Simon R. Blackburn. Jount work with Tuvi Etzion. 9th July 2012

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1 Grassmannian Codes Royal and Holloway Quasirandom logo Hypergraphs guidelines Simon R. Blackburn Jount work with Tuvi Etzion 9th July 2012 Standard logo The logo should be reproduced in the primary co Pantone 660c, on all publications printed in two more colours. Refer to the Branded merchandise sheet for guidelines on use on promotional items The text, University of London, is set as a 50% o blue background. Do not use a keyline see Non standard backgrounds for exceptions. The College name has been specially drawn; plea use the original digital artwork and do not try t re-set. Clear area Simon R. Blackburn (RHUL) Grassmannians and QR Hypergraphs No graphic or text should be placed in an area around the logo equivalent to the width 1 of / 10 the b

2 Grassmannian codes The Grassmannian G q (l, k) is the set of all k-dimensional subspaces of F l q. Simon R. Blackburn (RHUL) Grassmannians and QR Hypergraphs 2 / 10

3 Grassmannian codes The Grassmannian G q (l, k) is the set of all k-dimensional subspaces of F l q. For U, V G q (l, k), define d(u, V ) = 2k 2 dim(u V ). Simon R. Blackburn (RHUL) Grassmannians and QR Hypergraphs 2 / 10

4 Grassmannian codes The Grassmannian G q (l, k) is the set of all k-dimensional subspaces of F l q. For U, V G q (l, k), define d(u, V ) = 2k 2 dim(u V ). A set C G q (l, k) is a Grassmannian code of minimum distance d if d(u, V ) d for all U, V C. Simon R. Blackburn (RHUL) Grassmannians and QR Hypergraphs 2 / 10

5 Grassmannian codes The Grassmannian G q (l, k) is the set of all k-dimensional subspaces of F l q. For U, V G q (l, k), define d(u, V ) = 2k 2 dim(u V ). A set C G q (l, k) is a Grassmannian code of minimum distance d if d(u, V ) d for all U, V C. Combinatorial problem When l, k and a distance d are fixed, find the size A q (l, d, k) of the largest Grassmannian code of minimum distance d in G(l, k). Simon R. Blackburn (RHUL) Grassmannians and QR Hypergraphs 2 / 10

6 Grassmannian codes The Grassmannian G q (l, k) is the set of all k-dimensional subspaces of F l q. For U, V G q (l, k), define d(u, V ) = 2k 2 dim(u V ). A set C G q (l, k) is a Grassmannian code of minimum distance d if d(u, V ) d for all U, V C. Combinatorial problem When l, k and a distance d are fixed, find the size A q (l, d, k) of the largest Grassmannian code of minimum distance d in G(l, k). Koetter and Kschischang (2008) give applications to network coding. Simon R. Blackburn (RHUL) Grassmannians and QR Hypergraphs 2 / 10

7 Upper bounds Important observation A Grassmannian code C has minimum distance 2δ + 2 or more if and only if no subspace of dimension k δ is contained in distinct U, V C. Simon R. Blackburn (RHUL) Grassmannians and QR Hypergraphs 3 / 10

8 Upper bounds Important observation A Grassmannian code C has minimum distance 2δ + 2 or more if and only if no subspace of dimension k δ is contained in distinct U, V C. Proof. d(u, V ) = 2k 2 dim(u V ) so d(u, V ) 2δ if and only if dim(u V ) k δ. Simon R. Blackburn (RHUL) Grassmannians and QR Hypergraphs 3 / 10

9 Upper bounds Important observation A Grassmannian code C has minimum distance 2δ + 2 or more if and only if no subspace of dimension k δ is contained in distinct U, V C. Proof. d(u, V ) = 2k 2 dim(u V ) so d(u, V ) 2δ if and only if dim(u V ) k δ. Define a hypergraph (V, E) where V = G q (l, k δ) and E = G q (l, k). We say V E contains those U V with U V. Then C corresponds to a packing by hyperedges. Simon R. Blackburn (RHUL) Grassmannians and QR Hypergraphs 3 / 10

10 Upper bounds Write [ ] l k for the number of k-dimensional subspaces of an l-dimensional q vector space over F q. Simon R. Blackburn (RHUL) Grassmannians and QR Hypergraphs 4 / 10

11 Upper bounds Write [ ] l k for the number of k-dimensional subspaces of an l-dimensional q vector space over F q. Corollary (The packing bound) A q (l, 2δ + 2, k) [ l ] k δ q [ k ] k δ q Simon R. Blackburn (RHUL) Grassmannians and QR Hypergraphs 4 / 10

12 Upper bounds Write [ ] l k for the number of k-dimensional subspaces of an l-dimensional q vector space over F q. Corollary (The packing bound) A q (l, 2δ + 2, k) [ l ] k δ q [ k ] k δ q Proof. V = [ l k δ ]q, and each hyperedge contains [ k k δ ] q vertices. Simon R. Blackburn (RHUL) Grassmannians and QR Hypergraphs 4 / 10

13 Upper bounds Write [ ] l k for the number of k-dimensional subspaces of an l-dimensional q vector space over F q. Corollary (The packing bound) A q (l, 2δ + 2, k) [ l ] k δ q [ k ] k δ q Proof. V = [ l k δ ]q, and each hyperedge contains [ k k δ ] q vertices. A version of the iterated Johnson bound for constant weight codes is a little better, but its asymptotics are the same. Simon R. Blackburn (RHUL) Grassmannians and QR Hypergraphs 4 / 10

14 Hypergraphs Let V be a set of vertices, and E be a set of hyperedges. A hypergraph (V, E) is r-uniform if all hyperedges have cardinality r. Simon R. Blackburn (RHUL) Grassmannians and QR Hypergraphs 5 / 10

15 Hypergraphs Let V be a set of vertices, and E be a set of hyperedges. A hypergraph (V, E) is r-uniform if all hyperedges have cardinality r. Our example: r = [ ] k k δ q. Simon R. Blackburn (RHUL) Grassmannians and QR Hypergraphs 5 / 10

16 Hypergraphs Let V be a set of vertices, and E be a set of hyperedges. A hypergraph (V, E) is r-uniform if all hyperedges have cardinality r. Our example: r = [ ] k k δ. q The degree deg(v) of a vertex v is the number of hyperedges containing v. Simon R. Blackburn (RHUL) Grassmannians and QR Hypergraphs 5 / 10

17 Hypergraphs Let V be a set of vertices, and E be a set of hyperedges. A hypergraph (V, E) is r-uniform if all hyperedges have cardinality r. Our example: r = [ ] k k δ. q The degree deg(v) of a vertex v is the number of hyperedges containing v. The hypergraph (V, E) is d-regular if deg(v) = d for all v V. Simon R. Blackburn (RHUL) Grassmannians and QR Hypergraphs 5 / 10

18 Hypergraphs Let V be a set of vertices, and E be a set of hyperedges. A hypergraph (V, E) is r-uniform if all hyperedges have cardinality r. Our example: r = [ ] k k δ. q The degree deg(v) of a vertex v is the number of hyperedges containing v. The hypergraph (V, E) is d-regular if deg(v) = d for all v V. Our example: d = [ ] l k+δ δ q (large). Simon R. Blackburn (RHUL) Grassmannians and QR Hypergraphs 5 / 10

19 Hypergraphs Let V be a set of vertices, and E be a set of hyperedges. A hypergraph (V, E) is r-uniform if all hyperedges have cardinality r. Our example: r = [ ] k k δ. q The degree deg(v) of a vertex v is the number of hyperedges containing v. The hypergraph (V, E) is d-regular if deg(v) = d for all v V. Our example: d = [ ] l k+δ δ q (large). The codegree codeg(u, v) of a pair of vertices is the number of hyperedges containing both u and v. Simon R. Blackburn (RHUL) Grassmannians and QR Hypergraphs 5 / 10

20 Hypergraphs Let V be a set of vertices, and E be a set of hyperedges. A hypergraph (V, E) is r-uniform if all hyperedges have cardinality r. Our example: r = [ ] k k δ. q The degree deg(v) of a vertex v is the number of hyperedges containing v. The hypergraph (V, E) is d-regular if deg(v) = d for all v V. Our example: d = [ ] l k+δ δ q (large). The codegree codeg(u, v) of a pair of vertices is the number of hyperedges containing both u and v. Our example: number of k-dim subspaces containing u and v; small compared to degree. Simon R. Blackburn (RHUL) Grassmannians and QR Hypergraphs 5 / 10

21 Covers and packings of hypergraphs Suppose we have a large r-uniform d-regular hypergraph Γ. Suppose the codegree of a pair of vertices is always much smaller than d. Then there is a covering of the vertices by hyperedges that is about as small as we could hope for (about V /r hyperedges). Similarly there is a large hyperedge packing. Simon R. Blackburn (RHUL) Grassmannians and QR Hypergraphs 6 / 10

22 Covers and packings of hypergraphs Suppose we have a large r-uniform d-regular hypergraph Γ. Suppose the codegree of a pair of vertices is always much smaller than d. Then there is a covering of the vertices by hyperedges that is about as small as we could hope for (about V /r hyperedges). Similarly there is a large hyperedge packing. Theorem (Rödl) Fix an integer r and a positive real number δ. Then there exists an integer n 0 and a positive real number δ with the following property. Let Γ be an r-uniform hypergraph on n vertices, where n n 0. Suppose that all vertices of Γ have degree d for some integer d. Let c = max codeg(u, v), where the maximum is taken over all pairs of distinct vertices u, v Γ. If c δ d, then there exists a hyperedge cover consisting of at most (1 + δ)n/r hyperedges, and a hyperedge packing consisting of at least (1 δ)n/r hyperedges. Simon R. Blackburn (RHUL) Grassmannians and QR Hypergraphs 6 / 10

23 An existence result Theorem (SRB Etzion 2012) Let q, k and δ be fixed integers, with 0 δ k and such that q is a prime power. Then ] as l. A q (l, 2δ + 2, k) [ l k δ q [ k ] k δ q (1) Simon R. Blackburn (RHUL) Grassmannians and QR Hypergraphs 7 / 10

24 An existence result Theorem (SRB Etzion 2012) Let q, k and δ be fixed integers, with 0 δ k and such that q is a prime power. Then ] as l. A q (l, 2δ + 2, k) [ l k δ q [ k ] k δ q (1) Proof. Upper bound from packing bound. Lower bound from Rödl s theorem on quasirandom hypergraphs. Simon R. Blackburn (RHUL) Grassmannians and QR Hypergraphs 7 / 10

25 An existence result Theorem (SRB Etzion 2012) Let q, k and δ be fixed integers, with 0 δ k and such that q is a prime power. Then ] as l. A q (l, 2δ + 2, k) [ l k δ q [ k ] k δ q (1) Proof. Upper bound from packing bound. Lower bound from Rödl s theorem on quasirandom hypergraphs. Explicit families are within a constant of the optimal size. Examples due to Koetter and Kschischang; larger codes by generalising these due to Etzion and Silberstein. Simon R. Blackburn (RHUL) Grassmannians and QR Hypergraphs 7 / 10

26 k-radius sequences A 5-ary 2-radius sequence of length 7 is: 0, 1, 2, 3, 4, 0, 1 Simon R. Blackburn (RHUL) Grassmannians and QR Hypergraphs 8 / 10

27 k-radius sequences A 5-ary 2-radius sequence of length 7 is: 0, 1, 2, 3, 4, 0, 1 Definition (Jaromczyk, Lonc 2004) Let F = {0, 1,..., n 1}. An n-ary k-radius sequence is a finite sequence a 0, a 1,..., a m 1 over the alphabet F with the following property: For all x, y F, there exist i, j {0, 1,..., m 1} such that a i = x, a j = y and i j k. Simon R. Blackburn (RHUL) Grassmannians and QR Hypergraphs 8 / 10

28 Another existence result Let f k (q) be the shortest q-ary k-radius sequence. Simon R. Blackburn (RHUL) Grassmannians and QR Hypergraphs 9 / 10

29 Another existence result Let f k (q) be the shortest q-ary k-radius sequence. Theorem (SRB 2012) Let k be fixed and q. Then f k (q) 1 k ( ) q 2 Simon R. Blackburn (RHUL) Grassmannians and QR Hypergraphs 9 / 10

30 Another existence result Let f k (q) be the shortest q-ary k-radius sequence. Theorem (SRB 2012) Let k be fixed and q. Then f k (q) 1 k ( ) q 2 Proof. Upper bound: every position can be the left-hand entry of at most k pairs. Simon R. Blackburn (RHUL) Grassmannians and QR Hypergraphs 9 / 10

31 Another existence result Let f k (q) be the shortest q-ary k-radius sequence. Theorem (SRB 2012) Let k be fixed and q. Then f k (q) 1 k ( ) q 2 Proof. Upper bound: every position can be the left-hand entry of at most k pairs. Lower bound: from quasirandom hypergraph argument. Simon R. Blackburn (RHUL) Grassmannians and QR Hypergraphs 9 / 10

32 Another existence result Let f k (q) be the shortest q-ary k-radius sequence. Theorem (SRB 2012) Let k be fixed and q. Then f k (q) 1 k ( ) q 2 Proof. Upper bound: every position can be the left-hand entry of at most k pairs. Lower bound: from quasirandom hypergraph argument. Remark: Jaromczyk,Lonc,Truszczyński (2012) have recently given an explicit asymptotically tight construction. Simon R. Blackburn (RHUL) Grassmannians and QR Hypergraphs 9 / 10

33 Some Links This talk will appear soon on my home page: S.R. Blackburn and T. Etzion, The asymptotic behavior of Grassmannian codes, IEEE Trans. Information Theory, to appear. See: S.R. Blackburn, The existence of k-radius sequences, J. Comb. Theory - Series A, Vol. 119 (2012), pp See: Simon R. Blackburn (RHUL) Grassmannians and QR Hypergraphs 10 / 10

Distinct Difference Configurations

Distinct Difference Configurations Simon R. Blackburn Joint work with: Tuvi Etzion, Keith M. Martin, Maura B. Paterson Royal Holloway, University of London 16th July 009 S.R. Blackburn (RHUL) Distinct