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Reversible Menon-Hadamard Difference Sets in Abelian 2-groups Jordan D. Webster Mid Michigan Community College October 10, 2015 Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

SLIDE 1

Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

Jordan D. Webster Mid Michigan Community College October 10, 2015

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

SLIDE 2

Introduction

A (v, k, λ) difference set D in a group G of order v is a k-subset of G such that each group element other than the identity appears exactly λ times in the multiset {d1d−1

2

: d1, d2 ∈ D}. A Menon-Hadamard difference set has parameters (4m2, 2m2 − m, m2 − m) for some m ∈ N.

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

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Example

Example: {x, x3, y, y3, xy, x3y3} is a (16, 6, 2) Menon-Hadamard difference set in C4 × C4 =< x, y : x4 = y4 = [x, y] = 1 >. x x3 y y3 xy x3y3 x3 1 x2 x3y x3y3 y x2y3 x x2 1 xy xy3 x2y y3 y3 xy3 x3y3 1 y2 x x3y2 y xy x3y y2 1 xy2 x3 x3y3 y3 x2y3 x3 x3y2 1 x2y2 xy x2y y xy2 x x2y2 1

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

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Example

Example: {x, x3, y, y3, xy, x3y3} is a (16, 6, 2) Menon-Hadamard difference set in C4 × C4 =< x, y : x4 = y4 = [x, y] = 1 >. x x3 y y3 xy x3y3 x3 1 x2 x3y x3y3 y x2y3 x x2 1 xy xy3 x2y y3 y3 xy3 x3y3 1 y2 x x3y2 y xy x3y y2 1 xy2 x3 x3y3 y3 x2y3 x3 x3y2 1 x2y2 xy x2y y xy2 x x2y2 1

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

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Reversibility

Our Example D = {x, x3, y, y3, xy, x3y3} in C4 × C4 Notice for each element in D, the inverse element is also in D. A difference set D is Reversible if for each d ∈ D, d−1 ∈ D.

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

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What we know

Abelian non-2 groups do not have reversible difference sets Some Abelian 2 groups with reversible difference sets: C4, (C2r )(2), (C22r )(3) Direct products of groups that contain reversible difference sets have reversible difference sets. Some Abelian 2 groups don’t have reversible difference sets. C8 × C2, C64 × (C16)(2).

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

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What we don’t/didn’t know

Odd powers on groups with C2: Example (C32)(3) × C2 Odd powers on groups without C2: Example (C128)(3) × (C32)(3) Single cyclic group in direct product with no match Example (C32)(2) × C16

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

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What we don’t/didn’t know

Odd powers on groups with C2: Example (C32)(3) × C2 Odd powers on groups without C2: Example (C128)(3) × (C32)(3) Single cyclic group in direct product with no match Example (C32)(2) × C16

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

SLIDE 9

This Talk

Build a difference set in (C8)(3) × C2 Convince that the pattern extends to (C22r+1)(3) × C2

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

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The Group Ring

The group ring C[G]. This is the ring of all formal sums of the form

g∈G

agg where ag ∈ C. Addition in this ring is defined pointwise. (4g1 + 5g2) + (−2g1 + 8g2) = 2g1 + 13g2

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

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The Group Ring

Multiplication is done with consideration of both the multiplication in C and by the “multiplication” of the group G. (4g1 + 5g2)(2g3) = 8g1g3 + 10g2g3 Multiplication distributes over addition.

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

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D in the Group Ring

Difference set in the ring is D =

d∈D

d. Also Notation of D(−1) =

d∈D

d−1 DD(−1) = (k − λ) + λ(G)

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

SLIDE 13

D in the Group Ring

Difference set in the ring is D =

d∈D

d. If D is reversible then D = D(−1) DD = (k − λ) + λ(G)

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

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Transformation

We wish to substitute the difference set for something else. The Hadamard transform: D = G − 2D. If a (4m2, 2m2 − m, m2 − m) difference set D exists, then

D(−1) = 4m2 For remainder of talk, we say that a (4m2, 2m2 − m, m2 − m) difference set is an element of the group ring D with coefficients of ±1 and has the property that D D(−1) = 4m2.

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

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Character Values

For abelian groups, a character of the group is a homomophism into the set of complex numbers. If G is abelian, then the complete set of characters forms the dual group G ∗. Extend each character to C-algebra homomorphism. χ(

agg) =
agχ(g)

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

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Character Values

Theorem (Turyn 1965)

D is a Menon-Hadamard difference set in an abelian group of order

4m2 if and only if it is an element of the group ring with ±1 coefficients and for each χ ∈ G ∗, we have χ( D)χ( D(−1)) = 4m2.

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

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Character Example

The group C4 =< x : x4 = 1 >. Each character is homomorphism so each is defined by where it sends generator(s) of the group. If χ is a character, then (χ(x))4 = 1 χj(x) = (e

2πi 4 )j for 1 ≤ j ≤ 4.

The dual group C ∗

4 = {χj : 1 ≤ j ≤ 4}

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

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Character Values for Reversible

We know χ( D)χ( D(−1)) = 4m2 When D is reversible, D(−1) = D and (χ( D))2 = 4m2 χ( D) = ±2m

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

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Building a Better Basis

The ring C[G] has the standard basis {g : g ∈ G}. Create elements for each character value. eχ =

1 |G|

g∈G

χ(g)g−1 These idempotents form basis for C[G].

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

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The Better Part

In the basis {eχ; χ ∈ G ∗} we take advantage of character values. χ(eχ′) = δχ,χ′ Let Y =

χ∈G ∗

cχeχ So χ(Y ) = χ(cχ)

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

SLIDE 21

The Better Part

We write D =

χ∈G ∗

cχeχ. So χ( D) = χ(cχ) Each χ(cχ) must be a complex number of modulus 2m. If D is reversible then χ(cχ) = ±2m

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

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The Better Part

We have a Menon-Hadamard Difference set ( D) if

◮ ±1 are coefficients on each group element g ◮ Each coefficient, cχ, on eχ has the property that χ(cχ) is

a complex number of modulus 2m. Reversible if χ(cχ) = ±2m

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

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The Better Part (2)

Recall that each idempotent is 1 |G|

g∈G

χ(g)g−1 Difference set D has coefficients in {−1, 1}.

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

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The Better Part (2)

Idempotents with the same kernel may be added together and we may use the same coefficient for all of them. Sums of idempotents with the same kernel are called rational idempotents. [eχ] is be the rational idempotent containing eχ and all other eχ′ such that ker(χ) = ker(χ′). The [eχ] exist in the group ring Q[G]

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

SLIDE 25

The Better Part (2)

We have a Hadamard Difference set if

◮ ±1 are coefficients on each group element g ◮ Each coefficient, cχ, on [eχ] has the property that χ(cχ) is

a complex number of modulus 2m.

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

SLIDE 26

Tiles

A tile T is an element of the group ring with the following properties:

◮ Coefficients of g are all in the set {1, 0, −1} ◮ For every character χ either χ(T) = 0 or χ(T) is a complex

number of modulus 2m Sums of rational idempotents with appropriate aliases create tiles.

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

SLIDE 27

The Goal

The Goal is to create a set of tiles so that support lines up in the correct way. We create the tiles from the rational idempotents.

◮ Idempotents

C[G]

◮ Rational Idempotents(with aliases)

Q[G]

◮ Tiles

Z[G] (Coefficients in set {1, 0, −1})

◮ Menon-Hadamard Difference Set

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

SLIDE 28

Goal

Goal is to do this process in (C8)(3) × C2

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

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Notation

In (C8)(3) × C2. Elements of the group x8 = y8 = z8 = w2 = 1 Size of the group is 1024.

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

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Characters and Aliases

Group size 1024 Means χ(D) = ±32 for all characters. Means for each alias on an idempotent, we can set to be 32g where χ(g) = ±1 For this reason, we multiply each idempotent by 32.

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

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Notation

Rational idempotents denoted [ei1,i2,i3,i4] Associated with character χi1,i2,i3,i4 which does x → (e

2πi 8 )i1

y → (e

2πi 8 )i2

z → (e

2πi 8 )i3

w → (−1)i4 0 ≤ i1 ≤ 7, 0 ≤ i4 ≤ 1

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

SLIDE 32

The idempotents

This means there are 1024 idempotents. Combining into rational idempotents gives 296 rational idempotents. For convenience we will multiply rational idempotents by 32

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

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The rational idempotents

We further break up the 296 rational idempotents by what roots of unity they send the elements of the group. ζ8 = e

2πi 8

There are 224 rational idempotents where an element to g → ζ8 There are 56 rational idempotents where an element to g → ζ4 (no elements sent to ζ8) There are 16 rational idempotents where an element g → ±1

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

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The rational idempotents

Use the 224 rational idempotents which have g → ζ8. Combine to cover the portion of the group < x >< y >< z >< w > − < x2 >< y2 >< z2 >< w >

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

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The rational idempotents

Use the 72 rational idempotents g → ζ4 or → ±1. Combine to cover the portion of the group < x2 >< y2 >< z2 >< w >

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

SLIDE 36

The rational idempotents

Use the 224 rational idempotents → ζ8. Combine to cover < x >< y >< z >< w > − < x2 >< y2 >< z2 >< w > How many ways to send a group element to ζ8? Could send some power of x , some power of y, or some power of z to ζ8

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

SLIDE 37

x → ζ8

Look at rational idempotents sending x to ζ8. The elements y and z are not sent to ζ8 32[e1,0,0,0] = 1

8(1 − x4) < y >< z >< w >

Any time i2, i3, i4 even in 32[e1,i2,i3,i4] 32[e1,i2,i3,i4] = 1

8(1 − x4) < x8−i2y >< x8−i3z > (1 + (−1)i4w)

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

SLIDE 38

x → ζ8

x to ζ8. y and z not to ζ8 32[e1,0,0,0], 32[e1,4,0,0], 32[e1,0,4,0], 32[e1,4,4,0], 32[e1,0,0,1], 32[e1,4,0,1], 32[e1,0,4,1], 32[e1,4,4,1] 32[e1,2,0,0], 32[e1,6,0,0], 32[e1,2,4,0], 32[e1,6,4,0], 32[e1,2,0,1], 32[e1,6,0,1], 32[e1,2,4,1], 32[e1,6,4,1] 32[e1,0,2,0], 32[e1,4,2,0], 32[e1,0,6,0], 32[e1,4,6,0], 32[e1,0,2,1], 32[e1,4,2,1], 32[e1,0,6,1], 32[e1,4,6,1] 32[e1,2,2,0], 32[e1,6,2,0], 32[e1,2,6,0], 32[e1,6,6,0], 32[e1,2,2,1], 32[e1,6,2,1], 32[e1,2,6,1], 32[e1,6,6,1]

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

SLIDE 39

x → ζ8

x to ζ8. y and z not to ζ8 32[e1,0,0,0], 32[e1,4,0,0], 32[e1,0,4,0], 32[e1,4,4,0], 32[e1,0,0,1], 32[e1,4,0,1], 32[e1,0,4,1], 32[e1,4,4,1] 32[e1,2,0,0], 32[e1,6,0,0], 32[e1,2,4,0], 32[e1,6,4,0], 32[e1,2,0,1], 32[e1,6,0,1], 32[e1,2,4,1], 32[e1,6,4,1] 32[e1,0,2,0], 32[e1,4,2,0], 32[e1,0,6,0], 32[e1,4,6,0], 32[e1,0,2,1], 32[e1,4,2,1], 32[e1,0,6,1], 32[e1,4,6,1] 32[e1,2,2,0], 32[e1,6,2,0], 32[e1,2,6,0], 32[e1,6,6,0], 32[e1,2,2,1], 32[e1,6,2,1], 32[e1,2,6,1], 32[e1,6,6,1]

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

SLIDE 40

x → ζ8

x to ζ8. y and z not to ζ8 32[e1,0,0,0], 32[e1,4,0,0], 32[e1,0,4,0], 32[e1,4,4,0], 32[e1,0,0,1], 32[e1,4,0,1], 32[e1,0,4,1], 32[e1,4,4,1] Sum is A1 = (1 − x4) < y2 >< z2 > Group elements that can be sent to ±1 by all character values: y, z, yz, yw, zw, yzw

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

SLIDE 41

x → ζ8

x to ζ8. y and z not to ζ8 32[e1,2,0,0], 32[e1,6,0,0], 32[e1,2,4,0], 32[e1,6,4,0], 32[e1,2,0,1], 32[e1,6,0,1], 32[e1,2,4,1], 32[e1,6,4,1] Sum is A2 = (1 − x4) < x4y2 >< z2 > Group elements that can be sent to ±1 by all character values: x2y, z, x2yz, x2yw, zw, x2yzw,

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

SLIDE 42

x → ζ8

x to ζ8. y and z not to ζ8 32[e1,0,2,0], 32[e1,4,2,0], 32[e1,0,6,0], 32[e1,4,6,0], 32[e1,0,2,1], 32[e1,4,2,1], 32[e1,0,6,1], 32[e1,4,6,1] Sum is A3 = (1 − x4) < y2 >< x4z2 > Group elements that can be sent to ±1 by all character values: y, x2z, x2yz, yw, x2zw, x2yzw,

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

SLIDE 43

x → ζ8

x to ζ8. y and z not to ζ8 32[e1,2,2,0], 32[e1,6,2,0], 32[e1,2,6,0], 32[e1,6,6,0], 32[e1,2,2,1], 32[e1,6,2,1], 32[e1,2,6,1], 32[e1,6,6,1] Sum is A4 = (1 − x4) < x4y2 >< x4z2 > Group elements that can be sent to ±1 by all character values: x2y, x2z, yz, x2yw, x2zw, yzw,

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

SLIDE 44

x → ζ8

Lots of ways to combine and get a few results. For instance zA1 + zwA2 + x2zA3 + x2zwA4 has support z < x2 >< y2 >< z2 >< w > yA1 + ywA3 + x2yA2 + x2ywA4 has support y < x2 >< y2 >< z2 >< w > yzA1 + x2yzwA2 + x2yzA3 + yzwA4 has support yz < x2 >< y2 >< z2 >< w >

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

SLIDE 45

x → ζ8

Options for the rational idempotents with x → ζ8 . Support z < x2 >< y2 >< z2 >< w > y < x2 >< y2 >< z2 >< w > yz < x2 >< y2 >< z2 >< w >

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

SLIDE 46

g → ζ8

We can get support of a group element multiplied by H =< x2 >< y2 >< z2 >< w > each time we look at sets of rational idempotents which have generators sent to primitive 8th roots of unity Group elements g → ζ8 Possible elements times H x z, y, yz y z, x, xz z x, y, yx x y z, xy, xyz x z y, xz, xyz y z x, yz, xyz x y z xz, xy, yz

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

SLIDE 47

g → ζ8

We can get support of a group element multiplied by H =< x2 >< y2 >< z2 >< w > each time we look at sets of rational idempotents which have generators sent to primitive 8th roots of unity Group elements g → ζ8 Possible elements times H x z, y, yz y z, x, xz z x, y, yx x y z, xy, xyz x z y, xz, xyz y z x, yz, xyz x y z xz, xy, yz

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

SLIDE 48

g → ζ8

Gives support as < x >< y >< z >< w > − < x2 >< y2 >< z2 >< w > Group elements g → ζ8 Possible elements times H x z, y, yz y z, x, xz z x, y, yx x y z, xy, xyz x z y, xz, xyz y z x, yz, xyz x y z xz, xy, yz

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

SLIDE 49

g → ζ4

Similarly we can send all group elements to a power of ζ4. 32[e0,0,0,∗], 32[e4,0,0,∗], 32[e0,4,0,∗], 32[e0,0,4,∗], 32[e4,4,0,∗], 32[e4,0,4,∗], 32[e0,4,4,∗], 32[e4,4,4,∗] 32[e2,0,0,∗], 32[e2,4,0,∗], 32[e2,0,4,∗], 32[e2,4,4,∗] 32[e0,0,2,∗], 32[e4,0,2,∗], 32[e0,4,2,∗], 32[e4,4,2,∗], 32[e2,0,2,∗], 32[e2,0,6,∗], 32[e2,4,2,∗], 32[e2,4,6,∗] 32[e0,2,0,∗], 32[e4,2,0,∗], 32[e0,2,4,∗], 32[e4,2,4,∗], 32[e2,2,0,∗], 32[e2,6,0,∗], 32[e2,2,4,∗], 32[e2,6,4,∗] 32[e0,2,2,∗], 32[e0,2,6,∗], 32[e4,2,2,∗], 32[e4,2,6,∗], 32[e2,2,2,∗], 32[e2,2,6,∗], 32[e2,6,2,∗], 32[e2,6,6,∗]

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

SLIDE 50

g → ζ4

Similarly we can send all group elements to a power of ζ4. 32[e0,0,0,∗], 32[e4,0,0,∗], 32[e0,4,0,∗], 32[e0,0,4,∗], 32[e4,4,0,∗], 32[e4,0,4,∗], 32[e0,4,4,∗], 32[e4,4,4,∗] 32[e2,0,0,∗], 32[e2,4,0,∗], 32[e2,0,4,∗], 32[e2,4,4,∗] 32[e0,0,2,∗], 32[e4,0,2,∗], 32[e0,4,2,∗], 32[e4,4,2,∗], 32[e2,0,2,∗], 32[e2,0,6,∗], 32[e2,4,2,∗], 32[e2,4,6,∗] Let sums be Q1 , Q2

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

SLIDE 51

g → ζ4

Similarly we can send all group elements to a power of ζ4. 32[e0,2,0,∗], 32[e4,2,0,∗], 32[e0,2,4,∗], 32[e4,2,4,∗], 32[e2,2,0,∗], 32[e2,6,0,∗], 32[e2,2,4,∗], 32[e2,6,4,∗] 32[e0,2,2,∗], 32[e0,2,6,∗], 32[e4,2,2,∗], 32[e4,2,6,∗], 32[e2,2,2,∗], 32[e2,2,6,∗], 32[e2,6,2,∗], 32[e2,6,6,∗] Let sums be Q3 , Q4

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

SLIDE 52

g → ζ4

1Q1 + x2Q2 + x2wQ3 + wQ4 is a tile with support of < x2 >< y2 >< z2 >< w >

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

SLIDE 53

The difference set

Adding tiles we get support of (C8)(3) × C2 Since we chose aliases where characters send to ±32 We have a difference set D!

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

SLIDE 54

Generalization

(C8)(3) × C2 had a tiling structure. The breakdown on the support was < x >< y >< z >< w > − < x2 >< y2 >< z2 >< w > < x2 >< y2 >< z2 >< w >

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

SLIDE 55

Generalization

(C32)(3) × C2 has a tiling structure. The breakdown on the support was < x >< y >< z >< w > − < x2 >< y2 >< z2 >< w > < x2 >< y2 >< z2 >< w > − < x4 >< y4 >< z4 >< w > < x4 >< y4 >< z4 >< w > − < x8 >< y8 >< z8 >< w > < x8 >< y8 >< z8 >< w >

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

SLIDE 56

Generalization

(C22r+1)(3) × C2 has a tiling structure. The breakdown on the support was < x >< y >< z >< w > − < x2 >< y2 >< z2 >< w > < x2 >< y2 >< z2 >< w > − < x4 >< y4 >< z4 >< w > . . . < x22r−1 >< y22r−1 >< z22r−1 >< w >

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

SLIDE 57

What we know

Abelian non-2 groups have no reversible difference sets Some Abelian 2 groups with reversible difference sets: C4, (C2r )(2), (C22r )(3) Direct products of groups that contain reversible difference sets have reversible difference sets. Some Abelian 2 groups don’t have reversible difference sets. Ex: C8 × C2, C64 × (C16)(2).

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

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What we don’t/didn’t know

Odd powers on groups with C2: Example (C22r+1)(3) × C2 Odd powers on groups without C2: Example (C22r+1)(3) × (C22s+1)(3) Single cyclic group in direct product with no match Examples (C32)(2) × C16, (C32)(3) × C2

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

SLIDE 59

What we don’t/didn’t know

Odd powers on groups with C2: Example (C32)(3) × C2 Odd powers on groups without C2: Example (C128)(3) × (C8)(3) Single cyclic group in direct product with no match Examples (C32)(2) × C16, (C32)(3) × C2

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

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Possibilities

Odd powers on groups without C2: Example (C128)(3) × (C8)(3)

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

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Difficulties

Single cyclic group in direct product with no match Example (C32)(2) × C16

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

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Implications

Chipping away at Necessary and Sufficient conditions for Reversible difference sets in abelian groups.

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups

SLIDE 63

Thanks

Thank you!

Jordan D. Webster Mid Michigan Community College Reversible Menon-Hadamard Difference Sets in Abelian 2-groups