1. Equidistant Codes in the Grassmannian Netanel Raviv Equidistant Codes in the Grassmannian Netanel Raviv June 20141 Joint work with: Prof. Tuvi Etzion Technion, Israel June 19 th, 2014 Algebra, Codes and Networks, Bordeaux

2. Equidistant Codes in the Grassmannian Netanel Raviv Motivation – Subspace Codes for Network Coding June 2014 2 “The Butterfly Example” A and B are two information sources. A sends B sends A,B The values of A,B are the solution of:

3. Equidistant Codes in the Grassmannian Netanel Raviv Errors in Network Coding. Motivation – Subspace Codes for Network Coding June 2014 3 The values of A,B are the solution of: Solution: Both Wrong… A,B

4. Equidistant Codes in the Grassmannian Netanel Raviv Motivation – Subspace Codes for Network Coding 4 Received message Sent message Transfer matrix Error vectors MetricMetric SetTermSetting Coherent Network Coding known to the receiver. chosen by adversary. Kschischang, Silva 09’ Noncoherent Network Coding chosen by adversary. Koetter, Kshischang 08’

5. Equidistant Codes in the Grassmannian Netanel Raviv Equidistant Codes - Definitions 5 A t-Intersecting Code. Definition A code is called Equidistant if such that all distinct satisfy. Hamming Metric A binary constant weight equidistant code satisfies Subspace Metric A constant dimension equidistant code satisfies

6. Equidistant Codes in the Grassmannian Netanel Raviv Equidistant Codes - Motivation 6

7. Equidistant Codes in the Grassmannian Netanel Raviv Trivial Equidistant Codes 7 Definition A binary constant-weight equidistant code is called trivial if all words meet in the same coordinates. For subspace codes, similar… t A Sunflower.

8. Equidistant Codes in the Grassmannian Netanel Raviv If there exists a perfect partial spread of size. If, best known construction [Etzion, Vardy 2011] Construction of a t-intersecting sunflower from a spread - Trivial Equidistant Codes - Construction 8 Definition A 0-intersecting code is called a partial spread. Trivial codes are not at all trivial…

9. Equidistant Codes in the Grassmannian Netanel Raviv Bounds on Nontrivial Codes 9 Theorem [Deza, 73] Let be a nontrivial, intersecting binary code of constant weight. Then The bound is attained by Projective Planes: The Fano Plane Use Deza’s bound to attain a bound on equidistant subspace codes: The number of 1-subspaces of

10. Equidistant Codes in the Grassmannian Netanel Raviv Idea: Embed in a larger linear space. Let whose row space is, and map it to Problem: is not unique. Construction of a Nontrivial Code 10 Plücker Embedding However: Julius Plücker 1801-1868 M

11. Equidistant Codes in the Grassmannian Netanel Raviv Define: For Plücker Embedding 11 Theorem [Plücker, Grassmann ~1860] P is 1:1.

12. Equidistant Codes in the Grassmannian Netanel Raviv Consider the following table: Construction of a Nontrivial Code 12 00… 10… 00 …… 11 0/1 by inclusion Each pair of 1-subspaces is in exactly one 2-subspace. Any two rows have a unique common 1.

13. Equidistant Codes in the Grassmannian Netanel Raviv Construction of a Nontrivial Code 13 00… 10… 00 …… 11 00… 10… 00 …… 11 Define:

14. Equidistant Codes in the Grassmannian Netanel Raviv Construction of a Nontrivial Code 14. Lemma: is bilinear when applied over 2-row matrices. Proof:

15. Equidistant Codes in the Grassmannian Netanel Raviv Construction of a Nontrivial Code 15 Lemma: is bilinear when applied over 2-row matrices. Theorem: Proof:

16. Equidistant Codes in the Grassmannian Netanel Raviv Construction of a Nontrivial Code 16 00… 11… 00 …0 11 The Code: A 1-intersecting code in Size:

17. Equidistant Codes in the Grassmannian Netanel Raviv A network of servers, storing a file. Application in Distributed Storage Systems 17

18. Equidistant Codes in the Grassmannian Netanel Raviv Each storage vertex is associated with a subspace. Storage: each receives for some Repair: gets such that Extract Reconstruction: Reconstruct DSS – Subspace Interpretation [Hollmann 13’] 18

19. Equidistant Codes in the Grassmannian Netanel Raviv DSS from Equidistant Subspace Codes 19 For let and Claim 1: Allows good locality. Claim 2: If are a basis, then Allows low repair bandwidth.

20. Equidistant Codes in the Grassmannian Netanel Raviv DSS from Equidistant Subspace Codes 20

21. Equidistant Codes in the Grassmannian Netanel Raviv A rank-metric code (RMC) is a subset of Under the metric Construct an equidistant RMC from our code. Recall: Lemma: Construction: All spanning matrices of the form Equidistant Rank-Metric Codes 21

22. Equidistant Codes in the Grassmannian Netanel Raviv Linear – Constant rank - Equidistant Rank-Metric Codes 22

23. Equidistant Codes in the Grassmannian Netanel Raviv Conjecture [Deza]: A nontrivial equidistant satisfies Attainable by Attainable by our code. Using computer search: Open Problems 23

24. Equidistant Codes in the Grassmannian Netanel Raviv Close the gap: For a nontrivial equidistant Find an equidistant code in a smaller space. Equidistant rank-metric codes: Our code Linear equidistant rank-metric code in of size. Max size of equidistant rank-metric codes? Open Problems 24 Smaller?

25. Equidistant Codes in the Grassmannian Netanel Raviv Questions? 25 Thank you!