June 2014 June 19 th , 2014 Algebra, Codes and Networks, Bordeaux

Equidistant Codes in the Grassmannian

Netanel Raviv Joint work with: Prof. Tuvi Etzion Technion, Israel Netanel Raviv Equidistant Codes in the Grassmannian 1

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: Netanel Raviv Equidistant Codes in the Grassmannian June 2014 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 entire message.

Both Wrong… Netanel Raviv Equidistant Codes in the Grassmannian June 2014 3

Motivation – Subspace Codes for Network Coding Received message

Setting

Kschischang, Silva 09’ Koetter, Kshischang 08’ known to the receiver.

chosen by adversary.

chosen by adversary.

Transfer matrix Sent message

Term

Coherent Network Coding

Transfer matrix

Metric Set

Noncoherent Network Coding

Error vectors

Metric

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Equidistant Codes - Definitions 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 A t-Intersecting Code.

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Equidistant Codes - Motivation

Interesting Mathematical Structure

Distributed Storage

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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.

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Trivial Equidistant Codes - Construction Definition A 0-intersecting code is called a partial spread.

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 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 Plücker Embedding Idea: Embed in a larger linear space.

Let whose row space is , and map it to Julius Plücker 1801-1868 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.

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Construction of a Nontrivial Code Consider the following table: Each pair of 1-subspaces is in exactly one 2-subspace.

0 1 0 … 1 0 0 0 … 1 … … Any two rows have a unique common 1.

0/1 by inclusion Netanel Raviv Equidistant Codes in the Grassmannian 12

Construction of a Nontrivial Code 0 1 0 … 1 0 0 0 … 1 … … 0 1 0 … 1 0 0 0 … 1 … … Netanel Raviv Equidistant Codes in the Grassmannian Define: 13

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

Proof: 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 1 0 … 1 0 1 0 0 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

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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.

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DSS from Equidistant Subspace Codes

Low Bandwidth Low Update Complexity No Restriction on Field Size High Error Resilience Good Locality

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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

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Open Problems Conjecture [Deza]: A nontrivial equidistant satisfies Attainable by Attainable by our code .

Using computer search: 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: Smaller?

Our code Linear equidistant rank-metric code in of size .

Max size of equidistant rank-metric codes?

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Questions?

Thank you!

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