Fundamentals of Coding For Network Coding

Transcription

1 Fundamentals of Coding For Network Coding Marcus Greferath and Angeles Vazquez-Castro Aalto University School of Sciences, Finland Autonomous University of Barcelona, Spain and 2016

2 Structure of the Presentation Projective Geometry Traditional Designs q -Analogs of Designs Subspace Codes Computational Methods Further Results 2/41

3 Structure of the Presentation Projective Geometry Traditional Designs q -Analogs of Designs Subspace Codes Computational Methods Further Results 3/41

4 Has anybody ever seen this figure? One of the best known images in Discrete Mathematics It is an illustration of the so-called Fano plane. 4/41

5 Projective Geometry The Fano plane is the smallest non-trivial example of what is called a projective space. These geometries are incidence structures (P, G) consisting of points and lines, such that the following 3 axioms are satisfied: Any two (distinct) points are contained in a unique line. If a line intersects with two sides of a triangle (but not in the vertices), then it will also intersect with the third side. There are at least 4 points, no 3 of which are on a common line. 5/41

6 Projective Geometry A large class of projective spaces comes from vector spaces: Let V be a vector space of dimension at least 3 over the field F. Define (P, G) by P := {U V dim(u) = 1} G := {U V dim(u) = 2} Then the pair (P, G) indeed satisfies the above 3 axioms. More-over, if V = F 3 2, then the above image of the Fano plane describes the structure of (P, G). 6/41

7 Projective Geometry Finally, there is a notion of dimension, that arises as follows: A subspace is a subset A of P such that with every two points p, q A the entire line spanned by p and q will be also contained in A. It can be seen that arbitrary intersections of subspaces again form subspaces. If B is an arbitrary subset of P, then [B] := {B A P A subspace}, i.e. the smallest subspace containing B is called the subspace generated by B. 7/41

8 Projective Geometry The dimension of a subspace A is then the size of the smallest subset generating A, reduced by 1. This way, we have the following further concepts: Singletons are the subspaces of dimension 0. Lines are subspaces of dimension 1. Planes are subspaces of dimension Hyperplanes are subspaces of dimension n 1, provided the given projective space is of dimension n. 8/41

9 Projective Geometry The set of all subspaces of a projective space (P, G), partially ordered by set inclusion, is called the projective geometry of (P, G). 9/41

10 Structure of the Presentation Projective Geometry Traditional Designs q -Analogs of Designs Subspace Codes Computational Methods Further Results 10/41

11 Subset lattice Let X be a v -element set. ( X ) k := Set of all k -subsets of X. # ( ) ( X k = v k). Subsets of X form a Boolean lattice (wrt. ). Definition D ( X k ) is a t -(v, k, λ) (block) design if each T ( X t ) is contained in exactly λ blocks (elements of D ). If λ = 1: D is called a Steiner system For λ = 1, t = 2, k = 3, D is referred to as a Steiner triple system STS(v) 11/41

12 Example X = {1, 2, 3, 4, 5, 6, 7}, D = {{1, 2, 5}, {1, 4, 6} {1, 3, 7}, {2, 3, 6}, {2, 4, 7}, {3, 4, 5}, {5, 6, 7}} Fano plane D is a 2-(7, 3, 1) design, i.e an STS(7). 12/41

13 Lemma Let D be a t -(v, k, λ) design and i {0,..., t}. Then D is also an i -(v, k, λ i ) design with Example λ i = ( v i t i ( k i t i ) ) λ. In particular, #D = λ 0 Fano plane STS(7) (v = 7, k = 3, t = 2, λ = 1): λ 2 = 1, λ 1 = 3, λ 0 = 7 Corollary: Integrality conditions If a t -(v, k, λ) design exists, then λ 0, λ 1,..., λ t Z, i.e. all these Parameters must be admissible. 13/41

14 Remark All finite projective spaces are Steiner systems. The underlying 2-designs have parameters ( qn 1 q 1, q + 1, 1). Here q is the number of elements in the finite field, and n is the dimension of the vector space. Lemma STS(v) admissible v 1, 3 (mod 6). STS(v) for small v Fano plane STS(7) is smallest (non-trivial) example. Next admissible case is STS(9) which exists as affine plane of order 3. Theorem (Kirkman 1847) All admissible STS(v) do exist. 14/41

15 Subspace lattice Let V be a v -dimensional F q vector space. Grassmannian [ ] V k := Set of all k -dim. subspaces of V. q Gaussian Binomial coefficient [ ] [ ] V v # = = (qv 1)(q v 1 1)... (q v k+1 1) k k (q 1)(q 2 1)... (q k 1) q q Subspace lattice of V = projective geometry PG(v 1, q) [ Elements of V ] 1 are points. q [ Elements of V ] 2 are lines. q Elements of [ ] V 3 are planes. q [ Elements of V ] v 1 are hyperplanes. q 15/41

16 Structure of the Presentation Projective Geometry Traditional Designs q -Analogs of Designs Subspace Codes Computational Methods Further Results 16/41

17 How to obtain q -analogs in combinatorics? Replace subset lattice by subspace lattice! traditional q-analog v-element set X v-dim. F q vector spacev ) [ V ] k ( V k ( v k ) [ v k] cardinality dimension + The subset lattice corresponds to the limit when q 1. A groaner: This comes from the unary field F 1. q q 17/41

18 What is the definition of a q -analog of a design? We follow the above recipe and make the transit from sets to vector spaces: Definition (subspace design) Let V be a v -dimensional F q vector space. D [ ] V k q is a t -(v, k, λ) q (subspace) design if each T [ ] V t is contained in q exactly λ elements of D. If λ = 1: D is called a q -Steiner system If λ = 1, t = 2, k = 3: D is referred to as q -Steiner triple system STS q (v) Geometrically, the STS q (v) is a set of planes in PG(v 1, q) which cover each line exactly once. 18/41

19 Lemma Let D be a t -(v, k, λ) q design and i {0,..., t}. Then D is also an i -(v, k, λ i ) q design with λ i = [ v i ] t i q [ k i ] t i q λ. In particular, #D = λ 0. Corollary: Integrality conditions If a t -(v, k, λ) q design exists, then λ 0, λ 1,..., λ t Z, i.e. all of these Parameters must be admissible 19/41

20 Lemma STS q (v) admissible v 1, 3 (mod 6). STS q (v) for small v The famous q -analog of the Fano plane STS q (7). Problem: its existence is open for every field order q. STS q (9) which could be called the q -analog of the binary affine plane. Problem: its existence is open for every q. Good News: A non-trivial q -Steiner system is known to exist: STS 2 (13) and was discovered by Braun, Etzion, Ostergard, Vardy, Wassermann in /41

21 Focus on binary q -analog of the Fano plane STS 2 (7). λ 2 = 1, λ 1 = 21, λ 0 = 381 STS 2 (7) consists of λ 0 = 381 blocks which need to be found in a set of [ ] 7 3 = planes of that projective 2 space! This is a huge search space, as ( ) has 730 decimal digits. Additional properties and insight is required! Through each point P there are λ 1 = 21 blocks. Image in V /P = PG(5, 2) is a line spread, (cf. Mateva, Topalova 2009): There are types of spreads. More refined information by intersection numbers (see Kiermaier and Pavcevic 2014). 21/41

22 Structure of the Presentation Projective Geometry Traditional Designs q -Analogs of Designs Subspace Codes Computational Methods Further Results 22/41

23 Definition (Recall: Steiner system) D [ ] V k q is a t -(v, k, 1) q Steiner system if each T [ ] V t is contained in exactly one element of D. q Definition ((constant dimension) subspace code) C [ ] V k q is a (v, 2(k t + 1); k) q subspace code if each T [ ] V t is contained in at most one element of C. q q -Fano setting: (7, 4; 3) q subspace code C. For q = 2: #C 381 #C = 381 C is a STS2 (7) Task: Find maximum size A q (7, 4; 3) of (7, 4; 3) q subspace code! 23/41

24 History Silberstein 2008: A 2 (7, 4; 3) 289 Based on lifted rank metric codes. Vardy 2008: A 2 (7, 4; 3) 294 Kohnert, Kurz 2008: A 2 (7, 4; 3) 304 Prescribe group of order 21 Braun, Reichelt 2012: A 2 (7, 4; 3) 329 Prescribe group of order 15, modify large solutions. Liu, Honold 2014; Honold, K. 2015: explicit construction of #C = 329 via expurgation and augmentation of the lifted Gabidulin code More-over: A 3 (7, 4; 3) 6977 for q = 3 (which means STS 3 (7) would have size 7651.) 24/41

25 Recent approach joint work with Daniel Heinlein, Sascha Kurz and Alfred Wassermann. Systematically check G < GL(7, 2) for admitting large G-invariant codes. Found #G = 64 admitting #C = having a subgroup of order 32 admitting #C = having a subgroup of order 16 admitting #C = having a subgroup of order 4 admitting #C = 333. Code provided at subspacecodes.uni-bayreuth.de 25/41

26 Structure of the Presentation Projective Geometry Traditional Designs q -Analogs of Designs Subspace Codes Computational Methods Further Results 26/41

27 Automorphisms Designs over sets: S v : symmetric group σ S v is automorphism: B σ = B d c Example: 4 2 σ = (ad)(bc) a b Set of automorphisms: automorphism group Subspace designs: PGL(v, q) projective semilinear group GL(v, q) = {M F v v q : M invertible} σ PGL(v, q) automorphism: B σ = B 27/41

28 Brute force approach for construction Step 1: Build matrix Designs over sets: incidence matrix between t -subset T i and k -subsets K j : M t,k = (m i,j ), where m i,j = { 1 if T i K j 0 else Subspace designs: incidence matrix between t -subspaces T i and k -subspaces K j : M t,k = (m i,j ), where m i,j = M t,k = [ v t ]q [ ] v k q { 1 if T i K j 0 else 28/41

29 Brute force approach for construction Step 2: Solve system of Diophantine linear equations Solve M t,k x 1 x 2. x n = λ λ. λ Here x = [x 1,..., x n ] T is a binary vector. 29/41

30 Example 3 d a 1 design 1 3 d 5 4 a 6 1 c b c 2 b 4 d a design 2 c b 2 d a 5 6 design 3 M 1, a b c d design design design c b 30/41

31 Side remark: Linear codes Search for linear [n, k, d] q code Ambient space F k q Build incidence matrix between k 1-subspaces H i and 1-subspaces P j : { 1 if H i P j M k 1,1 = (m i,j ), where m i,j = 0 else Solve M t,k x n d n d. n d with 0/1-vector x, x 1 = n 31/41

32 Designs with prescribed automorphism group Construction of designs with prescribed automorphism group: choose group G acting on X, i.e. G S v search for t -designs D = (X, B) having G as a group of automorphisms, i.e. for all g G and K B = K g B construct D = (X, B) as union of orbits of G on k -subsets. 32/41

33 Example: cyclic symmetry d 4 a c 2 b a b c d d 5 c {1, 2, 3, 4} {5, 6} a 2 1 b 2 1 c 2 1 d 2 1 a design 3 6 b {1, 2, 3, 4} {5, 6} {a, b, c, d} /41

34 The method of Kramer and Mesner Definition K X and K = k : K G := {K g g G} T X and T = t : T G := {T g g G} Let and K G 1 K G 2... K G n T G 1 T G 2... T G m = ( ) X k ( ) X t M G t,k = (m i,j) where m i,j := {K K G j T i K } 34/41

35 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 λ Mt,k G x = λ. λ is a t -(v, k, λ) design having G as an automorphism group. Expected gain Brute force approach: M t,k = [ v t ]q [ ] v k q Kramer-Mesner: Mt,k G [v t] q G [v k] q G 35/41

36 Applications of Kramer-Mesner mostly by Betten, Braun, Kerber, Kiermaier, Kohnert, Kurz, Laue, Vogel, W., Zwanzger at Bayreuth University designs over sets subspace designs large sets of designs subspace codes linear codes self-orthogonal codes LCD codes ring-linear codes two-weight codes arcs, blocking sets in projective geometry 36/41

37 Solving algorithms t -designs with λ = 1: maximum clique algorithms (Östergård: cliquer) exact cover (Knuth: dancing links) t -designs with λ > 1: integer programming (CPLEX, Gurobi) heuristic algorithms Logic programming Gröbner bases lattice basis reduction + exhaustive enumeration (W. 1998, 2002) 37/41

38 Structure of the Presentation Projective Geometry Traditional Designs q -Analogs of Designs Subspace Codes Computational Methods Further Results 38/41

39 Subspace designs by computer construction (q = 2, t 2) Braun, Kerber, Laue (2005), S. Braun (2010), Braun, Etzion, Östergård, Vardy, W. (2013), M. Braun (2015) t-(v, k, λ; q) M G t,k λ 3-(8, 4, λ; 2) , 15 2-(13, 3, λ; 2) ,..., (11, 3, λ; 2) , (10, 3, λ; 2) , 30, 45, 60, 75, 90, 105, (9, 4, λ; 2) , 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) , 22, 42, 43, , 12, 19, 24, 31, 36, 43, 48, 55, (8, 4, λ; 2) , 35, 56, 70, 91, 105, 126, 140, 161, 175, 196, 210, 231, 245, 266, 280, 301, , 14, 49, 56, 63, 98, 105, 112, 147, 154, , 203, 210, 245, 252, 259, 294, 301, (8, 3, λ; 2) (7, 3, λ; 2) , 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 2-(6, 3, λ; 2) , 6 All beside 2-(13, 3, λ; 2) were found by solvediophant 39/41

40 Back to the (binary) Fano plane Does a 2-(7, 3, 1; q) design exist for q 2? Theorem (Braun, Kiermaier, Nakić (2016)) A binary Fano plane can have four non-trivial automorphism groups: One group of order 2, two groups of order 3, one group of order 4. Theorem (Kiermaier, Kurz, W. (2016)) The order of the automorphism group of a binary q-analog of the Fano plane is at most two. 40/41

41 Thanks to M. Kiermaier and A. Wassermann for their material and kind assistance in composing this tutorial 41/41