Equidistant Codes in the Grassmannian

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

2. Motivation – Subspace Codes for Network Coding “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. Motivation – Subspace Codes for Network Coding • Errors in Network Coding. A,B The values of A,B are the solution of: Even a single error may corrupt the entire message. Solution: Both Wrong…

4. Motivation – Subspace Codes for Network Coding Error vectors Received message Sent message Transfer matrix Transfer matrix

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

6. Equidistant Codes - Motivation Interesting Mathematical Structure Distributed Storage

7. Trivial Equidistant Codes 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. Trivial Equidistant Codes - Construction • 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 - Definition A 0-intersecting code is called a partial spread. Trivial codes are not at all trivial…

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

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

11. Plücker Embedding • Define: • For Theorem [Plücker, Grassmann~1860] P is 1:1.

12. Construction of a Nontrivial Code • Consider the following table: Each pair of 1-subspaces is in exactly one2-subspace. Any two rows have a unique common 1. 0/1 by inclusion

13. Construction of a Nontrivial Code Define:

14. Construction of a Nontrivial Code • . • Lemma: is bilinear when applied over 2-row matrices. • Proof:

15. Construction of a Nontrivial Code • Lemma: is bilinear when applied over 2-row matrices. • Theorem: • Proof:

16. Construction of a Nontrivial Code The Code: A 1-intersecting code in Size:

17. Application in Distributed Storage Systems • A network of servers, storing a file . Failure Resilient Reconstruction

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

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

20. DSS from Equidistant Subspace Codes Low Bandwidth Low Update Complexity No Restriction on Field Size High Error Resilience Good Locality

21. Equidistant Rank-Metric Codes • 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

22. Equidistant Rank-Metric Codes • Linear – • Constant rank - Linear, Equidistant, Constant Rank RMC

23. Open Problems • Conjecture [Deza]: • A nontrivial equidistant satisfies • Attainable by • Attainable by our code . • Using computer search:

24. Open Problems • 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? Smaller?

25. Questions? Thank you!