = (b0 , . . . , bm−1 ).
g=
i=0
m−1 X
bi xi .
i=0
If ai , bi ∈ {±1}, and f (x)f (x−1 ) + g(x)g(x−1 ) = 2m, then (a, b) is a Golay pair of length m. Golay pairs of length 2α 10β 26γ are known to exist [Turyn 1974]
Quaternary Golay Pairs
Definition If ai , bi are in {±1, ±i}, and f (x)f (x−1 ) + g(x)g(x−1 ) = 2m, then (a, b) is called a quaternary Golay pair of length m.
Quaternary Golay Pairs
Definition If ai , bi are in {±1, ±i}, and f (x)f (x−1 ) + g(x)g(x−1 ) = 2m, then (a, b) is called a quaternary Golay pair of length m. Quaternary Golay polynomials of length 3 and 11 exist.
Example of Quaternary Golay Pairs of Length 11 a = (1 i −1 1 −1 i −i −1 i i 1) b = (1 1 −i −i −i 1 1 i −1 1 −1)
Example of Quaternary Golay Pairs of Length 11 a = (1 i −1 1 −1 i −i −1 i i 1) b = (1 1 −i −i −i 1 1 i −1 1 −1)
f (x) = 1 + i − x2 + x3 − x4 + ix5 − ix6 − x7 + ix8 + ix9 + x10 g(x) = 1 + x − ix2 − ix3 − ix4 + x5 + x6 + ix7 − x8 + x9 − x10
Example of Quaternary Golay Pairs of Length 11 a = (1 i −1 1 −1 i −i −1 i i 1) b = (1 1 −i −i −i 1 1 i −1 1 −1)
f (x)f (x−1 ) + g(x)g(x−1 )
Example of Quaternary Golay Pairs of Length 11 a = (1 i −1 1 −1 i −i −1 i i 1) b = (1 1 −i −i −i 1 1 i −1 1 −1)
f (x)f (x−1 ) + g(x)g(x−1 ) = x10 + ix8 + (1 − i)x7 − x6 + ix4 + (1 − i)x3 − x2 + (−2 − 2i)x +(−2 + 2i)x−1 − x−2 + (1 + i)x−3 − ix−4 − x−6 + (1 + i)x−7 −ix−8 + x−10 + 11 −x10 − ix8 + (−1 + i)x7 + x6 − ix4 + (−1 + i)x3 + x2 +(2 + 2i)x + (2 − 2i)x−1 + x−2 + (−1 − i)x−3 + ix−4 + x−6 +(−1 − i)x−7 + ix−8 − x−10 + 11
Example of Quaternary Golay Pairs of Length 11 a = (1 i −1 1 −1 i −i −1 i i 1) b = (1 1 −i −i −i 1 1 i −1 1 −1)
f (x)f (x−1 ) + g(x)g(x−1 ) = x10 + ix8 + (1 − i)x7 − x6 + ix4 + (1 − i)x3 − x2 + (−2 − 2i)x +(−2 + 2i)x−1 − x−2 + (1 + i)x−3 − ix−4 − x−6 + (1 + i)x−7 −ix−8 + x−10 + 11 −x10 − ix8 + (−1 + i)x7 + x6 − ix4 + (−1 + i)x3 + x2 +(2 + 2i)x + (2 − 2i)x−1 + x−2 + (−1 − i)x−3 + ix−4 + x−6 +(−1 − i)x−7 + ix−8 − x−10 + 11 = 22
Example of Construction by Quaternary Golay Pairs
R = A + By Q(H, g)
(f, g) quaternary Golay pair length 11
Example of Construction by Quaternary Golay Pairs
R = A + By Q(H, g)
(f, g) quaternary Golay pair length 11
R 0 = A0 + B 0 y Q(H × Z22 , g) Z22 = hzi
Example of Construction by Quaternary Golay Pairs
R = A + By Q(H, g)
(f, g) quaternary Golay pair length 11
Evaluate f, g at z 2 and i 7→ y A0 = Af (z 2 ) − zB (−1) g(z 2 ) B 0 = zA(−1) g(z 2 ) + Bf (z 2 )
R 0 = A0 + B 0 y Q(H × Z22 , g) Z22 = hzi
Example of Construction by Quaternary Golay Pairs (f, g) quaternary Golay pair length 11
R = A + By Q(H, g)
Evaluate f, g at z 2 and i 7→ y A0 = Af (z 2 ) − zB (−1) g(z 2 ) B 0 = zA(−1) g(z 2 ) + Bf (z 2 ) A0 A0(−1) + B 0 B 0(−1)
R 0 = A0 + B 0 y Q(H × Z22 , g) Z22 = hzi
Example of Construction by Quaternary Golay Pairs
R = A + By Q(H, g)
(f, g) quaternary Golay pair length 11
R 0 = A0 + B 0 y Q(H × Z22 , g) Z22 = hzi
Evaluate f, g at z 2 and i 7→ y A0 = Af (z 2 ) − zB (−1) g(z 2 ) B 0 = zA(−1) g(z 2 ) + Bf (z 2 ) A0 A0(−1) + B 0 B 0(−1) = AA(−1) + BB (−1) f (z 2 )f (z −2 ) + g(z 2 )g(z −2 )
Main Result: Existence of (2m, 2, 2m, m) Relative Difference Sets I
Ito’s Construction
I
Golay Pairs (binary, quaternary)
I
Williamson Matrices
I
Building Sets
Main Result: Existence of (2m, 2, 2m, m) Relative Difference Sets I
Ito’s Construction
I
Golay Pairs (binary, quaternary)
I
Williamson Matrices
I
Building Sets
a+t+u+w+δ−+1 b c
m = x2
d
e
f
6 9 10 22 26
t Y i=1
qi2
u Y i=1
((ri +
s Y
i p4a i
i=1 w Y 1)/2) rivi si i=1
Main Result: Existence of (2m, 2, 2m, m) Relative Difference Sets I
Ito’s Construction
I
Golay Pairs (binary, quaternary)
I
Williamson Matrices
I
Building Sets
m= x2
a+t+u+w+δ−+1 b c
d
e
f
6 9 10 22 26
t Y i=1
qi2
u Y i=1
((ri +
s Y
pi4ai
i=1 w Y 1)/2) rivi si i=1
Main Result: Existence of (2m, 2, 2m, m) Relative Difference Sets I
Ito’s Construction
I
Golay Pairs (binary, quaternary)
I
Williamson Matrices
I
Building Sets
a+t+u+w+δ−+1 b c
m=x2
d
e
f
6 9 10 22 26
t Y i=1
qi2
u Y i=1
((ri +
s Y
pi4ai
i=1 w Y 1)/2) rivi si i=1
Main Result: Existence of (2m, 2, 2m, m) Relative Difference Sets I
Ito’s Construction
I
Golay Pairs (binary, quaternary)
I
Williamson Matrices
I
Building Sets
m = x2
a+t+u+w+δ−+1 b
c
d
e
f
6 9 10 22 26
s Y
i p4a i
i=1 t Y i=1
qi2
u Y i=1
((ri + 1)/2) rivi
w Y i=1
si
Main Result: Existence of (2m, 2, 2m, m) Relative Difference Sets I
Ito’s Construction
I
Golay Pairs (binary, quaternary)
I
Williamson Matrices
I
Building Sets
a+t+u+w+δ−+1 b c
m = x2
d
e
f
6 9 10 22 26
s Y
i p4a i
i=1 t Y i=1
qi2
u Y i=1
((ri + 1)/2) rivi
w Y i=1
si
Main Result: Existence of (2m, 2, 2m, m) Relative Difference Sets I
Ito’s Construction
I
Golay Pairs (binary, quaternary)
I
Williamson Matrices
I
Building Sets
m=x2
a+t+u+w+δ−+1
t Y i=1
qi2
b c
d
e
f
6 9 10 22 26
u Y i=1
((ri +
s Y
i p4a i
i=1 w Y 1)/2) rivi si i=1
Main Result: Existence of (2m, 2, 2m, m) Relative Difference Sets I I I I
Ito’s Construction Golay Pairs (binary, quaternary) Williamson Matrices Building Sets
m = x 2a+t+u+w+δ−+1 6b 9c 10d 22e 26f t Y i=1
qi2
u Y i=1
((ri +
s Y
i p4a i
i=1 w Y vi 1)/2) ri si i=1
Asymptotic Existence (Launey, Kharaghani) 10+t q j k 2 (q−1) t ≥ 8 log210 , q is odd natural number.
Main Result: Existence of (2m, 2, 2m, m) Relative Difference Sets I
Ito’s Construction
I
Golay Pairs (binary, quaternary)
I
Williamson Matrices
I
Building Sets
m=x2
a+t+u+w+δ−+1
t Y i=1
qi2
b c
d
e
f
6 9 10 22 26
u Y i=1
((ri +
s Y
i p4a i
i=1 w Y 1)/2) rivi si i=1
Table of abelian groups of order 4m, H such that I
no (2m, 2, 2m, m) RDS exists in Q(H, g),
I
existence of such RDS is unknown.
Table of abelian groups of order 4m, H such that I
no (2m, 2, 2m, m) RDS exists in Q(H, g),
I
existence of such RDS is unknown. Order
Group H
Exist?
12
(2)(2)(3)
N
24
(2)(2)(2)(3)
N
28
(2)(2)(7)
N
36
(2)(2)(9)
N
44
(2)(2)(11)
N
48
(2)(2)(2)(2)(3)
N
56
(2)(2)(2)(7)
N
60
(2)(2)(3)(5)
N
68
(2)(2)(17)
?
72
(2)(2)(2)(9)
N
Order
Group H
Exist?
76
(2)(2)(19)
N
84
(2)(2)(3)(7)
N
88
(2)(2)(2)(11)
N
92
(2)(2)(23)
N
96
(2)(2)(2)(2)(2)(3)
N
100
(2)(2)(25)
?
100
(2)(2)(5)(5)
?
108
(2)(2)(27)
N
108
(3)(4)(9)
?
108
(2)(2)(3)(9)
N
108
(3)(3)(3)(4)
?
108
(2)(2)(3)(3)(3)
N
112
(2)(2)(2)(2)(7)
N
Order
Group H
Exist?
116
(2)(2)(29)
?
120
(2)(2)(2)(3)(5)
N
124
(2)(2)(31)
N
132
(2)(2)(3)(11)
N
136
(2)(2)(2)(17)
?
140
(2)(2)(5)(7)
N
144
(2)(2)(2)(2)(9)
N
148
(2)(2)(37)
?
152
(2)(2)(2)(19)
N
156
(2)(2)(3)(13)
N
164
(2)(2)(41)
?
168
(2)(2)(2)(3)(7)
N
172
(2)(2)(43)
N
Order
Group
Exist?
176
(2)(2)(2)(2)(11)
N
180
(2)(2)(5)(9)
?
180
(2)(2)(3)(3)(5)
?
184
(2)(2)(2)(23)
N
188
(4)(47)
?
188
(2)(2)(47)
N
192
(2)(2)(2)(2)(2)(2)(3)
N
196
(2)(2)(49)
N
196
(4)(7)(7)
?
196
(2)(2)(7)(7)
N
Questions?
Thank you!
July 5, 2013
Outline
Introduction Hadamard Matrices (2m, 2, 2m, m) Relative Difference Sets
Constructions of (2m, 2, 2m, m) Relative Difference Sets Example of Construction by Quaternary Golay Pairs
Table
Hadamard Matrices Definition A square matrix H of order m with entries ±1 is an Hadamard matrix if it satisfies HH T = mI.
Hadamard Matrices Definition A square matrix H of order m with entries ±1 is an Hadamard matrix if it satisfies HH T = mI.
Example
1 1 1 1 1 1 −1 −1 1 −1 1 −1 1 −1 −1 1
Hadamard Matrices Definition A square matrix H of order m with entries ±1 is an Hadamard matrix if it satisfies HH T = mI.
Example
1 1 1 1 1 1 −1 −1 1 −1 1 −1 1 −1 −1 1
Hadamard Conjecture There exists an Hadamard matrix of order 4t for every positive integer t.
Hadamard Matrices Definition A square matrix H of order m with entries ±1 is an Hadamard matrix if it satisfies HH T = mI.
Example
1 1 1 1 1 1 −1 −1 1 −1 1 −1 1 −1 −1 1
Hadamard Conjecture There exists an Hadamard matrix of order 4t for every positive integer t. Smallest open case: 668
Hadamard Matrices
(2m, 2, 2m, m) Relative Difference Sets
Hadamard Groups
Cocyclic Hadamard Matrices
Hadamard Matrices
(2m, 2, 2m, m) Relative Difference Sets
Hadamard Groups
Cocyclic Hadamard Matrices
Hadamard Matrices
(2m, 2, 2m, m) Relative Difference Sets
Hadamard Groups
Cocyclic Hadamard Matrices
Hadamard Matrices
(2m, 2, 2m, m) Relative Difference Sets
Hadamard Groups
Ito’s Conjecture
Cocyclic Hadamard Matrices
Hadamard Matrices
Hadamard Groups
(2m, 2, 2m, m) Relative Difference Sets
Cocyclic Hadamard Matrices
Asymptotic Existence (Launey, Kharaghani) 10+t q k 2 (q−1) , q is odd natural number. t ≥ 8 log210
j
Hadamard Matrices
(2m, 2, 2m, m) Relative Difference Sets
Hadamard Groups
Cocyclic Hadamard Matrices
(2m, 2, 2m, m) Relative Difference Sets Definition Let G be a group of order 4m. Let N be a normal subgroup of G of order 2. Let R be a 2m-subset of G. Then R is a normal (2m, 2, 2m, m) relative difference set in G relative to N if I
every g ∈ G \ N has exactly m representations g = r1 r2−1 with r1 , r2 ∈ R
I
no non-identity element in N has such a representation.
N is called the forbidden subgroup.
Example of (2m, 2, 2m, m) Relative Difference Sets Quaternion group: Q8 = hx, y|x4 = y 4 = 1, x2 = y 2 , y −1 xy = x−1 i
Example of (2m, 2, 2m, m) Relative Difference Sets Quaternion group: Q8 = hx, y|x4 = y 4 = 1, x2 = y 2 , y −1 xy = x−1 i N = hy 2 i
Example of (2m, 2, 2m, m) Relative Difference Sets Quaternion group: Q8 = hx, y|x4 = y 4 = 1, x2 = y 2 , y −1 xy = x−1 i N = hy 2 i R = {1, x, y, xy} is a (4, 2, 4, 2) relative difference set of Q8 relative to N
Example of (2m, 2, 2m, m) Relative Difference Sets Quaternion group: Q8 = hx, y|x4 = y 4 = 1, x2 = y 2 , y −1 xy = x−1 i N = hy 2 i R = {1, x, y, xy} is a (4, 2, 4, 2) relative difference set of Q8 relative to N 1
x
y
xy
1
−
x3
x2 y x3 y
x
x
−
x3 y
y
y
y
xy
−
x3
xy xy x2 y
x
−
Example of (2m, 2, 2m, m) Relative Difference Sets Quaternion group: Q8 = hx, y|x4 = y 4 = 1, x2 = y 2 , y −1 xy = x−1 i N = hy 2 i R = {1, x, y, xy} is a (4, 2, 4, 2) relative difference set of Q8 relative to N 1
x
y
xy
1
−
x3
x2 y x3 y
x
x
−
x3 y
y
y
y
xy
−
x3
xy xy x2 y
x
−
Example of (2m, 2, 2m, m) Relative Difference Sets Quaternion group: Q8 = hx, y|x4 = y 4 = 1, x2 = y 2 , y −1 xy = x−1 i N = hy 2 i R = {1, x, y, xy} is a (4, 2, 4, 2) relative difference set of Q8 relative to N
R = 1 + x + y + xy RR(−1) = (1 + x + y + xy)(1 + x−1 + y −1 + (xy)−1 )
Example of (2m, 2, 2m, m) Relative Difference Sets Quaternion group: Q8 = hx, y|x4 = y 4 = 1, x2 = y 2 , y −1 xy = x−1 i N = hy 2 i R = {1, x, y, xy} is a (4, 2, 4, 2) relative difference set of Q8 relative to N
R = 1 + x + y + xy RR(−1) = (1 + x + y + xy)(1 + x−1 + y −1 + (xy)−1 ) = 4 + 2(Q8 − N )
Example of (2m, 2, 2m, m) Relative Difference Sets Quaternion group: Q8 = hx, y|x4 = y 4 = 1, x2 = y 2 , y −1 xy = x−1 i N = hy 2 i R = {1, x, y, xy} is a (4, 2, 4, 2) relative difference set of Q8 relative to N
R = 1 + x + y + xy RR(−1) = (1 + x + y + xy)(1 + x−1 + y −1 + (xy)−1 ) = 4 + 2(Q8 − N ) RR(−1) = 2m + m(G − N )
Hadamard Groups Definition A group of order 4m which contains a (2m, 2, 2m, m) relative difference set is called an Hadamard group.
Hadamard Groups Definition A group of order 4m which contains a (2m, 2, 2m, m) relative difference set is called an Hadamard group.
Example (Ito) The generalized quaternion group Q8t = hx, y|x4t = y 4 = 1, x2t = y 2 , y −1 xy = x−1 i is an Hadamard group for all t such that 2t − 1 or 4t − 1 is a prime power.
Hadamard Groups Definition A group of order 4m which contains a (2m, 2, 2m, m) relative difference set is called an Hadamard group.
Example (Ito) The generalized quaternion group Q8t = hx, y|x4t = y 4 = 1, x2t = y 2 , y −1 xy = x−1 i is an Hadamard group for all t such that 2t − 1 or 4t − 1 is a prime power.
Ito’s Conjecture Q8t is an Hadamard group for all t > 0 Smallest open case: t = 47
Partial Semidirect Product of Z4 and Abelian Groups
Q8t = hx, y|x4t = 1, y 4 = 1, y 2 = x2t , y −1 xy = x−1 i
Partial Semidirect Product of Z4 and Abelian Groups
Q8t = hx, y|x4t = 1, y 4 = 1, y 2 = x2t , y −1 xy = x−1 i Q(H, g) = hH, y|y 4 = 1, y 2 = g, y −1 hy = h−1
H is abelian group of order 2m. g is an element of order 2 in H.
∀h ∈ Hi
(2m, 2, 2m, m) RDS in Q(H, g)
Q(H, g) = hH, y|y 4 = 1, y 2 = g, yhy −1 = h−1 R = A + By
where A, B ∈ H
∀h ∈ Hi
(2m, 2, 2m, m) RDS in Q(H, g)
Q(H, g) = hH, y|y 4 = 1, y 2 = g, yhy −1 = h−1 R = A + By
Example Q8 = Q(Z4 , x2 ) N = hy 2 i
where A, B ∈ H
∀h ∈ Hi
(2m, 2, 2m, m) RDS in Q(H, g)
Q(H, g) = hH, y|y 4 = 1, y 2 = g, yhy −1 = h−1 R = A + By
where A, B ∈ H
Example Q8 = Q(Z4 , x2 ) N = hy 2 i R = 1 + x + y + xy = (1 + x) + (1 + x)y.
∀h ∈ Hi
(2m, 2, 2m, m) RDS in Q(H, g) Q(H, g) = hH, y|y 4 = 1, y 2 = g, yhy −1 = h−1 R = A + By
where A, B ∈ H
AA(−1) + BB (−1) = 2m + m(H − N )
Example Q8 = Q(Z4 , x2 ) N = hy 2 i R = 1 + x + y + xy = (1 + x) + (1 + x)y.
∀h ∈ Hi
Recursive Construction Methods for (2m, 2, 2m, m) Relative Difference Sets R = A + By Q(H, g)
Recursive Construction Methods for (2m, 2, 2m, m) Relative Difference Sets R = A + By Q(H, g)
R 0 = A0 + B 0 y Q(H 0 , g)
Recursive Construction Methods for (2m, 2, 2m, m) Relative Difference Sets R 0 = A0 + B 0 y Q(H 0 , g)
R = A + By Q(H, g)
I
Golay pairs (binary and quaternary)
I
Williamson matrices
I
Building sets
Golay Pairs Definition Let a = (a0 , . . . , am−1 ), Write f=
m−1 X
ai xi ,
b = (b0 , . . . , bm−1 ).
g=
i=0
m−1 X
bi xi .
i=0
If ai , bi ∈ {±1}, and f (x)f (x−1 ) + g(x)g(x−1 ) = 2m, then (a, b) is a Golay pair of length m.
Golay Pairs Definition Let a = (a0 , . . . , am−1 ), Write f=
m−1 X
ai xi ,
b = (b0 , . . . , bm−1 ).
g=
i=0
m−1 X
bi xi .
i=0
If ai , bi ∈ {±1}, and f (x)f (x−1 ) + g(x)g(x−1 ) = 2m, then (a, b) is a Golay pair of length m. Golay pairs of length 2α 10β 26γ are known to exist [Turyn 1974]
Quaternary Golay Pairs
Definition If ai , bi are in {±1, ±i}, and f (x)f (x−1 ) + g(x)g(x−1 ) = 2m, then (a, b) is called a quaternary Golay pair of length m.
Quaternary Golay Pairs
Definition If ai , bi are in {±1, ±i}, and f (x)f (x−1 ) + g(x)g(x−1 ) = 2m, then (a, b) is called a quaternary Golay pair of length m. Quaternary Golay polynomials of length 3 and 11 exist.
Example of Quaternary Golay Pairs of Length 11 a = (1 i −1 1 −1 i −i −1 i i 1) b = (1 1 −i −i −i 1 1 i −1 1 −1)
Example of Quaternary Golay Pairs of Length 11 a = (1 i −1 1 −1 i −i −1 i i 1) b = (1 1 −i −i −i 1 1 i −1 1 −1)
f (x) = 1 + i − x2 + x3 − x4 + ix5 − ix6 − x7 + ix8 + ix9 + x10 g(x) = 1 + x − ix2 − ix3 − ix4 + x5 + x6 + ix7 − x8 + x9 − x10
Example of Quaternary Golay Pairs of Length 11 a = (1 i −1 1 −1 i −i −1 i i 1) b = (1 1 −i −i −i 1 1 i −1 1 −1)
f (x)f (x−1 ) + g(x)g(x−1 )
Example of Quaternary Golay Pairs of Length 11 a = (1 i −1 1 −1 i −i −1 i i 1) b = (1 1 −i −i −i 1 1 i −1 1 −1)
f (x)f (x−1 ) + g(x)g(x−1 ) = x10 + ix8 + (1 − i)x7 − x6 + ix4 + (1 − i)x3 − x2 + (−2 − 2i)x +(−2 + 2i)x−1 − x−2 + (1 + i)x−3 − ix−4 − x−6 + (1 + i)x−7 −ix−8 + x−10 + 11 −x10 − ix8 + (−1 + i)x7 + x6 − ix4 + (−1 + i)x3 + x2 +(2 + 2i)x + (2 − 2i)x−1 + x−2 + (−1 − i)x−3 + ix−4 + x−6 +(−1 − i)x−7 + ix−8 − x−10 + 11
Example of Quaternary Golay Pairs of Length 11 a = (1 i −1 1 −1 i −i −1 i i 1) b = (1 1 −i −i −i 1 1 i −1 1 −1)
f (x)f (x−1 ) + g(x)g(x−1 ) = x10 + ix8 + (1 − i)x7 − x6 + ix4 + (1 − i)x3 − x2 + (−2 − 2i)x +(−2 + 2i)x−1 − x−2 + (1 + i)x−3 − ix−4 − x−6 + (1 + i)x−7 −ix−8 + x−10 + 11 −x10 − ix8 + (−1 + i)x7 + x6 − ix4 + (−1 + i)x3 + x2 +(2 + 2i)x + (2 − 2i)x−1 + x−2 + (−1 − i)x−3 + ix−4 + x−6 +(−1 − i)x−7 + ix−8 − x−10 + 11 = 22
Example of Construction by Quaternary Golay Pairs
R = A + By Q(H, g)
(f, g) quaternary Golay pair length 11
Example of Construction by Quaternary Golay Pairs
R = A + By Q(H, g)
(f, g) quaternary Golay pair length 11
R 0 = A0 + B 0 y Q(H × Z22 , g) Z22 = hzi
Example of Construction by Quaternary Golay Pairs
R = A + By Q(H, g)
(f, g) quaternary Golay pair length 11
Evaluate f, g at z 2 and i 7→ y A0 = Af (z 2 ) − zB (−1) g(z 2 ) B 0 = zA(−1) g(z 2 ) + Bf (z 2 )
R 0 = A0 + B 0 y Q(H × Z22 , g) Z22 = hzi
Example of Construction by Quaternary Golay Pairs (f, g) quaternary Golay pair length 11
R = A + By Q(H, g)
Evaluate f, g at z 2 and i 7→ y A0 = Af (z 2 ) − zB (−1) g(z 2 ) B 0 = zA(−1) g(z 2 ) + Bf (z 2 ) A0 A0(−1) + B 0 B 0(−1)
R 0 = A0 + B 0 y Q(H × Z22 , g) Z22 = hzi
Example of Construction by Quaternary Golay Pairs
R = A + By Q(H, g)
(f, g) quaternary Golay pair length 11
R 0 = A0 + B 0 y Q(H × Z22 , g) Z22 = hzi
Evaluate f, g at z 2 and i 7→ y A0 = Af (z 2 ) − zB (−1) g(z 2 ) B 0 = zA(−1) g(z 2 ) + Bf (z 2 ) A0 A0(−1) + B 0 B 0(−1) = AA(−1) + BB (−1) f (z 2 )f (z −2 ) + g(z 2 )g(z −2 )
Main Result: Existence of (2m, 2, 2m, m) Relative Difference Sets I
Ito’s Construction
I
Golay Pairs (binary, quaternary)
I
Williamson Matrices
I
Building Sets
Main Result: Existence of (2m, 2, 2m, m) Relative Difference Sets I
Ito’s Construction
I
Golay Pairs (binary, quaternary)
I
Williamson Matrices
I
Building Sets
a+t+u+w+δ−+1 b c
m = x2
d
e
f
6 9 10 22 26
t Y i=1
qi2
u Y i=1
((ri +
s Y
i p4a i
i=1 w Y 1)/2) rivi si i=1
Main Result: Existence of (2m, 2, 2m, m) Relative Difference Sets I
Ito’s Construction
I
Golay Pairs (binary, quaternary)
I
Williamson Matrices
I
Building Sets
m= x2
a+t+u+w+δ−+1 b c
d
e
f
6 9 10 22 26
t Y i=1
qi2
u Y i=1
((ri +
s Y
pi4ai
i=1 w Y 1)/2) rivi si i=1
Main Result: Existence of (2m, 2, 2m, m) Relative Difference Sets I
Ito’s Construction
I
Golay Pairs (binary, quaternary)
I
Williamson Matrices
I
Building Sets
a+t+u+w+δ−+1 b c
m=x2
d
e
f
6 9 10 22 26
t Y i=1
qi2
u Y i=1
((ri +
s Y
pi4ai
i=1 w Y 1)/2) rivi si i=1
Main Result: Existence of (2m, 2, 2m, m) Relative Difference Sets I
Ito’s Construction
I
Golay Pairs (binary, quaternary)
I
Williamson Matrices
I
Building Sets
m = x2
a+t+u+w+δ−+1 b
c
d
e
f
6 9 10 22 26
s Y
i p4a i
i=1 t Y i=1
qi2
u Y i=1
((ri + 1)/2) rivi
w Y i=1
si
Main Result: Existence of (2m, 2, 2m, m) Relative Difference Sets I
Ito’s Construction
I
Golay Pairs (binary, quaternary)
I
Williamson Matrices
I
Building Sets
a+t+u+w+δ−+1 b c
m = x2
d
e
f
6 9 10 22 26
s Y
i p4a i
i=1 t Y i=1
qi2
u Y i=1
((ri + 1)/2) rivi
w Y i=1
si
Main Result: Existence of (2m, 2, 2m, m) Relative Difference Sets I
Ito’s Construction
I
Golay Pairs (binary, quaternary)
I
Williamson Matrices
I
Building Sets
m=x2
a+t+u+w+δ−+1
t Y i=1
qi2
b c
d
e
f
6 9 10 22 26
u Y i=1
((ri +
s Y
i p4a i
i=1 w Y 1)/2) rivi si i=1
Main Result: Existence of (2m, 2, 2m, m) Relative Difference Sets I I I I
Ito’s Construction Golay Pairs (binary, quaternary) Williamson Matrices Building Sets
m = x 2a+t+u+w+δ−+1 6b 9c 10d 22e 26f t Y i=1
qi2
u Y i=1
((ri +
s Y
i p4a i
i=1 w Y vi 1)/2) ri si i=1
Asymptotic Existence (Launey, Kharaghani) 10+t q j k 2 (q−1) t ≥ 8 log210 , q is odd natural number.
Main Result: Existence of (2m, 2, 2m, m) Relative Difference Sets I
Ito’s Construction
I
Golay Pairs (binary, quaternary)
I
Williamson Matrices
I
Building Sets
m=x2
a+t+u+w+δ−+1
t Y i=1
qi2
b c
d
e
f
6 9 10 22 26
u Y i=1
((ri +
s Y
i p4a i
i=1 w Y 1)/2) rivi si i=1
Table of abelian groups of order 4m, H such that I
no (2m, 2, 2m, m) RDS exists in Q(H, g),
I
existence of such RDS is unknown.
Table of abelian groups of order 4m, H such that I
no (2m, 2, 2m, m) RDS exists in Q(H, g),
I
existence of such RDS is unknown. Order
Group H
Exist?
12
(2)(2)(3)
N
24
(2)(2)(2)(3)
N
28
(2)(2)(7)
N
36
(2)(2)(9)
N
44
(2)(2)(11)
N
48
(2)(2)(2)(2)(3)
N
56
(2)(2)(2)(7)
N
60
(2)(2)(3)(5)
N
68
(2)(2)(17)
?
72
(2)(2)(2)(9)
N
Order
Group H
Exist?
76
(2)(2)(19)
N
84
(2)(2)(3)(7)
N
88
(2)(2)(2)(11)
N
92
(2)(2)(23)
N
96
(2)(2)(2)(2)(2)(3)
N
100
(2)(2)(25)
?
100
(2)(2)(5)(5)
?
108
(2)(2)(27)
N
108
(3)(4)(9)
?
108
(2)(2)(3)(9)
N
108
(3)(3)(3)(4)
?
108
(2)(2)(3)(3)(3)
N
112
(2)(2)(2)(2)(7)
N
Order
Group H
Exist?
116
(2)(2)(29)
?
120
(2)(2)(2)(3)(5)
N
124
(2)(2)(31)
N
132
(2)(2)(3)(11)
N
136
(2)(2)(2)(17)
?
140
(2)(2)(5)(7)
N
144
(2)(2)(2)(2)(9)
N
148
(2)(2)(37)
?
152
(2)(2)(2)(19)
N
156
(2)(2)(3)(13)
N
164
(2)(2)(41)
?
168
(2)(2)(2)(3)(7)
N
172
(2)(2)(43)
N
Order
Group
Exist?
176
(2)(2)(2)(2)(11)
N
180
(2)(2)(5)(9)
?
180
(2)(2)(3)(3)(5)
?
184
(2)(2)(2)(23)
N
188
(4)(47)
?
188
(2)(2)(47)
N
192
(2)(2)(2)(2)(2)(2)(3)
N
196
(2)(2)(49)
N
196
(4)(7)(7)
?
196
(2)(2)(7)(7)
N
Questions?
Thank you!
