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

1. 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

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 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.

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 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.

9. 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

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* < 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. 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

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.