Low Complexity Encoding for Network Codes Yuval Cassuto Michelle Effros Sidharth Jaggi

Obligatory Example/History s t1t1 t2t2 b1b1 b2b2 b2b2 b2b2 b1b1 b1b1 b1b1 b1b1 b1b1 b1b1 (b 1,b 2 ) b 1 +b 2 (b 1,b 2 ) [ACLY00] [ACLY00] Characterization Non-constructive [LYC03], [KM02] Constructive (linear) Exp-time design [JCJ03], [SET03] Poly-time design Centralized design [HKMKE03], [JCJ03] Decentralized design SIMPLERSIMPLER... C=2 [This work] All the above, plus optimal implementation complexity.

Complexity b1b1 b2b2 bmbm β1β1 β2β2 βkβk F(2 m )-linear network [KM02],[HKMKE03],… Source:- Group together `m’ bits, Every node:- Perform linear combinations over finite field F(2 m )

Complexity “Thm”: For any algebraic code, at least half the β matrices in F(2 m ) have at least m 2 /2 non-zero elements Randomly chosen algebraic encoders require O(m 2 ) bit operations [HKMKE03],… = …,[JEHM04],… [KKHRM05]

Simplicity – Permute-and-add “Thm”: With “high” probability, permute-and-add codes have “almost” the same performance as algebraic codes Permute-and-add encoders require O(m) bit operations Tight! “Thm”: To achieve capacity, need O(m) bit operations = Permutation matrix (sparse)

Simplicity – Permute-and-add “Thm”: With “high” probability, permute-and-add codes have “almost” the same performance as algebraic codes Permute-and-add encoders require O(m) bit operations Tight! “Thm”: To achieve capacity, need O(m) bit operations = Loss of information

Permute-and-add Codes m “sufficiently” large  b1b1 b2b2 bmbm b’ 1 b’ 2 b’ m b’’ 1 b’’ 2 b’’ m  ’’  ’’ Uniformly at random b2b2 b1b1 bmbm b’ 1 b’ m b’ 2 b’’ m b’’ 1 b’’ 2

Permute-and-add Codes   ’’  ’’ Uniformly at random b2b2 b1b1 bmbm b’ 1 b’ m b’ 2 b’’ m b’’ 1 b’’ 2   Transfer matrix

Permute-and-add Codes Percolate transfer matrices across successive cutsets (in header) If each transfer matrix full rank, Final transfer matrices full rank Decode by inverting final transfer matrix, QED Not true, with probability c > 0

Permute-and-add Codes m “sufficiently” large Each transfer matrix “almost” full rank, Final transfer matrices “almost” full rank Decode by inverting final transfer matrix, QED Pr π [Row rank > (1-ε m ) fraction] > 1-2 -O(mε m ) Pr π [Row rank > (1-|E|ε m ) fraction] > 1-|T||E|2 -O(mε m ) R=C- |E|ε m - ε m Pr π [Final transform invertible] > 1-(|T||E|+1)2 -O(mε m ) (1-|E|ε m ) fraction (1-ε m ) fraction Thm: Permute-and-add codes achieve R=C-(|E|+1)ε m, Pr > 1-(|T||E|+1)2 -O(mε m )

Proof of Lemma Transform I L 1,L 2 Gaussian Elimination “Almost” full rank, w.h.p.

The End/Where now? Low-complexity decoding? Fewer packets encoded together at nodes? Same permutation at each node? Zero-error/Deterministic? Permute-and-add Vs. Algebraic [HKMKE03] RateAlmost Same (ε loss) Probability of errorAlmost same (smaller exp) Block-lengthAlmost same (1/ε increase) Simple Distributed DesignDitto Implementation ComplexityQuadratically better Randomness requiredQuadratically better