Sunflowers and Primitive SCIDs

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Allan Woods
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1 (joint work with R.D. Barrolleta, L. Storme and E. Suárez-Canedo) July 15, 2016 (joint work with R.D. Barrolleta, L. Storme and E. Suárez-Canedo)

2 Definitions Background 1 Definitions Background 2

3 Subspace Codes Definitions Background Definition An n-subspace code is a set of n-spaces in V (m, q) = F m q, for some fixed n. In other words: it is any subset of any Grassmanian. Definition An (n, m)-scid is an n-subspace code with Constant Intersection Dimension m, i.e. a set of n-spaces intersecting pairwise in dimension m.

4 Sunflowers Definitions Background Example Let Π be an (n k)-space. Let S be a collection of n-spaces pairwise intersecting in Π. Then S is an (n, n k)-scid. Definition Any (n, n k)-scid where all spaces intersect pairwise in the same (n k)-space, is called a sunflower. Example Any set of vector planes in F 3 q not all through the same vector line, forms a non-sunflower (2, 1)-SCID.

5 Definitions Background 1 Definitions Background 2

6 Background Definitions Background Subspace codes have been around for long but received a lot of attention recently because of a theorem by Koetter and Kschischang. Theorem (Koetter, Kschischang; 2008) Subspace codes are exactly what is needed for error correction in random network coding. Remark The best known subspace codes for this purpose are the SCIDs.

7 Definitions Background What is random network coding? We want to transmit each of two messages a, b (from different senders) to each of two recipients. We have three routes as depicted below. Standard techniques require two steps to transmit these messages (e.g. first a, then b over the middle channel).

8 Random Network Codes Definitions Background Random network coding however, does it in one step. The example above transmits two messages from two different senders to each of two different receivers over only three routes (rather than 4).

9 1 Definitions Background 2

10 on Size A famous theorem by Etzion and Raviv states that the largest SCIDs are sunflowers. Theorem (Etzion, Raviv; 2015) If a (k, t)-scid in a vector space over F q has more than ( ) 2 q k q t + q k q t + 1 elements, then it is a sunflower. q 1 q 1 In most cases, it is not known if this bound is sharp.

11 On the Ambient Space This inspired us to find a similar result about the dimension spanned. Definition A sunflower of maximum dimension is a sunflower (k, t)-scid such that any element π meets the subspace generated by all the other elements in precisely the common t-space. It is easy to see that this name is well-chosen and that its dimension is nt + (k t), where n = S. But what about the (dimensionwise) largest non-sunflower?

12 First Example Example Choose integers n 3 and k, t such that 3 t k 1. Let V be a vector space over a (sufficiently large) field F and let X 1,..., X n and V be linearly independent subspaces of V such that dim V = k t + 2 and dim X i = t 1. Let W 1,..., W n be n distinct (k t + 1)-spaces in V, not all through a common (k t)-space. We define π 1 = X 1, W 1, π 2 = X 2, W 2,..., π n = X n, W n. This example has dim π 1,..., π n = k + (n 1)(t 1) + 1 and is a non-sunflower (k, k t)-scid.

13 First Example π 1 = X 1, W 1, π 2 = X 2, W 2,..., π n = X n, W n

14 Second Example Example Choose integers n 3 and k, t such that 3 t k 1 and let m be an integer with 2 m min{t + 1, n 1}. Let V be a vector space over a (sufficiently large) field F. Let V, X, N 1,..., N m and M m+1,..., M n 1 be linearly independent subspaces of V such that dim V = k t, dim X = t + 1 m, dim N i = t and dim M i = t 1. Let n 1,..., n m be 1-spaces in N 1,..., N m respectively. Let p m+1,..., p n 1 be distinct 1-spaces in n 1,..., n m \ {n 1,..., n m }. Then we define the sets π 1,..., π n as follows. π 1 = V, N 1, π 2 = V, N 2,..., π m = V, N m, π m+1 = V, M m+1, p m+1,..., π n 1 = V, M n 1, p n 1, π n = W, X, n 1,..., n m. This example has dim π 1,..., π n = k + (n 1)(t 1) + 1 and is a non-sunflower (k, k t)-scid.

15 Second Example n 1,..., n m N 1,..., N m ; p m+1,..., p n 1 n 1,..., n m \ {n 1,..., n m } π 1 = V, N 1, π 2 = V, N 2,..., π m = V, N m, π m+1 = V, M m+1, p m+1,..., π n 1 = V, M n 1, p n 1, π n = W, X, n 1,..., n m.

16 Are there more large examples? We found two constructions with total dimension k + (t 1)(n 1) + 1, denote this number by N k,t,n. Can we do better? Did we find all? 3-wise intersection of spaces is not strong enough here (doesn t even distinguish all examples mutually or from the sunflower). Classical counting techniques also seem to fail However, using the extremal principle and a rearrangement argument we were able to show that these are the smallest examples possible.

17 Classification Theorem Let S = {π 1,..., π n } be a (k, k t)-scid with 3 t < k. If dim S > N k,t,n, then S is a sunflower. If dim S = N k,t,n, then S is either a sunflower, or isomorphic to one of the previous two constructions. The trick in the proof is to consider δ i = dim π 1,..., π i dim π 1,..., π i 1 for i = 1,..., n and rearrange the spaces such that (δ i ) i=1,...,n is nondecreasing. The smallest δ n among all such rearrangements, and the largest m for which δ m = t, determine the non-sunflower structure up to isomorphism.

18 1 Definitions Background 2

19 Primitive SCIDs A class of nontrivial SCIDs was defined by Eisfeld in Definition (Eisfeld, 2002) A (k, m)-scid S in the vector space V is called primitive if it satisfies the following properties: S = V ; no nonzero vector is contained in all of the elements of S; each element π of S is spanned by {π σ σ S \ {π}}; dim(v ) 2k + 2.

20 A problem In ongoing joint work between D. Bartoli, A. Riet, L. Storme and P. Vandendriessche, a problem with this definition was revealed. Problem Eisfeld incorrectly argued that dim(v ) 2k is justified because of dualization, but the third condition is not self-dual. It is not clear what s the right way to fix this, and it was only used in the motivation, so we decided to keep this definition and just remove the claim that it is self-dual.

21 Eisfeld s Theorem (Eisfeld, 2002) For every field F there is a unique primitive (3, 1)-SCID up to isomorphism, residing in an 8-dimensional vector space. Theorem (Eisfeld, 2002) There are no primitive (n, n 2)-SCIDS for n > 2. Open Problem (Eisfeld, 2002) Do (n, n 3) exist for n 2? When does (n, n k) exist? No constructions are known for k 3.

22 A New Construction Theorem Let L be a set of k-subsets of a finite set Ω, with L = t n k + 1 and Ω = m n + k + 1 for an integer n k + 1, such that every element of Ω is contained in at least two elements of L, such that any two different elements in L have exactly one element of Ω in common, and such that no element of Ω is contained in all the elements of L. If V is an (m + n k + 1)-dimensional vector space over a (sufficiently large) field F, then a primitive (n, n k)-scid of size t in V exists. Corollary Substructures of projective planes yield a myriad of constructions for primitive (n, n k)-scids with a large spectrum of parameters.

23 Primitive (n, n 3)-SCIDs Corollary Primitive (4, 1)-SCIDs exist by the previous construction. Construction L = {{1, 2, 4, 7, 8}, {1, 2, 5, 9, 10}, {1, 2, 6, 11, 12}, {1, 3, 4, 9, 12}, {1, 3, 5, 7, 11}, {1, 3, 6, 8, 10}, {2, 3, 4, 5, 6}}, S = { e v, e w, e x, e y, e z {v, w, x, y, z} L}. Then S is a primitive (5, 2)-SCID of V = e 1,..., e 12. Open Problem For n 6, do primitive (n, n 3)-SCIDs exist?

24 Thank you for your attention!

MTH5102 Spring 2017 HW Assignment 3: Sec. 1.5, #2(e), 9, 15, 20; Sec. 1.6, #7, 13, 29 The due date for this assignment is 2/01/17.

MTH5102 Spring 2017 HW Assignment 3: Sec. 1.5, #2(e), 9, 15, 20; Sec. 1.6, #7, 13, 29 The due date for this assignment is 2/01/17. Sec. 1.5, #2(e). Determine whether the following sets are linearly dependent