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

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

10th Canadian Workshop on Information TheoryJune 7, 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 TheoryJune 7, 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 TheoryJune 7, 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]

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

10th Canadian Workshop on Information TheoryJune 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 TheoryJune 7, 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

10th Canadian Workshop on Information TheoryJune 7, 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!

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

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

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

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

10th Canadian Workshop on Information TheoryJune 7, 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

10th Canadian Workshop on Information TheoryJune 7, 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

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

10th Canadian Workshop on Information TheoryJune 7, – 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

10th Canadian Workshop on Information TheoryJune 7, 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