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

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

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.

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.

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.

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.

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]

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.

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 = � 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.