1. 10th Canadian Workshop on Information Theory June 7, 2007 Rank-Metric Codes for Priority Encoding Transmission with Network Coding Danilo Silva and Frank R. Kschischang University of Toronto

2. 10th Canadian Workshop on Information TheoryJune 7, 20072 Outline Motivation – Priority Encoding Transmission – Random Network Coding – What happens when we combine both? A rank-metric approach Conclusions

3. 10th Canadian Workshop on Information TheoryJune 7, 20073 Priority Encoding Transmission Approaches to erasure correction (packet loss): – Rateless codes/retransmission: requires acknowledgement introduce delay – Classical erasure codes: rate decided a priori bandwidth waste if rate smaller than capacity low performance if rate higher than capacity

4. 10th Canadian Workshop on Information TheoryJune 7, 20074 Why Priority Encoding Transmission? – Priority encoding transmission: better trade-off between performance and rate requires source signal than can be partitioned into layers of unequal importance apply unequal error protection to layers

5. 10th Canadian Workshop on Information TheoryJune 7, 20075 Priority Encoding Transmission Deterministic PET: – Input: layers L i with priority levels k i · n (smaller k i = higher importance) – Output: n packets such that: any K of these packets are sufficient to recover all layers that have priority level · K [A. Albanese et al., “ Priority encoding transmission, ” 1996]

6. 10th Canadian Workshop on Information TheoryJune 7, 20076 Priority Encoding Transmission packets information symbols parity symbols layers encoding (MDS code) Example:

7. 10th Canadian Workshop on Information TheoryJune 7, 20077 Random Network Coding Network coding: – Generalizes routing in communication networks – Can increase the throughput of traditional networks (achieves the multicast capacity) Random network coding: – A practical way to perform network coding – Many practical advantages over solutions based on routing [Ho et al., “ A random linear network coding approach to multicast, ” ]

8. 10th Canadian Workshop on Information TheoryJune 7, 20078 Random Network Coding Each block (generation) of the information stream is partitioned into n packets Nodes form outgoing packets as random linear combinations of incoming packets headerpayload “ mixed ” data

9. 10th Canadian Workshop on Information TheoryJune 7, 20079 Erasures in Network Coding What if not enough packets can reach the destination? – An erasure in network coding is more severe than a classical erasure since one erased packet may “ contaminate ” other packets – Classical erasure correcting codes will not work! no packets can be recovered!

10. 10th Canadian Workshop on Information TheoryJune 7, 200710 Combining PET and Network Coding One possible solution to combine PET and RNC: [P.A. Chou, Y. Wu, and K. Jain, “ Practical network coding, ” 2003] – However, the guarantees are probabilistic.

11. 10th Canadian Workshop on Information TheoryJune 7, 200711 Combining PET and Network Coding Example in : k=2 nonsingular linearly dependent linearly independent

12. 10th Canadian Workshop on Information TheoryJune 7, 200712 Combining PET and Network Coding Our goal: – Obtain a deterministic PET system that is compatible with network coding Observation: – Classical erasures are special cases of network coding erasures  must use MDS codes Approach: – Are there MDS codes that can also correct network coding erasures?

13. 10th Canadian Workshop on Information TheoryJune 7, 200713 Traditional FEC and Network Coding Suppose packets are encoded with a RS code: RS encoder message codeword transmitted packets

14. 10th Canadian Workshop on Information TheoryJune 7, 200714 Traditional FEC and Network Coding received packets not necessarily invertible! e.g., in After packet mixing and one packet erasure:

15. 10th Canadian Workshop on Information TheoryJune 7, 200715 Linearized Polynomials Is there a polynomial f(x) that satisfies...? If this is true, then ? are three evaluation points for f(x)

16. 10th Canadian Workshop on Information TheoryJune 7, 200716 Linearized Polynomials Linearized polynomials: The property that gives their name: – An evaluation of a linearized polynomial is a map from to itself that is linear over

17. 10th Canadian Workshop on Information TheoryJune 7, 200717 Gabidulin Codes Encoding packets with a Gabidulin code: encoder message codeword transmitted packets

18. 10th Canadian Workshop on Information TheoryJune 7, 200718 Decoding Gabidulin Codes After packet mixing and one packet erasure: q 3 distinct evaluation points for f(x) of degree < q 3 can find f(x) using Lagrangian interpolation

19. 10th Canadian Workshop on Information TheoryJune 7, 200719 Rank-Metric Codes [E.M. Gabidulin, “ Theory of codes with maximum rank distance, ” Probl. Inform. Transm., 1985] Reed-Solomon codesGabidulin codes Hamming distance metricRank distance metric PolynomialsLinearized polynomials MDSMRD (maximum rank distance) errors and erasures “ rank errors ” and “ rank erasures ” Berlekamp-Massey algorithmmodified Berlekamp-Massey algorithm

20. 10th Canadian Workshop on Information TheoryJune 7, 200720 Main implications: – Need m symbols in to make a symbol in – Field size is exponentially larger: Example: A Rank-Metric PET System...

21. 10th Canadian Workshop on Information TheoryJune 7, 200721 – Can also correct errors introduced by a jammer: A Rank-Metric PET System [D. Silva and F.R. Kschischang, “ Using rank-metric codes for error correction in random network coding, ” ISIT 2007] all received packets are corrupt only one rank error

22. 10th Canadian Workshop on Information TheoryJune 7, 200722 Conclusions Combining PET and RNC is a promising approach to low-latency multicast Existing PET systems are either probabilistic or incompatible with RNC We propose a PET system based on rank-metric codes that is compatible with RNC and provides deterministic guarantees of recovery Our system can also correct packet errors introduced by a jammer