10th canadian workshop on information theory june 7, 2007 rank-metric codes for priority encoding...
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10th Canadian Workshop on Information TheoryJune 7, 2007
Rank-Metric Codes forPriority Encoding
Transmissionwith Network Coding
Danilo Silvaand
Frank R. KschischangUniversity of Toronto
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Outline
• Motivation– Priority Encoding Transmission– Random Network Coding– What happens when we combine both?
• A rank-metric approach
• Conclusions
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Priority Encoding Transmission
• Approaches to erasure correction (packet
loss):– Rateless codes/retransmission:
• requires acknowledgement• introduce delay
– Classical erasure codes:• rate decided a priori• bandwidth waste if rate smaller than
capacity• low performance if rate higher than capacity
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Why Priority Encoding Transmission?
– Priority encoding transmission:• better trade-off between performance and
rate• requires source signal than can be
partitioned into layers of unequal importance
• apply unequal error protection to layers
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Priority Encoding Transmission
• Deterministic PET:– Input: layers Li with priority levels ki · n
(smaller ki = higher importance)
– Output: n packets such that:any K of these packets are sufficient to recover all layers that have priority level · K
[A. Albanese et al., “Priority encoding transmission,” 1996]
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Priority Encoding Transmission
packets
information
symbols
paritysymbol
s
layers
encoding(MDS code)
• Example:
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Random Network Coding
• Network coding:– Generalizes routing in communication
networks– Can increase the throughput of traditional
networks (achieves the multicast capacity)
• Random network coding:– A practical way to perform network coding– Many practical advantages over solutions
based on routing[Ho et al., “A random linear network coding approach to multicast,”]
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Random Network Coding
• Each block (generation) of the information stream is partitioned into n packets
• Nodes form outgoing packets as random linear combinations of incoming packets
header payload“mixed” data
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Erasures in Network Coding
• What if not enough packets can reach the destination?
– An erasure in network coding is more severe than a classical erasure since one erased packet may “contaminate” other packets
– Classical erasure correcting codes will not work!
no packets canbe recovered!
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Combining PET and Network Coding
• One possible solution to combine PET and RNC:
[P.A. Chou, Y. Wu, and K. Jain, “Practical network coding,” 2003]
– However, the guarantees are probabilistic.
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Combining PET and Network Coding
• Example in :
k=2
nonsingular
linearly dependentlinearly
independent
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Combining PET and Network Coding
• Our goal:– Obtain a deterministic PET system that is
compatible with network coding
• Observation:– Classical erasures are special cases of
network coding erasures must use MDS codes
• Approach:– Are there MDS codes that can also correct
network coding erasures?
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Traditional FEC and Network Coding
• Suppose packets are encoded with a RS code:
RS encoder
messagecodeword
transmittedpackets
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Traditional FEC and Network Coding
received packetsnot necessarily invertible!e.g., in
• After packet mixing and one packet erasure:
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Linearized Polynomials
• Is there a polynomial f(x) that satisfies...?
If this is true, then
?
are three evaluation points for f(x)
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Linearized Polynomials
• Linearized polynomials:
• The property that gives their name:
– An evaluation of a linearized polynomial is a map from to itself that is linear over
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Gabidulin Codes
• Encoding packets with a Gabidulin code:
encoder
message
codewordtransmitted
packets
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Decoding Gabidulin Codes
• After packet mixing and one packet erasure:
q3 distinct evaluation points for f(x) of degree < q3
can find f(x) using Lagrangian interpolation
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Rank-Metric Codes
[E.M. Gabidulin, “Theory of codes with maximum rank distance,” Probl. Inform. Transm., 1985]
Reed-Solomon codes Gabidulin codes
Hamming distance metric
Rank distance metric
Polynomials Linearized polynomials
MDS MRD (maximum rank
distance)
errors and erasures “rank errors” and“rank erasures”
Berlekamp-Massey algorithm
modified Berlekamp-Massey algorithm
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• Main implications:– Need m symbols in to make a symbol in
– Field size is exponentially larger:
Example:
A Rank-Metric PET System
..
.
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– Can also correct errors introduced by a jammer:
A Rank-Metric PET System
[D. Silva and F.R. Kschischang, “Using rank-metric codes for error correction in random network coding,” ISIT 2007]
all received packets are corrupt
only one rank error
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Conclusions
• Combining PET and RNC is a promising approach to low-latency multicast
• Existing PET systems are either probabilistic or incompatible with RNC
• We propose a PET system based on rank-metric codes that is compatible with RNC and provides deterministic guarantees of recovery
• Our system can also correct packet errors introduced by a jammer