An Easy-to-Decode Network Coding Scheme for Wireless Broadcasting

1
An Easy-to-Decode Network Coding Scheme for
Wireless Broadcasting

• Ho Yuet Kwan (CityU, CUHK - INC)
• Joint work with
• K. Shum(CUHK - INC)
• and Chi Wan (Albert) Sung (CityU)

2
Contents

• Abstract
• Background
• System model
• Generation of encoding vectors and its complexity
• Performance evaluation
• Conclusion

3
Abstract

• We propose an easy-to-decode network coding
  scheme to offer reliable wireless broadcasting.
• Based on feedback, our scheme generates the
  encoding vectors that are sparse enough to make
  the decoding process much simpler.
• The delay performance of our scheme in binary
  cases is comparable to that of the Random Linear
  Network Coding (RLNC) scheme with a large finite
  field size, but requires less decoding
  operations.

4
Background

• In wireless broadcasting, a set of packets is
  broadcast to all receivers in wireless channels.
• Due to channel erasure, some packets would be
  lost. Retransmissions are required.
• A trivial way is to retransmit the whole set
  again.
• Linear network coding provides an excellent
  solution to the wireless broadcasting problem in
  terms of channel utilization (transmit less).

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A trivial way
Users may have different erasure patterns
P1
P2
P3
P1
P2
P3
P1
P2
P3
P1
P2
P3
U1
6 packets are transmitted for all users to
receive the intact file
P1
P2
P3
S
P1
P2
P3
U2
U3
P1
P2
P3
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With Linear Network coding
Where P P1 P2 P3
P
header tells how we encode a packet
P1
P2
P3
Only 4 packets are transmitted for all users to
receive the intact file!
U1
S
U2
U3
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Terminology

• A header tells decoders how we encode a packet.
• An encoding vector shows the coefficients for
  linearly combining the source packets to form an
  encoded packets .
• e.g. If P P1 P2 P3

x Encoding vector
P Packet vector
Encoded packet
a header contains an encoding vector x
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An innovative packet

• An encoded packet is innovative to a user if the
  corresponding encoding vector is linearly
  independent of all the encoding vectors already
  received by that user.
• (The packet only brings new information to that
  user)
• If an encoded packet is innovative, it is
  innovative to all users. (The packet brings new
  information to all)

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An innovative packet

• e.g.

Packets received by U1 with x1 1 0 0 and
x2 0 1 0
Packets received by U2 with x2 0 1 0 and
x3 0 0 1
A new packet received by U1 and U2 with xa 1
1 1
A new packet received by U1 and U2 with xa 0
1 1
xa is linearly independent of x1, x2 and x3
xa is innovative to U1 only!
xa is innovative!
xa is not innovative!
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Previous work
Binary encoding vector
Encoding vector with large finite field size
Fountain codes (Luby 2002)
Without feedback
Random Linear Network Code (RLNC) (Heide et al.
(2009))
Random Linear Network Code (RLNC) (Heide et al.
(2009) etc.)
Lu et al. (2010)
Durvy et al. (2007)
With feedback
The Cofactor Method
The Cofactor Method
Many of them put less emphasis on reducing the
decoding complexity.
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Challenges

• If all the encoded packets are innovative, the
  total number of packet transmissions for all
  users to receive all packets can be minimized.
• How to generate innovative packets?
• How to generate innovative packets so that we can
  decode them easily?

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System model

• A broadcasting system A transmitter S and K
  receivers
• where
  
• A complete file is packetized into N equal-size
  packets.
• Assume all wireless channels are independent
  erasure channels with probability of erasure
  .
• A perfect feedback channel is provided for each
  receiver.

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System model

• A file is broadcast to all receivers in two
  phases

No
No
Retransmission Phase
Initial Phase
All finished?
All finished?
Yes
Yes
End
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Initial Phase

• S broadcasts N source packets (uncoded) one by
  one.
• An acknowledgement (ACK) is sent when a receiver
  finds that a received packet is innovative to
  itself.
• If sends an ACK to S, the corresponding
  encoding vector will be added to its encoding
  matrix at S.
• Encoding matrix stores innovative encoding
  vectors received so far by .

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Initial Phase

• e.g.
• For , received all packets
  except the 3rd and
• the 5th packet. After the initial phase, the
  encoding
• matrix at S would be

Rank( ) N
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Retransmission Phase

• S bases on all current , generate an
  encoding
• vector and the encoded packet.
• An ACK is sent when a receiver finds that an
  encoded packet is innovative to itself.
• If sends an ACK to S, will be
  added to its encoding matrix at S.
  (Updating at S)
• Only the innovative encoding vector will be added
  to
• S repeats the above until all users finish
  receiving all packets.

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Generation of encoding vectors

• If every encoded packet is innovative in
• retransmission phase, the number of
• retransmissions can be minimized.
• Objective
• Given and all encoding matrices
  ,
• generate an encoding vector that is innovative.

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Single-user case

• Objective
• Given and an encoding matrix ,
• generate an encoding vector that is innovative
• to one user .

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Single-user case

• All rows in are linearly independent.
• Assume that there are linearly independent
  rows in , where
  .
• Given the encoding matrix , S is looking
  for
• that
  is linearly independent of all row vectors in
  .

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Basic Idea

• Our idea is based on the fact that, given a
  matrix,
• its row rank its column rank.
• Given q 2 and we have an encoding matrix
  with
• 3 linearly independent rows

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Basic Idea

• We can always find 3 linearly independent columns
  and
• mark those 3 columns first.

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Basic Idea

• To find such
  that is linearly
• independent of all rows of .
• Append to the
  bottom of to
• form

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Basic Idea

• If we can find such that we have 4 linearly
• independent columns in , then
• is linearly independent of all row vectors
  in and

X
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Basic Idea

• Now looking for 4 linearly independent columns in
  
• Select columns in according to the same
• chosen column indices in .

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Basic Idea

• Choose an arbitrary columns from the rest in
• together with the chosen columns to form
• (we want 4 columns)

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Basic Idea

• We form in this way because we can have
  the
• first 3 rows are linearly independent.
• For to have 4 linearly independent rows,
• it only depends on the choices of

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Basic Idea

• If we can find
  such that contains 4
• linearly independent rows then contains 4
• linearly independent rows too. ( is a
  submatrix of )

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Basic Idea

• Observe that is a square matrix, all rows
  in
• are linearly independent iff its determinant
  .
• can be expressed in terms of
• If we can solve for , then
• 
  is linearly independent of all rows in .

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Basic Idea

• We transform our problem of linear
• independence into solving a linear inequality.

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Single-user case
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Single-user case

• can be expressed by the Laplace
  expansion
• on the last row of
• where is the cofactor of

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Single-user case

• Find
  such that
• Solve for
  , where q 2
  
• Possible sol.
• What about ?

Source of sparsity
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Single-user case

• Random Linear Network code scheme is not likely
  to generate a sparse encoding vector.
• Each encoding matrix (not empty , rank ltN)
• has its corresponding determinant inequality.

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General case

• Objective
• Given and all encoding matrices
  ,
• generate an encoding vector that is innovative.
• i.e. generate a common
• that satisfies linear simultaneous
• determinant inequalities

35
General case

• There are many ways to solve the problems.
• We may have many possible solutions.
• We prefer a solution that has more zero
  entries.
• Fewer additions and multiplications are needed in
  the decoding process.
• We apply the steps shown in single user case and
  propose the cofactor method to solve the problem.

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The cofactor method

• Consider q 5, K 4 and N 5
• Find a common
• that satisfies the following determinant
  inequality set
  

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The cofactor method

• Arrange the 4 inequalities in the following
  manner
• each column contains only one unknown
• put the inequality whose largest unknown index is
  the smallest one among the whole inequality set
  to the top and so on

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The cofactor method

• Pay attention to the unknown with the largest
  index in each inequality

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The cofactor method

• Those unknowns are to be determined, while for
  the rest, we are free to set them to zero (source
  of sparsity).

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The cofactor method

• For the first inequality, we can set

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The cofactor method

• For the second inequality, is the
  remaining unknown, also no other inequality
  contains only
• we can solve the second inequality and get

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The cofactor method

• For the third inequality, is the remaining
  unknown, it is also the case for the fourth
  inequality.
• We solve for the two simultaneous
  inequality and get

q 5
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The cofactor method

• So we have that
  satisfies all inequalities
• We summarized the above idea and develop so
  called the cofactor method.

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The cofactor method

• If , the cofactor method guarantees
  that the encoding vector so generated is always
  innovative.
• If , may not have a solution in
  general.
• Our method will still output an encoding vector
  by setting the unknowns to zero, but it may not
  be innovative.
• It can be shown that the total complexity of the
  cofactor method is .
  

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Performance

• A file is packetized into 32 packets. (
  )
• Erasure channel is assumed. ( )
• The worse case delay
• The average of total number of transmissions for
  S to ensure that all users receive an intact file
  over 1000 random realizations.
• The average delay
• The average number of transmissions for S so
  that an intact file can be received by a user.

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Performance
47
Performance
48
Performance

• For the decoding complexity, we count the number
  of additions and multiplications in decoding.
• An addition operation involving two non-zero
  operands is counted.
• A multiplication operation is counted when none
  of the two operands is 1 or 0.

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Performance
50
Performance
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Future work

• The sparsity of our network coding solution has
  yet been exploited in our current results.
• A corresponding simple decoding scheme tailored
  for our network coding solution will be
  considered.
• Simple binary network coding schemes for
  minimizing the total number of required packet
  transmissions will be investigated.
• Multi-source broadcasting problem can also be
  studied.

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Conclusion

• A new encoding scheme, called the cofactor
  method, is developed.
• When the finite field size is no smaller than the
  number of users, our method is thus delay
  optimal.
• Simulations show that it outperforms the RLNC
  scheme in both the large and small finite field
  size (i.e. q 2 and q 101) cases.

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References

• M. Luby, LT codes, in Proceedings of IEEE
  Symposium on Foundations of Computer Science
  (FOCS), November 2002, pp.
• 271282.
• J. Heide, M. V. Pedersen, F. H. P. Fitzek, and T.
  Larsen, Network coding for mobile devices
  systematic binary random rateless codes, in IEEE
  Int. Conf. Comm. Workshops (ICC Workshop 2009),
  Dresden, Jun. 2009, pp. 16.
• L. Lu, M. Xiao, M. Skoglund, L. K. Rasmussen, G.
  Wu, and S. Li,
• Efficient network coding for wireless
  broadcasting, in IEEE Wireless Comm. and
  networking conf. (WCNC 10), Sydney, Apr. 2010,
  pp. 16.
• M. Durvy, C. Fragouli, and P. Thiran, Towards
  reliable broadcasting using ACKs, in Proc. IEEE
  Int. Symp. Inform. Theory, 2007.

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q lt K

• Given q 2 , K 3 and N 2
• If q 3 , K 3 and N 2

No innovative binary encoding vector can be found
which is an innovative encoding vector