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Fountain Codes Amin Shokrollahi EPFL and Digital Fountain, Inc.

Fountain Codes

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Amin Shokrollahi. EPFL and Digital Fountain, Inc. Fountain Codes. BEC. BEC(p 1 ). BEC(p 2 ). BEC(p 3 ). BEC(p 4 ). BEC(p 5 ). BEC(p 6 ). Communication on Multiple Unknown Channels. Example: Popular Download. Example: Peer-to-Peer. Example: Satellite. - PowerPoint PPT Presentation

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Page 1: Fountain Codes

Fountain Codes

Amin Shokrollahi

EPFLand

Digital Fountain, Inc.

Page 2: Fountain Codes

BEC

Page 3: Fountain Codes

BEC(p1)

BEC(p2)

BEC(p3)

BEC(p4)

BEC(p5)

BEC(p6)

Communication on Multiple Unknown Channels

Page 4: Fountain Codes

Example: Popular Download

Page 5: Fountain Codes

Example: Peer-to-Peer

Page 6: Fountain Codes

Example: Satellite

Page 7: Fountain Codes

The erasure probabilities are unknown.

Want to come arbitrarily close to capacity on each of the erasure channels, with minimum amount of feedback.

Traditional codes don’t work in this setting since their rate is fixed.

Need codes that can adapt automatically to the erasure rate of the channel.

Page 8: Fountain Codes

Original Content

Blocks

Orig

inal

Orig

inal

Orig

inal

Red

unda

n t

Red

unda

n t

Red

u nda

n t

Traditional FEC

Page 9: Fountain Codes

• Fraction of losses must be less than K/(N+K)

• Worst user dictates amount of redundancy

• Loss provisioning is complicated and leads to overhead

Problems with FEC

Page 10: Fountain Codes

Originalcontent

Encoded packetsUsers reconstruct Original content as soon as they receive enough packets

Encoding

Engine

Transmission

Reconstruction time should depend only on size of content

What we Really Want

Page 11: Fountain Codes

Content

Enc

Digital buckets

Page 12: Fountain Codes

Fountain Codes

Sender sends a potentially limitless stream of encoded bits.

Receivers collect bits until they are reasonably sure that they can recover the content from the received bits, and send STOP feedback to sender.

Automatic adaptation: Receivers with larger loss rate need longer to receive the required information.

Want that each receiver is able to recover from the minimum possible amount of received data, and do this efficiently.

Page 13: Fountain Codes

Distribution on

Fountain Codes

Page 14: Fountain Codes

Universality and Efficiency

[Universality] Want sequences of Fountain Codes for which the overhead is arbitrarily small

[Efficiency] Want per-symbol-encoding to run in close to constant time, and decoding to run in time linear in number of output symbols.

Page 15: Fountain Codes

2

Insert header, and send

XOR Choose weight

Choose 2Random originalsymbols

Input symbols

Weight Prob

1 0.055

0.0004

0.32

0.13

0.084

100000

Weight table

The Fountain Coding Process

Page 16: Fountain Codes

Decoding

Page 17: Fountain Codes

Decoding

Page 18: Fountain Codes

Decoding

Page 19: Fountain Codes

Decoding

Page 20: Fountain Codes

Decoding

Page 21: Fountain Codes

Decoding

Page 22: Fountain Codes

Decoding

Page 23: Fountain Codes

Decoding

Page 24: Fountain Codes

Decoding