Equidistant Codes in the GrassmannianNetanel Raviv

Equidistant Codes in the GrassmannianNetanel Raviv

June 2014 1

Joint work with:Prof. Tuvi Etzion

Technion, Israel

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

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Motivation – Subspace Codes for Network Coding

June 2014

“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:

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Errors in Network Coding.

Motivation – Subspace Codes for Network Coding

June 2014

The values of A,B are the solution of:

Solution:

Both Wrong…

A,B

Even a single error may corrupt the entire message.

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Motivation – Subspace Codes for Network Coding

Received message

Sent message

Transfer matrix

Transfer matrix

Error vectors

Metric Metric Set Term Setting

Coherent Network Coding known to the receiver. chosen by adversary.

Kschischang, Silva 09’

Noncoherent Network Coding

chosen by adversary.

Koetter, Kshischang

08’

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Equidistant Codes - Definitions

A t-Intersecting Code.

DefinitionA 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

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Equidistant Codes - Motivation

Interesting Mathematical

Structure

Distributed Storage

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Trivial Equidistant Codes

DefinitionA 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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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

DefinitionA 0-intersecting code is called a partial spread.

Trivial codes are not at all trivial…

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Bounds on Nontrivial Codes

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

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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

Plücker Embedding

However:

Julius Plücker1801-1868

M

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Define: For

Plücker Embedding

Theorem [Plücker, Grassmann ~1860]P is 1:1.

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Consider the following table:

Construction of a Nontrivial Code

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.

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Construction of a Nontrivial Code

0 0 …1 0 …0 0… …1 1

0 0 …1 0 …0 0… …1 1

Define:

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Construction of a Nontrivial Code

• . • Lemma: is bilinear when applied over 2-row

matrices.• Proof:

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Construction of a Nontrivial Code

• Lemma: is bilinear when applied over 2-row matrices.

• Theorem: • Proof:

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Construction of a Nontrivial Code

0 0 …1 1 …0 0… 01 1

The Code:

A 1-intersecting code in

Size:

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A network of servers, storing a file .

Application in Distributed Storage Systems

Failure Resilient

Reconstruction

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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’]

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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

No Restriction on Field Size

Good Locality

Low Bandwidth

High Error Resilience

Low Update Complexity

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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

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Linear –

Constant rank -

Equidistant Rank-Metric Codes

Linear, Equidistant, Constant Rank RMC

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Conjecture [Deza]:A nontrivial equidistant satisfies Attainable by

Attainable by our code .Using computer search:

Open Problems

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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

Smaller?

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Questions?

Thank you!