Fundamentals of Coding For Network Coding

Fundamentals of Coding For Network Coding Marcus Greferath and Angeles Vazquez-Castro Aalto University School of Sciences, Finland Autonomous University of Barcelona, Spain marcus.greferath@aalto.fi and angeles.vazquez@uab.es, 2016

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

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

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

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

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

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

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 2.... Hyperplanes are subspaces of dimension n 1, provided the given projective space is of dimension n. 8/41

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

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

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

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

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

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

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

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

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

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

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

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 2013. 20/41

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 = 11811 planes of that projective 2 space! This is a huge search space, as ( ) 11811 381 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 131044 types of spreads. More refined information by intersection numbers (see Kiermaier and Pavcevic 2014). 21/41

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

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

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

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 = 319.... having a subgroup of order 32 admitting #C = 327.... having a subgroup of order 16 admitting #C = 329.... having a subgroup of order 4 admitting #C = 333. Code provided at subspacecodes.uni-bayreuth.de 25/41

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

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

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

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

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

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

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

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

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

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

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

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

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

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) 105 217 11, 15 2-(13, 3, λ; 2) 105 30705 1,..., 2047 2-(11, 3, λ; 2) 31 2263 245, 252 2-(10, 3, λ; 2) 20 633 15, 30, 45, 60, 75, 90, 105, 120 2-(9, 4, λ; 2) 11 725 21, 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, 1323 2-(9, 3, λ; 2) 31 529 21, 22, 42, 43, 63 28 408 7, 12, 19, 24, 31, 36, 43, 48, 55, 60 40 460 49 2-(8, 4, λ; 2) 15 217 21, 35, 56, 70, 91, 105, 126, 140, 161, 175, 196, 210, 231, 245, 266, 280, 301, 315 13 231 7, 14, 49, 56, 63, 98, 105, 112, 147, 154, 161 196, 203, 210, 245, 252, 259, 294, 301, 308 2-(8, 3, λ; 2) 43 381 21 2-(7, 3, λ; 2) 21 93 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 2-(6, 3, λ; 2) 77 155 3, 6 All beside 2-(13, 3, λ; 2) were found by solvediophant 39/41

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

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