Tiling groups with difference sets Vedran Kr cadinac 1/17 Joint - PowerPoint PPT Presentation

Tiling groups with difference sets Vedran Krˇ cadinac 1/17 Joint work with: University of Zagreb, Croatia krcko@math.hr Ante ´ Custi´ c Yue Zhou Simon Fraser University Otto-von-Guericke University Surrey, Canada Magdeburg, Germany acustic@sfu.ca yue.zhou.ovgu@gmail.com D edicated to the memory of Axel Kohnert. ◭ ◮ Back FullScr

Let G be an additively written group of order v . A ( v, k, λ ) difference set in G is a k -subset D ⊆ G such that every nonzero element of G can be expressed as a difference x − y with x, y ∈ D in exactly λ ways. 2/17 ◭ ◮ Back FullScr

Let G be an additively written group of order v . A ( v, k, λ ) difference set in G is a k -subset D ⊆ G such that every nonzero element of G can be expressed as a difference x − y with x, y ∈ D in exactly λ ways. 2/17 Tiling = partition of G into disjoint ( v, k, λ ) difference sets. ◭ ◮ Back FullScr

Let G be an additively written group of order v . A ( v, k, λ ) difference set in G is a k -subset D ⊆ G such that every nonzero element of G can be expressed as a difference x − y with x, y ∈ D in exactly λ ways. 2/17 Tiling = partition of G into disjoint ( v, k, λ ) difference sets. Theorem 1. It is not possible to tile G by difference sets. ◭ ◮ Back FullScr

Let G be an additively written group of order v . A ( v, k, λ ) difference set in G is a k -subset D ⊆ G such that every nonzero element of G can be expressed as a difference x − y with x, y ∈ D in exactly λ ways. 2/17 Tiling = partition of G into disjoint ( v, k, λ ) difference sets. Theorem 1. It is not possible to tile G by difference sets. � k − 1 � λ + 1 λ ( v − 1) = k ( k − 1) ⇒ v = k · . Proof. k However, k − 1 λ + 1 k can never be an integer. ◭ ◮ Back FullScr

Let G be an additively written group of order v . A ( v, k, λ ) difference set in G is a k -subset D ⊆ G such that every nonzero element of G can be expressed as a difference x − y with x, y ∈ D in exactly λ ways. 2/17 Tiling = partition of G into disjoint ( v, k, λ ) difference sets. Theorem 1. It is not possible to tile G by difference sets. � k − 1 � λ + 1 λ ( v − 1) = k ( k − 1) ⇒ v = k · . Proof. k However, k − 1 λ + 1 k can never be an integer. Let’s partition G \ { 0 } instead! ◭ ◮ Back FullScr

Definition. Let G be a finite group of order v with neutral element 0 . A ( v, k, λ ) tiling of G is a collection { D 1 , . . . , D t } of mutually disjoint ( v, k, λ ) difference sets such that D 1 ∪ · · · ∪ D t = G \ { 0 } . 3/17 ◭ ◮ Back FullScr

Definition. Let G be a finite group of order v with neutral element 0 . A ( v, k, λ ) tiling of G is a collection { D 1 , . . . , D t } of mutually disjoint ( v, k, λ ) difference sets such that D 1 ∪ · · · ∪ D t = G \ { 0 } . 3/17 Example 1. A (7 , 3 , 1) tiling of Z 7 : D 1 = { 1 , 2 , 4 } , D 2 = { 3 , 5 , 6 } . ◭ ◮ Back FullScr

Definition. Let G be a finite group of order v with neutral element 0 . A ( v, k, λ ) tiling of G is a collection { D 1 , . . . , D t } of mutually disjoint ( v, k, λ ) difference sets such that D 1 ∪ · · · ∪ D t = G \ { 0 } . 3/17 Example 1. A (7 , 3 , 1) tiling of Z 7 : D 1 = { 1 , 2 , 4 } , D 2 = { 3 , 5 , 6 } . Example 2. A (31 , 6 , 1) tiling of Z 31 : D 1 = { 1 , 5 , 11 , 24 , 25 , 27 } , D 2 = { 2 , 10 , 17 , 19 , 22 , 23 } , D 3 = { 3 , 4 , 7 , 13 , 15 , 20 } , D 4 = { 6 , 8 , 9 , 14 , 26 , 30 } , D 5 = { 12 , 16 , 18 , 21 , 28 , 29 } . ◭ ◮ Back FullScr

An application: 4/17 ◭ ◮ Back FullScr

An application: 4/17 ◭ ◮ Back FullScr

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In the terms of the previous presentation, this matrix is a 6 -mosaic 2 - (31 , 6 , 1) ⊕ · · · ⊕ 2 - (31 , 6 , 1) ⊕ 2 - (31 , 1 , 0) . � �� � 5 times 7/17 ◭ ◮ Back FullScr

In the terms of the previous presentation, this matrix is a 6 -mosaic 2 - (31 , 6 , 1) ⊕ · · · ⊕ 2 - (31 , 6 , 1) ⊕ 2 - (31 , 1 , 0) . � �� � 5 times 7/17 Generally, the development of a ( v, k, λ ) tiling of G by t difference sets is a ( t + 1) -mosaic of symmetric designs 2 - ( v, k, λ ) ⊕ · · · ⊕ 2 - ( v, k, λ ) ⊕ 2 - ( v, 1 , 0) . � �� � t times ◭ ◮ Back FullScr

In the terms of the previous presentation, this matrix is a 6 -mosaic 2 - (31 , 6 , 1) ⊕ · · · ⊕ 2 - (31 , 6 , 1) ⊕ 2 - (31 , 1 , 0) . � �� � 5 times 7/17 Generally, the development of a ( v, k, λ ) tiling of G by t difference sets is a ( t + 1) -mosaic of symmetric designs 2 - ( v, k, λ ) ⊕ · · · ⊕ 2 - ( v, k, λ ) ⊕ 2 - ( v, 1 , 0) . � �� � t times Another application of ( v, k, λ ) tilings: hopping sequences for multi- channel wireless networks. • F. Hou, L.X. Cai, X. Shen, and J. Huang, Asynchronous multichannel MAC design with difference-set-based hopping sequences , IEEE Transactions on Vehicular Technology, 60 (2011), no. 4, 1728–1739. • K. Wu, F. Han, F. Han, and D. Kong, Rendezvous sequence construction in cognitive radio ad- ◭ hoc networks based on difference sets , 2013 IEEE 24th International Symposium on Personal ◮ Indoor and Mobile Radio Communications, IEEE 2013, p. 1840–1845. Back FullScr

Necessary existence conditions The number of difference sets in a ( v, k, λ ) tiling: t = v − 1 = k − 1 8/17 λ ( v − 1) = k ( k − 1) ⇒ k λ ◭ ◮ Back FullScr

Necessary existence conditions The number of difference sets in a ( v, k, λ ) tiling: t = v − 1 = k − 1 8/17 λ ( v − 1) = k ( k − 1) ⇒ k λ Lemma 1. If a ( v, k, λ ) tiling exists, then k divides v − 1 . ◭ ◮ Back FullScr

Necessary existence conditions The number of difference sets in a ( v, k, λ ) tiling: t = v − 1 = k − 1 8/17 λ ( v − 1) = k ( k − 1) ⇒ k λ Lemma 1. If a ( v, k, λ ) tiling exists, then k divides v − 1 . The parameters (16 , 6 , 2) are ruled out by this criterion. ◭ ◮ Back FullScr

Necessary existence conditions The number of difference sets in a ( v, k, λ ) tiling: t = v − 1 = k − 1 8/17 λ ( v − 1) = k ( k − 1) ⇒ k λ Lemma 1. If a ( v, k, λ ) tiling exists, then k divides v − 1 . The parameters (16 , 6 , 2) are ruled out by this criterion. Only parameters ( v, k, λ ) satisfying Lemma 1 and groups G in which there is at least one ( v, k, λ ) difference set are considered admissible for tilings. ◭ ◮ Back FullScr

Necessary existence conditions The number of difference sets in a ( v, k, λ ) tiling: t = v − 1 = k − 1 8/17 λ ( v − 1) = k ( k − 1) ⇒ k λ Lemma 1. If a ( v, k, λ ) tiling exists, then k divides v − 1 . The parameters (16 , 6 , 2) are ruled out by this criterion. Only parameters ( v, k, λ ) satisfying Lemma 1 and groups G in which there is at least one ( v, k, λ ) difference set are considered admissible for tilings. ◭ The parameters (31 , 10 , 3) are not admissible because there is no (31 , 10 , 3) ◮ difference set in Z 31 (a nontrivial result!). Back FullScr

Translates of a ( v, k, λ ) difference set D ⊆ G are the sets D + x , x ∈ G . They form a symmetric ( v, k, λ ) block design. 9/17 ◭ ◮ Back FullScr

Translates of a ( v, k, λ ) difference set D ⊆ G are the sets D + x , x ∈ G . They form a symmetric ( v, k, λ ) block design. Lemma 2. If { D 1 , . . . , D t } is a ( v, k, λ ) tiling of a group G , then the 9/17 difference sets D 1 , . . . , D t are not translates of each other. ◭ ◮ Back FullScr

Translates of a ( v, k, λ ) difference set D ⊆ G are the sets D + x , x ∈ G . They form a symmetric ( v, k, λ ) block design. Lemma 2. If { D 1 , . . . , D t } is a ( v, k, λ ) tiling of a group G , then the 9/17 difference sets D 1 , . . . , D t are not translates of each other. Proposition 1. A (21 , 5 , 1) tiling of Z 21 does not exist. ◭ ◮ Back FullScr

Translates of a ( v, k, λ ) difference set D ⊆ G are the sets D + x , x ∈ G . They form a symmetric ( v, k, λ ) block design. Lemma 2. If { D 1 , . . . , D t } is a ( v, k, λ ) tiling of a group G , then the 9/17 difference sets D 1 , . . . , D t are not translates of each other. Proposition 1. A (21 , 5 , 1) tiling of Z 21 does not exist. Proposition 2. A (57 , 8 , 1) tiling of Z 57 does not exist. ◭ ◮ Back FullScr

Translates of a ( v, k, λ ) difference set D ⊆ G are the sets D + x , x ∈ G . They form a symmetric ( v, k, λ ) block design. Lemma 2. If { D 1 , . . . , D t } is a ( v, k, λ ) tiling of a group G , then the 9/17 difference sets D 1 , . . . , D t are not translates of each other. Proposition 1. A (21 , 5 , 1) tiling of Z 21 does not exist. Proposition 2. A (57 , 8 , 1) tiling of Z 57 does not exist. Example 3. A (57 , 8 , 1) tiling of the non-abelian group of order 57 , G = � a, b | a 3 = b 19 = 1 , ab 7 = ba � : ◭ ◮ Back FullScr

D 1 = { a, b, a 2 , b 2 , ab 4 , ab 10 , b 13 , b 18 } , D 2 = { ab, ab 5 , a 2 b 6 , a 2 b 13 , b 15 , a 2 b 14 , ab 15 , ab 18 } , D 3 = { a 2 b, a 2 b 7 , a 2 b 8 , ab 9 , ab 12 , b 14 , ab 14 , a 2 b 16 } , 10/17 D 4 = { ab 2 , b 4 , a 2 b 3 , b 9 , a 2 b 9 , b 11 , b 12 , a 2 b 18 } , D 5 = { b 3 , a 2 b 2 , b 5 , b 8 , a 2 b 10 , a 2 b 11 , ab 17 , a 2 b 17 } , D 6 = { ab 3 , b 6 , ab 6 , ab 8 , b 10 , b 16 , a 2 b 15 , b 17 } , D 7 = { a 2 b 4 , a 2 b 5 , b 7 , ab 7 , ab 11 , a 2 b 12 , ab 13 , ab 16 } . ◭ ◮ Back FullScr