On designs and Steiner systems over finite fields

Transcription

1 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

2 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

3 Network coding

4 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 sink sink In the following: c e = 1 ƒ e {0, 1}

5 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.

6 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

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

8 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

9 Example: 1 source, 2 sinks source b b b 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 b b sink

10 Example: 1 source, 2 sinks source b b 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

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

12 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

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

14 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

15 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 P (o t) = m j=1 j P (in) j + E

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

17 Error correction in noncoherent network coding Global view: The overall network can be viewed as a point-to-point channel X 1 X 2 Source: X = X, Y j F q Transmission:. X k sink: Y = Y 1 Y 2. Y k X Y = A X + B E, where A, B, E are unknown

18 Key observation X Y = A X + B E In case E = 0: X Y = A X rows of A X X 1, X 2,..., X k (= row space of X)

19 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

20 Network codes ambient space V = F q constant dimension (network) code: C {U F : dim U = k} q Grassmannian: G q (, k) := {U F : dim U = k} q H. Graßmann

21 Subspace lattice of F G 2 (4, 4) G 2 (4, 3) G 2 (4, 2) 0001 G 2 (4, 1) 0000 G 2 (4, 0)

22 Subspace lattice G q (, k) = k Gaussian coefficient: k lim q 1 k q q = (q 1)(q 1 1) (q k+1 1) (q k 1)(q k 1 1) (q 1) q = k

23 Subspace distance subspace distance for U, V G q (, k) d(u, V) = dim U + dim V 2 dim U V = 2k 2 dim U V =: 2δ minimum distance d(c) := min{d(u, V) : U, V C, U = V}

24 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)

25 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}

26 Upper bounds Sphere packing bound: A q (, k, 2δ) G q(, k) B k (δ 1) Singleton bound: A q (, k, 2δ) δ+1 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 Aq (, k, 2δ) Johnson type bounds: A q (, k, 2δ) q k+δ 1 δ 1 q q 1 = k δ+1 q k k δ+1 q q k 1 A q( 1, k 1, 2δ)

27 Previous bounds for A 2 (, 3, 4) Ref [K] [B] [B] [E] [K] [B] [B] [B] [K] Kohnert, Kurz (2008) [E] Etzion, Vardy (2008) [B] Braun, Reichelt (2013) U V dim 3 = k {0} dim 1

28 Constant dimension codes U, V G q (, k): d(u, V) = 2k 2 dim U V = 2δ Let t 1 := k δ U V dim k δ W U V dim t dim t 1 d(c) = 2δ: dim U V t 1 for all U, V C, U = V For all W G q (, t): {U C : W U} 1

29 Extremal case C G q (, k) For all W G q (, t): {U C : W U} 1

30 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

31 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: t C = q k t C: perfect diameter code q = k δ+1 q k k δ+1 q

32 Design theory

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

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

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

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

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

38 t-(, k, λ; q) designs B = λ [ t] q [ k t] q Necessary conditions: λ = λ t q k t q Z for = 0,..., t Example: t = 2, k = 3, λ = 1 1, 3 (mod 6)

39 Related design parameters t-(, k, λ; q) design supplemented design: t-(, k, t k t q λ; q) complementary design: t-(, k, λ t reduced design: (t 1)-(, k, λ t+1 1 derived design: (t 1)-( 1, k 1, λ; q) k q / t k t q / k t+1 1 residual design: (t 1)-( 1, k, λ q k 1 q t+1 1 ; q) Kiermaier, Laue (2013) Open problem: t (t + 1)? q ; q) q ; q)

40 t-designs (over sets) V: set of points, V =. B: set of k-subsets K (blocks) (V, B): t-(, k, λ) design K V and K = k Every t-subset T V is contained in exactly λ blocks of B. t-(, k, 1) design: Steiner system S(t, k, ) J. Plücker 1835 T. Kirkman 1844 J. Steiner 1853 R. Fisher 1926

41 Example 4 d a c b 2 Task: Cover every vertex (1- subset) by exactly one edge (2-subset): 1-(4, 2, 1) design

42 Example 4 d a c b 2 Task: Cover every vertex (1- subset) by exactly one edge (2-subset): 1-(4, 2, 1) design 3 d a 1 design 1 c b d 4 a design 2 c b 2 d 5 c a 6 b design 3

43 t-designs (over sets) designs over finite fields are also called q-analogs related design parameters t (t + 1) Ajoodani-Namini (1996)

44 Large sets of designs (over sets) the set of all k-subsets is a t-(, k, t k t ) design: trivial design a partition of the trivial design into N disjoint t-(, k, λ) designs is called large set N λ = t k t LS[N](t, k, ) Sylvester (1860): packing large set of disjoint designs, Lindner, Rosa (1975)

45 Large sets of designs over finite fields G q (, k) is a t-(, k, n t ; q) design k t q Large set LS q [N](t, k, ): partition of G q (, k) into N disjoint t-(, k, λ; q) designs LS 2 [7](1, 2, 4) Necessary: N λ = t k t q

46 Symmetry

47 Automorphisms Designs over sets: S : symmetric group σ S is automorphism: B σ = B d c Example: 4 2 σ = ( d)(bc) a b Set of automorphisms: automorphism group

48 Automorphisms Designs over sets: S : symmetric group σ S is automorphism: B σ = B d c Example: 4 2 σ = ( d)(bc) a b Set of automorphisms: automorphism group Designs over finite fields: P L(, q) projective semilinear group GL(, q) = {M F : M invertible} q σ P L(, q) automorphism: B σ = B

49 Automorphisms of designs over finite fields Singer cycle: take F q as an element of F q (Fq \ {0}, ) is a cyclic group G of order q 1, i.e. G = σ G GL(, q) is called Singer cycle Frobenius automorphism: ϕ : Fq F q, U U q ϕ = σ, ϕ = (q 1) odd prime: σ, ϕ maximal subgroup in GL(, q) (Kantor 1980, Dye 1989)

50 Computer construction

51 Brute force approach for construction incidence matrix between t-subset and k-subsets: 1 if T K j M t,k = (m,j ), where m,j = 0 else solve M t,k = λ λ. λ for 0/1-vector

52 Example d 3 c d 4 c 2 d 5 c a 1 design 1 b a design 2 b 6 a design 3 b 4 d a c b 2 M 1, a b c d design design design 3 1 1

53 Designs with prescribed automorphism group Construction of designs with prescribed automorphism group choose group G acting on V, i.e. G S search for t-designs D = (V, B) having G as a group of automorphisms, i.e. for all construct D = (V, B) as g G and K B = K g B. union of orbits of G on k-subsets.

54 Example: cyclic symmetry d 4 a d c 2 b c a b c d {1, 2, 3, 4} {5, 6} a 2 1 b 2 1 c 2 1 d 2 1 a 6 b {1, 2, 3, 4} {5, 6} {a, b, c, d} 2 1 design 3

55 The method of Kramer and Mesner Definition K V and K = k: K G := {K g g G} T V and T = t: T G := {T g g G} Let V K G 1 KG 2... KG n k and T G 1 TG 2... TG m = V t M G t,k = (m,j) where m,j := {K K G j T K}

56 The method of Kramer and Mesner Theorem (Kramer and Mesner, 1976) The union of orbits corresponding to the 1s in a {0, 1} vector which solves M G t,k = is a t-(, k, λ) design having G as an automorphism group. λ λ. λ

57 Expected gain Brute force approach: M t,k = t k Kramer-Mesner: M G t,k ( t) ( k) G G

58 Solving algorithms t-designs with λ > 1: integer programming (CPLEX, Gurobi) lattice basis reduction + exhaustive enumeration (W. 1998, 2002) heuristic algorithms t-designs with λ = 1: maximum clique algorithms (Östergård: cliquer) exact cover (Knuth: dancing links)

59 Applications of Kramer-Mesner in Bayreuth (Betten, Braun, Kerber, Kiermaier, Kohnert, Kurz, Laue, W., Vogel, Zwanzger) designs over sets designs over finite fields large sets of designs linear codes self-orthogonal codes ring-linear codes two-weight codes arcs, blocking sets in projective geometry

60 Known designs over finite fields

61 Families of designs Thomas (1987): 2-(, 3, 7; 2) for 7 and ±1 (mod 6) Suzuki (1989): 2-(, 3, q 2 + q + 1; q) for 7 and ±1 (mod 6) Miyakawa, Munemasa, Yoshiara (1995): transitive designs 2-(7, 3, λ; q) for q = 2, 3 Itoh (1998): From 2-(, 3, q 3 (q 5 1)/(q 1); q) to 2-(m, 3, q 3 (q 5 1)/(q 1); q)

62 Designs over F 2 by computer construction Braun, Kerber, Laue (2005), S. Braun (2010) t-(, k, λ; q) G M G t,k λm x λ 3-(8, 4, λ; 2) σ, ϕ , 15 2-(10, 3, λ; 2) σ, ϕ , 30, 45, 60, 75, 90, 105, (9, 4, λ; 2) σ, ϕ , 63, 84, 126, 147, 189, 210, 252, 273, 315, 336, 378, 399, 441, 462, 504, 525, 567, 576, 588, 630, 651, 693, 714, 756, 777, 819, 840, 882, 903, 945, 966, 1008, 1029, 1071, 1092, 1134, 1155, 1197, 1218, 1260, 1281, (9, 3, λ; 2) σ, ϕ , 22, 42, 43, 63 2-(8, 4, λ; 2) σ, ϕ , 35, 56, 70, 91, 105, 126, 140, 161, 175, 196, 210, 231, 245, 266, 280, 301, (8, 3, λ; 2) σ (7, 3, λ; 2) σ , 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 2-(6, 3, λ; 2) σ , 6 σ: Singer cycle, ϕ: Frobenius automorphism

63 Projective geometry

64 Projective geometry projective space PG( 1, q) spread in PG( 1, q): set of lines that partitions the points, i.e. S q [1, 2, ]

65 Projective geometry projective space PG( 1, q) spread in PG( 1, q): set of lines that partitions the points, i.e. S q [1, 2, ] (k 1)-spread in PG( 1, q): S q [1, k, ] (k 1)-spreads exist iff k divides

66 Projective geometry projective space PG( 1, q) spread in PG( 1, q): set of lines that partitions the points, i.e. S q [1, 2, ] (k 1)-spread in PG( 1, q): S q [1, k, ] (k 1)-spreads exist iff k divides (t 1, k 1)-spreads in PG( 1, q): S q [t, k, ] also called (t, k 1)-systems in PG(, q), Ceccherini (1967), Tallini (1975)

67 Projective geometry spreads

68 Projective geometry spreads Motivation: André, Bose, Bruck construction (1954): spreads translation planes Spread codes and spread decoding in network codes (Manganiello, Gorla, Rosenthal 2008) Large set of spreads: parallelism, packing

69 Projective geometry (s, r)-spreads Beutelspacher 1978: Es scheint unbekannt zu sein, ob in einem endlichen projektiven Raum der Dimension d eine (s, r)-faserung existieren kann, wenn 0 < s < r < d gilt. Conjecture (Metsch 1999): (s, r)-spreads in finite projective spaces do not exist for s > 0.

70 Projective geometry (s, r)-spreads (s, r)-spreads in finite projective spaces do not exist for s > 0. translates to S q [t, k, ] Steiner systems over finite fields do not exist for t > 1.

71 New results

72 S 2 [2, 3, 13] does exist Braun, Etzion, Östergård, Vardy, W. (2013) submitted

73 S 2 [2, 3, 13] 13 = # blocks: 13 2 / 3 = Kramer-Mesner with group G = ϕ, σ ϕ = , σ = G = 13 (2 13 1) = all orbits are of full length G

74 S 2 [2, 3, 13] Kramer-Mesner matrix M G 2,3 = M G = ,3 # columns containing 0, 1 only = use dancing links by Knuth to solve the system up to now: 1030 non-isomorphic solutions 630 disjoint solutions i.e. 2-(13, 3, λ; 2) exist for λ = 1, 2,..., 630

75 Why Singer cycle + Frobenius? Transitive designs: A group G acts t-transitively on a vector space V if the set of t-subspaces is a single orbit. Theorem (Cameron, Kantor 1979) If G GL(, q) is t-transitive with t 2 then G is also k-transitive for t < k.

76 Miyakawa, Munemasa, Yoshiara 1995 Theorem (Hering 1974, Liebeck 1987) If G GL(, q) acts transitively on the 1-subspaces of F with 6, then one of the following holds: q G σ, ϕ SL (q n/ ) G, where, 2 Sp 2 (q /2 ) G, where 2 G 2 (q /6 ) G < Sp 6 (q /6 ), few sporadic cases for = 6 where q = 2 m and 6

77 The first large sets for t 2 LS 2 [3](2, 3, 8) does exist 2 Consists of three disjoint 2-(8, 3, 21; 2) designs Group: Singer cycle in GL(8, 2) of order 255 LS2 [3](2, 5, 8) does exist (complementary design) LS 3 [2](2, 3, 6) does exist Consists of two disjoint 2-(6, 3, 20; 3) designs LS 5 [2](2, 3, 6) does exist Consists of two disjoint 2-(6, 3, 20; 5) designs 2 Braun, Kohnert, Östergård, W. (2013) submitted

78 Summary

79 Bounds for A 2 (, 3, 4) Ref [K], [H] [B] [B] [E] [K] [B] [B] [B] [K] Kohnert, Kurz (2008) [E] Etzion, Vardy (2008) [B] Braun, Reichelt (2013) [H] Honold, Kiermaier, Kurz (2013) U V dim 3 = k dim 2 = t dim 1 {0}

80 Designs over sets vs. finite fields designs over sets designs over finite fields constructions for t 9 constructions for t = 2, 3 designs exist for all t (Teirlinck 1986) designs exist for all t (Fazely, Lovett, Vardy 2013) t-design (t + 1)-designs? Steiner systems are known for t 5 t > 5? Steiner systems are known for t = 1 (k-spreads) and S 2 [2, 3, 13] S(2, 3, ) direct constructions? recursive constructions: S(2, 3, ) S(2, 3, 2 + 1) S(2, 3, ) S(2, 3, 3 ) S(2, 3, ), S(2, 3, ) S(2, 3, )?

81 Open problems Computer free description for S 2 [2, 3, 13] known: n = 13 is the smallest possible case having a Singer cycle as automorphism group (computer search) open: Are there S 2 [2, 3, ] for other groups? S 2 [2, 3, 7]? Infinite series? Problems on q-analogs in Coding Theory, T. Etzion (2013)

82 Thank you for listening!