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Long Erasure Codes: the New Frontier for Zero-Loss in Space Applications? Enrico Paolini, University of Bologna epaolini@deis.unibo.it Gian Paolo Calzolari, ESA/ESOC Gian.Paolo.Calzolari@esa.int Marco Chiani, University of Bologna mchiani@deis.unibo.it SpaceOps 2006, Rome, Italy, 19-23 June

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long erasure codes the new frontier for zero loss in space applications

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

Enrico Paolini, University of Bologna

epaolini@deis.unibo.it

Gian Paolo Calzolari, ESA/ESOC

Gian.Paolo.Calzolari@esa.int

Marco Chiani, University of Bologna

mchiani@deis.unibo.it

SpaceOps 2006, Rome, Italy, 19-23 June

outline
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
packet erasures
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

?

traditional techniques
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 couldn’t 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.
long erasure correcting lec codes
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).
space link protocols model
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

outline7
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
iterative decoding the basic idea
Iterative Decoding: the Basic Idea

a bit-wise single parity-check constraint

  • 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.
iterative decoding for ldpc codes
Iterative Decoding for LDPC Codes

received packet

  • 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

?

Check nodes:

parity-checks

received packet

Variable nodes:

encoded packets

decoding threshold
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* < 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).
outline11
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
ira codes
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)

tornado codes
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.
protograph codes
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:
generalized ldpc gldpc codes
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

outline16
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
burst erasure correcting lec codes

xn xi ? … ? ? x2 x1

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

outline18
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
memory less pec performance
Memory-less PEC Performance
  • Performance in terms of decoding failure rate VS channel packet erasure probability.
  • Compromise between waterfall and error floor performance.
memory less pec performance20
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:
conclusions
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.