RankMetric Codes for Priority Encoding Transmission with Network Coding

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Rank-Metric Codes forPriority Encoding
Transmissionwith Network Coding

• Danilo SilvaandFrank R. Kschischang
• University 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 thatany 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

• Example

layers
packets
encoding(MDS code)
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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

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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

k2
nonsingular
linearlyindependent
linearly dependent
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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

message
codeword
transmittedpackets
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Traditional FEC and Network Coding

• After packet mixing and one packet erasure

received packets
not necessarily invertible!e.g., in
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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
codeword
transmittedpackets
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Decoding Gabidulin Codes

• After packet mixing and one packet erasure

q3 distinct evaluation points for f(x) of degree
lt 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
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A Rank-Metric PET System

• Main implications
• Need m symbols in to make a symbol in
• Field size is exponentially larger
• Example

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A Rank-Metric PET System

• Can also correct errors introduced by a jammer

all received packets are corrupt
only one rank error
D. Silva and F.R. Kschischang, Using
rank-metric codes for error correction in random
network coding, ISIT 2007
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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