Upload
moriah
View
83
Download
0
Tags:
Embed Size (px)
DESCRIPTION
June 19 th , 2014. Algebra, Codes and Networks, Bordeaux. Equidistant Codes in the Grassmannian. Netanel Raviv. Joint work with:. Prof. Tuvi Etzion. Technion , Israel. Motivation – Subspace Codes for Network Coding. “The Butterfly Example” A and B are two information sources. A sends - PowerPoint PPT Presentation
Citation preview
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
Equidistant Codes in the GrassmannianNetanel Raviv 2
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:
Equidistant Codes in the GrassmannianNetanel Raviv 3
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.
Equidistant Codes in the GrassmannianNetanel Raviv 4
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’
Equidistant Codes in the GrassmannianNetanel Raviv 5
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
Equidistant Codes in the GrassmannianNetanel Raviv 6
Equidistant Codes - Motivation
Interesting Mathematical
Structure
Distributed Storage
Equidistant Codes in the GrassmannianNetanel Raviv 7
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.
Equidistant Codes in the GrassmannianNetanel Raviv 8
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…
Equidistant Codes in the GrassmannianNetanel Raviv 9
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
Equidistant Codes in the GrassmannianNetanel Raviv 10
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
Equidistant Codes in the GrassmannianNetanel Raviv 11
Define: For
Plücker Embedding
Theorem [Plücker, Grassmann ~1860]P is 1:1.
Equidistant Codes in the GrassmannianNetanel Raviv 12
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.
Equidistant Codes in the GrassmannianNetanel Raviv 13
Construction of a Nontrivial Code
0 0 …1 0 …0 0… …1 1
0 0 …1 0 …0 0… …1 1
Define:
Equidistant Codes in the GrassmannianNetanel Raviv 14
Construction of a Nontrivial Code
• . • Lemma: is bilinear when applied over 2-row
matrices.• Proof:
Equidistant Codes in the GrassmannianNetanel Raviv 15
Construction of a Nontrivial Code
• Lemma: is bilinear when applied over 2-row matrices.
• Theorem: • Proof:
Equidistant Codes in the GrassmannianNetanel Raviv 16
Construction of a Nontrivial Code
0 0 …1 1 …0 0… 01 1
The Code:
A 1-intersecting code in
Size:
Equidistant Codes in the GrassmannianNetanel Raviv 17
A network of servers, storing a file .
Application in Distributed Storage Systems
Failure Resilient
Reconstruction
Equidistant Codes in the GrassmannianNetanel Raviv 18
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’]
Equidistant Codes in the GrassmannianNetanel Raviv 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.
Equidistant Codes in the GrassmannianNetanel Raviv 20
DSS from Equidistant Subspace Codes
No Restriction on Field Size
Good Locality
Low Bandwidth
High Error Resilience
Low Update Complexity
Equidistant Codes in the GrassmannianNetanel Raviv 21
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
Equidistant Codes in the GrassmannianNetanel Raviv 22
Linear –
Constant rank -
Equidistant Rank-Metric Codes
Linear, Equidistant, Constant Rank RMC
Equidistant Codes in the GrassmannianNetanel Raviv 23
Conjecture [Deza]:A nontrivial equidistant satisfies Attainable by
Attainable by our code .Using computer search:
Open Problems
Equidistant Codes in the GrassmannianNetanel Raviv 24
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?
Equidistant Codes in the GrassmannianNetanel Raviv 25
Questions?
Thank you!