Rank-Metric Codes for Priority Encoding Transmission with Network Coding

1. Rank-Metric Codes forPriority Encoding Transmissionwith Network Coding Danilo SilvaandFrank R. Kschischang University of Toronto 10th Canadian Workshop on Information Theory

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

3. 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 10th Canadian Workshop on Information Theory

4. 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 10th Canadian Workshop on Information Theory

5. Priority Encoding Transmission • Deterministic PET: • Input: layers Li with priority levelski·n(smaller ki = 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] 10th Canadian Workshop on Information Theory

6. Priority Encoding Transmission information symbols parity symbols • Example: layers packets encoding(MDS code) 10th Canadian Workshop on Information Theory

7. 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,”] 10th Canadian Workshop on Information Theory

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

9. 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 canbe recovered! 10th Canadian Workshop on Information Theory

10. 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. 10th Canadian Workshop on Information Theory

11. Combining PET and Network Coding • Example in : k=2 nonsingular linearlyindependent linearly dependent 10th Canadian Workshop on Information Theory

12. 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? 10th Canadian Workshop on Information Theory

13. Traditional FEC and Network Coding RS encoder • Suppose packets are encoded with a RS code: message codeword transmittedpackets 10th Canadian Workshop on Information Theory

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

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

16. 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 10th Canadian Workshop on Information Theory

17. Gabidulin Codes • Encoding packets with a Gabidulin code: encoder message codeword transmittedpackets 10th Canadian Workshop on Information Theory

18. Decoding Gabidulin Codes • After packet mixing and one packet erasure: q3 distinct evaluation points for f(x) of degree <q3 can find f(x) using Lagrangian interpolation 10th Canadian Workshop on Information Theory

19. Rank-Metric Codes [E.M. Gabidulin, “Theory of codes with maximum rank distance,” Probl. Inform. Transm., 1985] 10th Canadian Workshop on Information Theory

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

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

22. 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 10th Canadian Workshop on Information Theory