On designs and Steiner systems over finite fields Alfred Wassermann - PowerPoint PPT Presentation

On designs and Steiner systems over finite fields Alfred Wassermann Department of Mathematics, Universität Bayreuth, Germany Special Days on Combinatorial Construction Using Finite Fields, Linz 2013

Outline ◮ Network coding ◮ Design theory ◮ Symmetry ◮ Computer construction ◮ Projective geometry ◮ New results (joint work with M. Braun, T. Etzion, A. Kohnert, P . Östergård, A. Vardy) ◮ Summary

Network coding

Flow network source ◮ directed graph, with sources and sinks ◮ each edge e has a capacity c e ◮ each edge receives a non-negative flow ƒ e ≤ c e ◮ the net flow into any non-source non-sink vertex is zero In the following: ◮ c e = 1 ◮ ƒ e ∈ { 0 , 1 } sink sink

Flow networks Theorem (Ford, Fulkerson 1956, Elias, Feinstein, Shannon 1956) In a flow network, the maximum amount of flow passing from a source s to a sink t is equal to the minimum capacity, which when removed, separates s from t. Theorem (Menger 1927) Maximum number of edge-disjoint paths from s to t in a directed graph is equal to the minimum s-t cut.

Example: 1 source, 1 sink source ◮ cut-capacity = 2 ◮ min-cut = 2 = max-flow ◮ Menger’s theorem: two edge-disjoint paths ◮ route packets  and b along these paths sink

Example: 1 source, 1 sink source  b ◮ cut-capacity = 2 b ◮ min-cut = 2 = max-flow ◮ Menger’s theorem: two edge-disjoint paths  b ◮ route packets  and b along these paths b sink

Example: 1 source, 2 sinks source ◮ cut-capacity = 2 ◮ can route 2 packets to one sink, 1 packet to the other ◮ and vice-versa ◮ Time-sharing between these two strategies can achieve a multicast rate of 1.5 packets per use of the network. sink sink

Example: 1 source, 2 sinks source  b ◮ cut-capacity = 2 b ◮ can route 2 packets to one sink, 1 packet to the other ◮ and vice-versa  ◮ Time-sharing between these b two strategies can achieve a multicast rate of 1.5 packets per use of the network. b b sink sink

Example: 1 source, 2 sinks source  b ◮ cut-capacity = 2  ◮ can route 2 packets to one sink, 1 packet to the other ◮ and vice-versa  ◮ Time-sharing between these b two strategies can achieve a multicast rate of 1.5 packets per use of the network.   sink sink

Example: 1 source, 2 sinks source ◮ perform coding at the  b bottle-neck ◮  and b are packets of bits  b ◮  ⊕ b =  + b over F 2 ◮  ⊕ (  ⊕ b ) = b b ⊕ (  ⊕ b ) =  ◮ both sinks can recover both   ⊕ b b messages ◮ Network coding achieves a multicast rate of 2 packets per use of the network  ⊕ b  ⊕ b ◮ best possible sink sink

Network coding – essence ◮ R. Ahlswede, N. Cai, S.-Y . R. Li, R. W. Yeung 2000 ◮ packets can be mixed with each other – rather than just routed or replicated ◮ a higher throughput can be achieved

Error correction in noncoherent network coding R. Kötter F . Kschischang ◮ Kötter, Kschischang (2008) ◮ Silva, Kötter, Kschischang (2008)

Error correction in noncoherent network coding Possible error sources: ◮ Random errors that could not be detected at the physical layer ◮ Corrupt packets injected at the application level by a malicious user

Error correction in noncoherent network coding Possible error sources: ◮ Random errors that could not be detected at the physical layer ◮ Corrupt packets injected at the application level by a malicious user Local view at routing node: ◮ Randomly combine incoming packets linearly ◮ A corrupt packet is modeled as the addition of an error packet to a genuine packet m � P ( ot )  j P ( in ) = + E   j j = 1

Error propagation ◮ Packet mixing makes network coding highly prone to error propagation. This essentially rules out classical error correction.

Error correction in noncoherent network coding Global view: ◮ The overall network can be viewed as a point-to-point channel  X 1   Y 1  X 2 Y 2 ◮ Source: X =     sink: Y = . .  .   .  . .     X k Y k ′ ◮ X  , Y j ∈ F  q ◮ Transmission: �→ Y = A · X + B · E, X where A , B , E are unknown

Key observation X �→ Y = A · X + B · E In case E = 0: X �→ Y = A · X rows of A · X (= row space of X ) ∈ 〈 X 1 , X 2 , . . . , X k 〉

Random linear network coding ◮ Randomly combine information vectors at intermediate nodes ◮ Codewords are subspaces of a finite vector space ◮ Convenient: all codewords have same dimension k

Network codes ◮ ambient space V = F  q ◮ constant dimension (network) code: C ⊆ { U ≤ F  q : d im U = k } ◮ Grassmannian: G q ( , k ) : = { U ≤ F  q : dim U = k } H. Graßmann

Subspace lattice of F 4 2 1000 0100 G 2 ( 4 , 4 ) 0010 0001 0100 G 2 ( 4 , 3 ) 0010 0001 0010 G 2 ( 4 , 2 ) 0001 G 2 ( 4 , 1 ) 0001 G 2 ( 4 , 0 ) 0000

Subspace lattice �  � ◮ | G q ( , k ) | = k q ◮ Gaussian coefficient: ( q  − 1 )( q  − 1 − 1 ) · · · ( q  − k + 1 − 1 ) �  � = ( q k − 1 )( q k − 1 − 1 ) · · · ( q − 1 ) k q �  �  � � ◮ lim q → 1 q = k k

Subspace distance ◮ subspace distance for U, V ∈ G q ( , k ) d im U + dim V − 2 dim U ∩ V d ( U, V ) = 2 k − 2 dim U ∩ V = = : 2 δ ◮ minimum distance d ( C ) : = min{ d ( U, V ) : U, V ∈ C , U � = V }

Subspace distance in F 4 2 G 2 ( 4 , 4 ) G 2 ( 4 , 3 ) G 2 ( 4 , 2 ) G 2 ( 4 , 1 ) G 2 ( 4 , 0 )

Problems ◮ maximize | C | for given  , k , d ◮ determine upper and lower bounds for A q ( , k, d ) : = mx{ | C | : C ⊆ G q ( , k ) , d ( C ) ≥ d }

Upper bounds | G q ( , k ) | ◮ Sphere packing bound: A q ( , k, 2 δ ) ≤ | B k ( δ − 1 ) | �  − δ + 1 ◮ Singleton bound: A q ( , k, 2 δ ) ≤ � k − δ + 1 q ◮ Anticode bound: ◮ Anticode of diameter e : set of subspaces U ∈ G q ( , k ) such that all pairwise distances are ≤ e �   � � � k − δ + 1 k q q ◮ A q ( , k, 2 δ ) ≤ = �  − k + δ − 1 k � � � δ − 1 k − δ + 1 q q ◮ Johnson type bounds: � q  − 1 � A q ( , k, 2 δ ) ≤ · A q (  − 1 , k − 1 , 2 δ ) q k − 1

Previous bounds for A 2 ( , 3 , 4 ) Ref ≥ ≤  6 77 81 [K] 7 329 381 [B] 8 1312 1493 [B] 9 5694 6205 [E] 10 21483 24698 [K] 11 92411 99718 [B] 12 385515 398385 [B] 13 1490762 1597245 14 5996178 6387029 [B] d im 3 = k U V ◮ [K] Kohnert, Kurz (2008) ◮ [E] Etzion, Vardy (2008) dim 1 ◮ [B] Braun, Reichelt (2013) {0}

Constant dimension codes ◮ U, V ∈ G q ( , k ) : d ( U, V ) = 2 k − 2 d im U ∩ V = 2 δ ◮ Let t − 1 : = k − δ U V dim k δ W dim t U ∩ V dim t − 1 ◮ d ( C ) = 2 δ : for all U, V ∈ C , U � = V dim U ∩ V ≤ t − 1 ◮ For all W ∈ G q ( , t ) : | { U ∈ C : W ≤ U } | ≤ 1

Extremal case ◮ C ⊆ G q ( , k ) ◮ For all W ∈ G q ( , t ) : | { U ∈ C : W ≤ U } | ≤ 1

Extremal case ◮ C ⊆ G q ( , k ) ◮ For all W ∈ G q ( , t ) : | { U ∈ C : W ≤ U } | ≤ 1 ◮ Extremal case: for all W ∈ G q ( , t ) | { U ∈ C : W ≤ U } | = 1

Extremal case ◮ C ⊆ G q ( , k ) ◮ For all W ∈ G q ( , t ) : | { U ∈ C : W ≤ U } | ≤ 1 ◮ Extremal case: for all W ∈ G q ( , t ) | { U ∈ C : W ≤ U } | = 1 ◮ In this case, | C | meets anticode bound and Johnson bound: �   � � � k − δ + 1 t q q | C | = = � k k � � � k − δ + 1 t q q ◮ C : perfect diameter code

Design theory

Design theory ◮ Cameron (1974), Delsarte (1976) P . Cameron ◮ B ⊆ G q ( , k ) : set of k -subspaces (blocks) ◮ ( F  q , B ) : q -Steiner system S q [ t, k,  ] each t -subspace of F  q is contained in exactly one block of B

Design theory ◮ Cameron (1974), Delsarte (1976) P . Cameron ◮ B ⊆ G q ( , k ) : set of k -subspaces (blocks) ◮ ( F  q , B ) : q -Steiner system S q [ t, k,  ] each t -subspace of F  q is contained in exactly one block of B More general: ◮ B ⊆ G q ( , k ) : set of k -subspaces (blocks) ◮ ( F  q , B ) : t - ( , k, λ ; q ) design over F q each t -subspace of F  q is contained in exactly λ blocks of B

Design theory ◮ B set: simple design ◮ B multiset: non-simple design

Design theory ◮ B set: simple design ◮ B multiset: non-simple design �  − t � ◮ B = G q ( , k ) is a t - ( , k, q ; q ) design: trivial k − t design trivial 1- ( 4 , 2 , 7; 2 ) design

Design theory ◮ B set: simple design ◮ B multiset: non-simple design �  − t � ◮ B = G q ( , k ) is a t - ( , k, q ; q ) design: trivial k − t design trivial 1- ( 4 , 2 , 7; 2 ) design 1- ( 4 , 2 , 1; 2 ) design

t - ( , k, λ ; q ) designs  ◮ | B | = λ [ t ] q k [ t ] q ◮ Necessary conditions: �  −  � t −  q for  = 0 , . . . , t ∈ Z λ  = λ � k −  � t −  q ◮ Example: t = 2, k = 3 , λ = 1  ≡ 1 , 3 ( mod 6 ) ⇒