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