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June 19th, 2014 Algebra, Codes and Networks, Bordeaux Equidistant Codes in the Grassmannian Netanel Raviv Joint work with: Prof. Tuvi Etzion Technion, Israel June 2014 Netanel Raviv Equidistant Codes in the Grassmannian 1 Motivation – Subspace Codes for Network Coding “The Butterfly Example” • • • A,B A and B are two information sources. A sends B sends The values of A,B are the solution of: June 2014 Netanel Raviv Equidistant Codes in the Grassmannian 2 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 Solution: the may corrupt entire message. Both Wrong… June 2014 Netanel Raviv Equidistant Codes in the Grassmannian 3 Motivation – Subspace Codes for Network Coding Received message Error vectors Transfer matrix Kschischang, Silva 09’ Koetter, Kshischang 08’ Sent message Setting Term known to the receiver. chosen by adversary. Coherent Network Coding chosen by adversary. Transfer matrix Metric Set Metric Noncoherent Network Coding Netanel Raviv Equidistant Codes in the Grassmannian 4 Equidistant Codes - Definitions Definition A code distinct is called Equidistant if satisfy Hamming Metric A binary constant weight equidistant code satisfies such that all . Subspace Metric A constant dimension equidistant code satisfies A t-Intersecting Code. Netanel Raviv Equidistant Codes in the Grassmannian 5 Equidistant Codes - Motivation Interesting Mathematical Structure Distributed Storage Netanel Raviv Equidistant Codes in the Grassmannian 6 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. Netanel Raviv Equidistant Codes in the Grassmannian 7 Trivial Equidistant Codes - Construction Definition A 0-intersecting code is called a partial spread. If If there exists a perfect partial spread of size . , best known construction [Etzion, Vardy 2011] Construction of a t-intersecting sunflower from a spread - Trivial codes are not at all trivial… Netanel Raviv Equidistant Codes in the Grassmannian 8 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 bound is attained by Projective Planes: The Fano Plane The number of 1-subspaces of Netanel Raviv Equidistant Codes in the Grassmannian 9 Construction of a Nontrivial Code Julius Plücker 1801-1868 Plücker Embedding Idea: Embed in a larger linear space. Let whose row space is , and map it to Problem: is not unique. However: M Netanel Raviv Equidistant Codes in the Grassmannian 10 Plücker Embedding Define: For Theorem [Plücker, Grassmann ~1860] P is 1:1. Netanel Raviv Equidistant Codes in the Grassmannian 11 Construction of a Nontrivial Code Consider the following table: 0 0 … 1 0 … 0 0 … … 1 1 0/1 by inclusion Each pair of 1-subspaces is in exactly one 2-subspace. Any two rows have a unique common 1. Netanel Raviv Equidistant Codes in the Grassmannian 12 Construction of a Nontrivial Code 0 0 … 0 0 … 1 0 … 1 0 … 0 0 0 0 … … … … 1 1 1 1 Netanel Raviv Equidistant Codes in the Grassmannian Define: 13 Construction of a Nontrivial Code • • Lemma: matrices. • Proof: . is bilinear when applied over 2-row Netanel Raviv Equidistant Codes in the Grassmannian 14 Construction of a Nontrivial Code • Lemma: is bilinear when applied over 2-row matrices. • Theorem: • Proof: Netanel Raviv Equidistant Codes in the Grassmannian 15 Construction of a Nontrivial Code The Code: 0 0 … 1 1 … 0 0 … 0 1 1 A 1-intersecting code in Size: Netanel Raviv Equidistant Codes in the Grassmannian 16 Application in Distributed Storage Systems A network of servers, storing a file . Failure Resilient Reconstruction Netanel Raviv Equidistant Codes in the Grassmannian 17 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 Netanel Raviv Equidistant Codes in the Grassmannian 18 DSS from Equidistant Subspace Codes • For let and • Claim 1: • • Allows good locality. • Claim 2: • If are a basis, then • Allows low repair bandwidth. Netanel Raviv Equidistant Codes in the Grassmannian 19 DSS from Equidistant Subspace Codes Low Bandwidth Low Update Complexity No Restriction on Field Size High Error Resilience Good Locality Netanel Raviv Equidistant Codes in the Grassmannian 20 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 Netanel Raviv Equidistant Codes in the Grassmannian 21 Equidistant Rank-Metric Codes Linear – Constant rank - Linear, Equidistant, Constant Rank RMC Netanel Raviv Equidistant Codes in the Grassmannian 22 Open Problems Conjecture [Deza]: A nontrivial equidistant Attainable by Attainable by our code Using computer search: satisfies . Netanel Raviv Equidistant Codes in the Grassmannian 23 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 Max size of equidistant rank-metric codes? Smaller? of size . Netanel Raviv Equidistant Codes in the Grassmannian 24 Questions? Thank you! Netanel Raviv Equidistant Codes in the Grassmannian 25