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
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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
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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
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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
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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
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Equidistant Codes - Motivation
Interesting
Mathematical
Structure
Distributed
Storage
Netanel Raviv Equidistant Codes in the Grassmannian
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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.
Netanel Raviv Equidistant Codes in the Grassmannian
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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
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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
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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
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Plücker Embedding
Define:
For
Theorem [Plücker, Grassmann ~1860]
P is 1:1.
Netanel Raviv Equidistant Codes in the Grassmannian
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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
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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
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Construction of a Nontrivial Code
• Lemma: is bilinear when applied over 2-row
matrices.
• Theorem:
• Proof:
Netanel Raviv Equidistant Codes in the Grassmannian
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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
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Application in Distributed Storage Systems
A network of servers, storing a file
.
Failure Resilient
Reconstruction
Netanel Raviv Equidistant Codes in the Grassmannian
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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
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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.
Netanel Raviv Equidistant Codes in the Grassmannian
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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
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Equidistant Rank-Metric Codes
Linear –
Constant rank -
Linear, Equidistant,
Constant Rank RMC
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Open Problems
Conjecture [Deza]:
A nontrivial equidistant
Attainable by
Attainable by our code
Using computer search:
satisfies
.
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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
.
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Questions?
Thank you!
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