Duality for Simple Multiple Access Networks Iwan Duursma Department - PowerPoint PPT Presentation

Duality for Simple Multiple Access Networks Iwan Duursma Department of Mathematics and Coordinated Science Laboratory U of Illinois at Urbana-Champaign DIMACS Workshop on Network Coding December 15-17, 2015 Iwan Duursma (U Illinois) Duality DIMACS - Dec 2015 1 / 29

Outline 1 - Reliability and Security 2 - Efficient repair 3 - Constrained codes and Duality Tail-biting trellises Simple multiple access networks Iwan Duursma (U Illinois) Duality DIMACS - Dec 2015 2 / 29

1 - Reliability and Security                                       High rank submatrices protect against erasures and eavesdroppers Iwan Duursma (U Illinois) Duality DIMACS - Dec 2015 3 / 29

Details Erasure channel Encoding using G C yields a vector with entropy H ( C ) . For vectors observed outside the erased positions E ⊂ [ n ] , H ( C ) = H ( C |E ) (information gain) + I ( C ; E ) (equivocation) Wiretep channel II Decoding using G T D distinguishes vectors with entropy H ( D ) . For vectors observed in the eavesdropped positions E ⊂ [ n ] , H ( D ) = I ( D ; E ) (information gain) + H ( D |E ) (equivocation) � � H ( C |E ) , H ( D |E ) = rank for E = , Iwan Duursma (U Illinois) Duality DIMACS - Dec 2015 4 / 29

Protection against erasures AND eavesdroppers Nested codes Combine encoding via G C with decoding via G T D Transmission rate reduces from H ( C ) to H ( C | D ⊥ ) in return for a higher threshold for the eavesdropper. We may assume wlog that D ⊥ ⊂ C (nested codes) For vectors observed outside E ⊂ [ n ] (legitimate receiver), H ( C | D ⊥ ) = H ( C ) − H ( D ⊥ ) = H ( C |E ) − H ( D ⊥ |E ) (information gain) + I ( C ; E ) − I ( D ⊥ ; E ) (equivocation) Iwan Duursma (U Illinois) Duality DIMACS - Dec 2015 5 / 29

Main example (Reed-Solomon)  1 1 1 1 1 1 · · · · · · · · ·  x y z u v w   x 2 y 2 z 2 u 2 v 2 w 2     G RS =  x 3 y 3 z 3 u 3 v 3 w 3      x 4 y 4 z 4 u 4 v 4 w 4   x 5 y 5 z 5 u 5 v 5 w 5 · · · · · · · · ·     B = rank  = 6 .      Iwan Duursma (U Illinois) Duality DIMACS - Dec 2015 6 / 29

2 - Efficient repair Reed-Solomon codes provide maximum protection of a message against erasures.    is full rank    However repair using RS-codes is inefficient. For RS-codes,   repair bandwith = rank     . Other codes are more suitable when erasure repair is important (e.g. in distributed storage). Iwan Duursma (U Illinois) Duality DIMACS - Dec 2015 7 / 29

MSR construction (Rashmi-Shah-Kumar 2010)  1 0 1 0 1 0 · · · · · · · · ·  x 1 y 1 z 1     0 x 0 y 0 z   G MSR =  x 2 y 2 z 2  0 0 0    x 3 x 2 y 3 y 2 z 3 z 2    x 3 y 3 z 3 0 0 0 · · · · · · · · ·    = 6 . B = rank ( k = 3 , α = 2 )  γ = repair bandwith = 4 ( d = 4 , β = 1 ) (modify if char � = 2) Iwan Duursma (U Illinois) Duality DIMACS - Dec 2015 8 / 29

MBR construction (Rashmi-Shah-Kumar 2010)  1 0 0 1 0 0 · · · · · · · · ·  x 1 0 y 1 0     0 x 0 0 y 0   G MBR =   x 1 0 y 1 0     x 2 y 2 2 x 1 2 y 1   x 2 y 2 0 x 0 y · · · · · · · · ·    = 5 . B = rank ( k = 2 , α = 3 )  γ = repair bandwith = 3 ( d = 3 , β = 1 ) Iwan Duursma (U Illinois) Duality DIMACS - Dec 2015 9 / 29

Storage vs bandwith trade-off MSR minimizes storage per disk. MBR minimizes repair bandwith. For exact repair solutions in between MBR and MSR, the optimal trade-offs are an open problem. Case n = k + 1 = d + 1 is solved Tian; Sasidharan, Senthoor, Kumar; D; Tian, Sasidharan, Aggarwal, Vaishampayan, Kumar; Mohajer, Tandon; Prakash, Krishnan; D’ Iwan Duursma (U Illinois) Duality DIMACS - Dec 2015 10 / 29

Storing four bits on four disks x , z + t y , t + x x , y , z , t z , x + y t , y + z B = 4 n = 4 Iwan Duursma (U Illinois) Duality DIMACS - Dec 2015 11 / 29

Reading four bits from any two disks x , z + t y , t + x x , z + t t , y + z z , x + y t , y + z α = 2 k = 2 Iwan Duursma (U Illinois) Duality DIMACS - Dec 2015 12 / 29

Disk repair with help form any three disks x , z + t z + t y , t + x t + x + y z , x + y z , x + y t , y + z t β = 1 d = 3 Iwan Duursma (U Illinois) Duality DIMACS - Dec 2015 13 / 29

Repair matrix The repair matrix summarizes storage and repair for a regenerating code. W 1 S 1 → 3 S 1 → 2 S 1 → 4 W 2 S 2 → 1 S 2 → 3 S 2 → 4 W 3 S 3 → 1 S 3 → 2 S 3 → 4 W 4 S 4 → 1 S 4 → 2 S 4 → 3 W i = data stored at node i S i → j = helper information from node i to node j Iwan Duursma (U Illinois) Duality DIMACS - Dec 2015 14 / 29

[D - arxiv 2014] Theorem 1 Let B q = H ( W 1 ) + · · · + H ( W q ) + � H ( S i → j ) , such that B ≤ B q . Let q , q 1 , . . . , q m − 2 , r , s > 0 such that (explicit condition). Then m − 2 � mB ≤ B q + B q i + B r + s − rs β. i = 1 Theorem 2 Let B q = H ( W 1 ) + · · · + H ( W q ) + � H ( S M → L ) , such that B ≤ B q . For each ( M , L ) , let ℓ = | L | , m = | M | , and let r ≥ ℓ . Then � � B + ℓ B ≤ B q + ( B r + m − 1 + ( ℓ − 1 )( B r + m − 2 − β )) . ( M , L ) ( M , L ) Iwan Duursma (U Illinois) Duality DIMACS - Dec 2015 15 / 29

[D arXiv 2015] Theorem 3 For any set of parameters ( n , k , d ) , and for 0 ≤ ℓ ≤ k , 0 ≤ v , � v + 2 � � v + 1 � � ℓ � B ≤ B k + ( v + 1 ) B k − ℓ − v β. 2 2 2 Independently (special cases) Prakash-Krishnan, arXiv 2015 Mohajer-Tandon, ITA/ISIT 2015a, ISIT2015b. Iwan Duursma (U Illinois) Duality DIMACS - Dec 2015 16 / 29

3 - Constrained codes David Forney (Talk at Allerton ’97) Does the Golay code have a generator matrix of the form  ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ 00 00 00 00 00 00 00  00 ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ 00 00 00 00 00 00     00 00 ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ 00 00 00 00 00     00 00 00 ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ 00 00 00 00       00 00 00 00 ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ 00 00 00     00 00 00 00 00 ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ 00 00     00 00 00 00 00 00 ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ 00     00 00 00 00 00 00 00 ∗∗ ∗∗ ∗∗ ∗∗ ∗∗       00 00 00 00 00 00 00 ∗∗ ∗∗ ∗∗ ∗∗ ∗∗     ∗∗ ∗∗ 00 00 00 00 00 00 00 ∗∗ ∗∗ ∗∗     ∗∗ ∗∗ ∗∗ 00 00 00 00 00 00 00 ∗∗ ∗∗   ∗∗ ∗∗ ∗∗ ∗∗ 00 00 00 00 00 00 00 ∗∗ Answer: Yes (Calderbank-Forney-Vardy 1999) Iwan Duursma (U Illinois) Duality DIMACS - Dec 2015 17 / 29

Characteristic matrices (Koetter-Vardy 2003) A set of characteristic generators for the row space row G is a subset of n vectors such that 1) Spans of vectors start and end in distinct positions, and 2) The sum of the spanlengths of the vectors is minimal. A square matrix is called a characteristic matrix for G if its rows form a set of characteristic generators. Example  1 1 1 0 0   1 0 1 1 1  0 1 1 0 1 1 1 0 0 1         X = 0 0 1 1 0 Y = 0 1 1 1 0         0 1 0 1 1 0 1 1 1 0     1 0 0 0 1 1 0 1 1 1 Iwan Duursma (U Illinois) Duality DIMACS - Dec 2015 18 / 29

Dual characteristic matrices Question Under what conditions is a pair of characteristic matrices in duality, i.e. when does a pair define dual trellises? Conjecture (KV 2003) For a choice of lexicographically first characteristic generators for G and for a matching choice of lexicographically first characteristic generators for H , the obtained tail-biting trellises are in duality. (Gleussen-Larssing and Weaver 2011) Counterexample to the conjecture. Characterization of dual characteristic matrices in terms of local duality of trellises Iwan Duursma (U Illinois) Duality DIMACS - Dec 2015 19 / 29

(GLW 2011) Example  1 1 0 0   1 1 0 1  0 1 1 1 1 1 0 1     X = Y =     1 0 1 1 1 1 1 0     0 1 1 1 0 0 1 1     1 1 0 0 1 1 1 0 0 1 1 1 1 1 0 1 X ′ = Y ′ =         1 0 1 1 1 1 1 0     1 0 1 1 0 0 1 1 Conjecture: X ∼ Y . Local duality: X ∼ Y ′ and X ′ ∼ Y . Iwan Duursma (U Illinois) Duality DIMACS - Dec 2015 20 / 29