Long Erasure Codes: the New Frontier for ZeroLoss in Space Applications

1
Long Erasure Codes the New Frontier for
Zero-Loss in Space Applications?

• Enrico Paolini, University of Bologna
• epaolini_at_deis.unibo.it
• Gian Paolo Calzolari, ESA/ESOC
• Gian.Paolo.Calzolari_at_esa.int
• Marco Chiani, University of Bologna
• mchiani_at_deis.unibo.it
• SpaceOps 2006, Rome, Italy, 19-23 June

2
Outline

• Packet erasure correction in space / satellite
  communications ARQ and FEC techniques
• Long erasure correcting (LEC) codes and iterative
  erasure correction algorithm
• Structures for LEC codes
• Correction of bursts of erasures
• Numerical results

3
Packet Erasures

• In space / satellite communications, traditional
  error correction and detection techniques only
  deliver the data units for which integrity can be
  guaranteed.
• From the point of view of the the upper layers,
  uncorrectable data units are lost.
• The upper layers have typically to face data
  units (i.e. packet) erasures.
• Packet erasure channel (PEC)
• Causes of packet losses brief outage conditions
  due to weather, shadowing, loss of frame
  synchronization
• Erasures can be correlated and bursts of erasures
  can take place.

Transmitted packet
Correctly received packet
Erased packet
?
4
Traditional Techniques

• ARQ (automatic repeat / retransmission query)
  not always possible in space communications
• Long round trip delay in deep space missions
• Feedback channel not always available
• In the satellite broadcast, the satellite is not
  able to manage several retransmission requests
• Limited on board memory persistency of the data
  couldnt be guaranteed.
• FEC (forward error correction)
• Reed-Solomon codes usually exploited (bounded
  distance decoding)
• Codeword length limited by complexity issues
  (typical value n 255)
• Limitation to the code performance
• Limitation to the maximal correctable erasure
  burst length
• Impossibility to encode a long file as a unique
  codeword.

5
Long Erasure Correcting (LEC) Codes

• They are able to overcome the complexity
  limitations of Reed-Solomon codes, while
  preserving good or very good erasure correction
  capability.
• Linear encoding and decoding complexity
  iterative decoding.
• Long codeword lengths can be exploited.
• Extremely good performance, outperforming the
  performance of maximum distance
• possibility to encode long files as an unique
  codeword
• possibility to face long bursts of erasures.
• Currently under investigation within the CCSDS
  Bird of Feather (LEC-BOF).

6
Space Link Protocols Model

• A LEC code code can be in principle implemented
  at different layer in the protocol stack.
• The term LEC packet assumed different meanings
  depending on the way the code is implemented.

Possible layers at which long erasure codes can
be implemented
7
Outline

• Packet erasure correction in space / satellite
  communications ARQ and FEC techniques
• Long erasure correcting (LEC) codes and iterative
  erasure correction algorithm
• Structures for LEC codes
• Correction of erasure bursts
• Numerical results

8
Iterative Decoding the Basic Idea

• The q packets x1,,xq must satisfy a bit-wise
  single parity-check constraint.
• If any of the q packets x1,,xq is unknown, it
  can be reconstructed if the others are known.
• A single parity-check (SPC) code can correct at
  most one erasure.

a bit-wise single parity-check constraint
9
Iterative Decoding for LDPC Codes

• Bipartite graph representation
• Degree of a variable (check) node.
• (?, ?) edge degree distribution.
• ?i (?i) fraction of edges towards the
• variable (check) nodes with degree i.
• Information packets, encoded packets, code rate
  R.
• Iterative decoding
• The previously described decoding rule
• is iteratively applied to all the check nodes.
• Equivalent description as a message
• passing decoding algorithm (belief-propagation).
• Repetition codes and SPC codes.

received packet
?
received packet
received packet
received packet
?
Check nodes parity-checks
received packet
Variable nodes encoded packets
10
Decoding Threshold

• Threshold of a degree distribution (?,?) maximum
  fraction of erased messages that an infinitely
  long LDPC code with degree distribution (?,?) is
  able to correct (under iterative decoding).
• The asymptotic performance of LDPC codes under
  message passing decoder only depends on the edge
  degree distribution of the underlying bipartite
  graph.
• From the channel coding theorem p lt 1 R, for
  a LDPC code with code rate R.
• Known result the iterative decoding of LDPC
  codes can achieve the memory-less erasure channel
  capacity (capacity achieving degree
  distributions).

11
Outline

• Packet erasure correction in space / satellite
  communications ARQ and FEC techniques
• Long erasure correcting (LEC) codes and iterative
  erasure correction algorithm
• Structures for LEC codes
• Correction of erasure bursts
• Numerical results

12
IRA Codes

• Class of LDPC codes with linear complexity
  encoding.
• Systematic encoding
• x1 u1, , xk uk
• Redundant packet p1 is generated as bit-wise XOR
  of some information packets.
• Redundant packet pi is generated as bit-wise XOR
  of pi-1 and some information packets.
• Codeword
• u1, , uk, p1, , pn-k

Redundant packets
Systematic packets (information packets)
13
Tornado Codes

• Special class of LDPC codes, whose structure
  allows for linear complexity and systematic
  encoding.
• Several layers of encoded packets
• packets in the first layer are the encoded
  packets
• packets in layer i are computed from packets in
  the layer i 1.
• Decoding process can be performed in the same way
  as for LDPC codes, or starting from the last
  layer to the first.

14
Protograph Codes

• The bipartite graph of a protograph code is
  obtained starting from a bipartite graph with a
  small number of edges and nodes (the protograph).
• The final bipartite graph is obtained from a
  certain number of repetitions of the protograph,
  in order to achieve the desired codeword length.
• Possibility to perform the analysis and the
  design on the protograph.
• Protograph codes have been proposed by NASA/JPL
  within the LEC BOF.
• Examples

15
Generalized LDPC (GLDPC) Codes

• Some check nodes are allowed to be (n, k) generic
  block linear codes (not SPC codes).
• Increased erasure correction capability at the
  generalized check nodes.
• bounded distance decoding (correct up to dmin
  1 erasures)
• maximum a posteriori (MAP) decoding (most
  powerful decoding algorithms)
• Possibility to improve the threshold with respect
  to LDPC codes.

n1 edges
SPC code
(n1,k1) block linear code
repetition codes
16
Outline

• Packet erasure correction in space / satellite
  communications ARQ and FEC techniques
• Long erasure correcting (LEC) codes and iterative
  erasure correction algorithm
• Structures for LEC codes
• Correction of erasure bursts
• Numerical results

17
Burst Erasure Correcting LEC Codes

• Packet erasures are usually correlated, and
  bursts of erasures can take place.
• Packet erasures can be due due to weather,
  shadowing, or loss of frame synchronization.
• An algorithm has been developed which permits to
  optimize the performance of LEC codes on (single)
  burst erasure channels, with no sacrifice on the
  performance on memory-less packet erasure
  channel.
• Optimization of Lmax maximum guaranteed erasure
  burst length.
• Example
• n 2000, R ½
• p n 921
• Lmax 904

18
Outline

• Packet erasure correction in space / satellite
  communications ARQ and FEC techniques
• Long erasure correcting (LEC) codes and iterative
  erasure correction algorithm
• Structures for LEC codes
• Correction of erasure bursts
• Numerical results

19
Memory-less PEC Performance

• Performance in terms of decoding failure rate VS
  channel packet erasure probability.
• Compromise between waterfall and error floor
  performance.

20
Memory-less PEC Performance

• Performance in terms of decoding failure rate.
• The two codes have the same performance on
  memory-less packet erasure channel.
• Channel model constant length burst erasure
  channel

21
Conclusions

• LE codes are currently under investigation within
  the CCSDS Long Erasure Codes Bird of Feather
  (LEC-BOF).
• Some possible codes structures and encoding /
  decoding algorithms have been recalled.
• Low complexity iterative decoding algorithm,
  which can asymptotically achieve the erasure
  channel capacity.
• Very good finite length performance, possibility
  to exploit long codeword lengths (up to thousands
  of packets).
• LE codes can be in principle implemented at
  different layers in the protocol stack, and offer
  flexibility in the choice of the packet length.