Different Flavors Hoang Dau University of Illinois at - PowerPoint PPT Presentation

Coding with Constraints: Different Flavors Hoang Dau University of Illinois at Urbana-Champaign 1 Email: hoangdau@uiuc.edu DIMACS Workshop on Network Coding: the Next 15 Years Rutgers University, NJ, 2015

Part I Coding with Constraints: A Quick Survey

Coding with Constraints: Definition 𝑑 1 𝑑 2 𝑦 1 𝑦 2 𝑜 coded 𝑙 data ENCODED symbols symbols 𝑦 𝑙 𝑑 𝑜 CONVENTIONAL CODE 𝑑 𝑘 = 𝑑 𝑘 (𝑦 1 , 𝑦 2 , … , 𝑦 𝑙 ) 3

Coding with Constraints: Definition 𝑑 1 𝑑 2 𝑦 1 𝑦 2 𝑜 coded 𝑙 data ENCODED symbols symbols 𝑦 𝑙 𝑑 𝑜 CODE WITH CONSTRAINTS CONVENTIONAL CODE 𝑑 𝑘 = 𝑑 𝑘 ({𝑦 𝑗 : 𝑗 ∈ 𝐷 𝑘 }) , 𝐷 𝑘 ⊆ {1,2, … , 𝑙} 𝑑 𝑘 = 𝑑 𝑘 (𝑦 1 , 𝑦 2 , … , 𝑦 𝑙 ) 4

Coding with Constraints: Example = 𝑑 1 (𝑦 1 , 𝑦 2 ) 𝑑 1 𝑦 1 = 𝑑 2 (𝑦 1 , 𝑦 3 ) 𝑑 2 𝑦 2 𝑑 3 = 𝑑 3 (𝑦 2 ) 𝑦 3 𝑑 4 = 𝑑 4 (𝑦 2 , 𝑦 3 ) 𝑑 5 = 𝑑 5 (𝑦 1 , 𝑦 3 ) 5

Coding with Constraints: Example = 𝑑 1 (𝑦 1 , 𝑦 2 ) 𝑑 1 𝑦 1 = 𝑑 2 (𝑦 1 , 𝑦 3 ) 𝑑 2 𝑦 2 𝑑 3 = 𝑑 3 (𝑦 2 ) 𝑦 3 𝑑 4 = 𝑑 4 (𝑦 2 , 𝑦 3 ) 𝑑 5 = 𝑑 5 (𝑦 1 , 𝑦 3 ) Linear code: 𝑑 1 , 𝑑 2 , 𝑑 3 , 𝑑 4 , 𝑑 5 = 𝑦 1 , 𝑦 2 , 𝑦 3 𝑯 where the generator matrix 𝑯 is ? ? 0 0 ? 𝑯 = ? 0 ? ? 0 0 ? 0 ? ? 6

Coding with Constraints: Main Problem Given the constraints 𝑑 𝑘 = 𝑑 𝑘 ({𝑦 𝑗 : 𝑗 ∈ 𝐷 𝑘 }) , 𝐷 𝑘 ⊆ {1,2, … , 𝑙} how to construct codes that – achieve the optimal minimum distance – over small field size 𝑟 ≈ poly 𝑜

Coding with Constraints: Main Problem Given the constraints 𝑑 𝑘 = 𝑑 𝑘 ({𝑦 𝑗 : 𝑗 ∈ 𝐷 𝑘 }) , 𝐷 𝑘 ⊆ {1,2, … , 𝑙} how to construct codes that – achieve the optimal minimum distance – over small field size 𝑟 ≈ poly 𝑜 Linear case: given ? ? 0 0 ? 𝑯 = ? 0 ? ? 0 0 ? 0 ? ? how to replace “?” -entries by elements of 𝐺 𝑟 ( 𝑟 ≈ poly(𝑜) ) so that 𝑯 generate a code with optimal distance

Coding with Constraints: Upper Bound Upper Bound (Halbawi-Thill- Hassibi’15, Song -Dau- Yuen’15) 𝑒 ≤ 𝑒 𝑛𝑏𝑦 = 1 + ∅≠𝐽⊆ 1,…,𝑙 ( ∪ 𝑗∈𝐽 𝑆 𝑗 − |𝐽|) min where 𝑆 𝑗 = {𝑘: 𝑗 ∈ 𝐷 𝑘 }

Coding with Constraints: Upper Bound Upper Bound (Halbawi-Thill- Hassibi’15, Song -Dau- Yuen’15) 𝑒 ≤ 𝑒 𝑛𝑏𝑦 = 1 + ∅≠𝐽⊆ 1,…,𝑙 ( ∪ 𝑗∈𝐽 𝑆 𝑗 − |𝐽|) min where 𝑆 𝑗 = {𝑘: 𝑗 ∈ 𝐷 𝑘 } Properties – 𝑒 𝑛𝑏𝑦 can be found in time poly(𝑜) 𝑜 – codes with 𝑒 = 𝑒 𝑛𝑏𝑦 always exists over fields of size ≈ 𝑒 − 1

Coding with Constraints: Upper Bound Upper Bound (Halbawi-Thill- Hassibi’15, Song -Dau- Yuen’15) 𝑒 ≤ 𝑒 𝑛𝑏𝑦 = 1 + ∅≠𝐽⊆ 1,…,𝑙 ( ∪ 𝑗∈𝐽 𝑆 𝑗 − |𝐽|) min where 𝑆 𝑗 = {𝑘: 𝑗 ∈ 𝐷 𝑘 } Properties – 𝑒 𝑛𝑏𝑦 can be found in time poly(𝑜) 𝑜 – codes with 𝑒 = 𝑒 𝑛𝑏𝑦 always exists over fields of size ≈ 𝑒 − 1 Question of interest: how about fields of size poly(𝑜) ?

Coding with Constraints: Review (Small field) MDS Case: 𝑒 𝑛𝑏𝑦 = 𝑜 − 𝑙 + 1 – Optimal codes exist in a few special cases (Halbawi-Ho-Yao- Duursma ’14, Dau-Song-Yuen ’14 , Yan-Sprintson- Zelenko’ 14)

Coding with Constraints: Review (Small field) MDS Case: 𝑒 𝑛𝑏𝑦 = 𝑜 − 𝑙 + 1 – Optimal codes exist in a few special cases (Halbawi-Ho-Yao- Duursma ’14, Dau-Song-Yuen ’14 , Yan-Sprintson- Zelenko’ 14) • 𝑙 ≤ 4 (every 𝑜 )

Coding with Constraints: Review (Small field) MDS Case: 𝑒 𝑛𝑏𝑦 = 𝑜 − 𝑙 + 1 – Optimal codes exist in a few special cases (Halbawi-Ho-Yao- Duursma ’14, Dau-Song-Yuen ’14 , Yan-Sprintson- Zelenko’ 14) • 𝑙 ≤ 4 (every 𝑜 ) • rows of 𝑯 partitioned into ≤ 3 groups, each has same “?” -pattern

Coding with Constraints: Review (Small field) MDS Case: 𝑒 𝑛𝑏𝑦 = 𝑜 − 𝑙 + 1 – Optimal codes exist in a few special cases (Halbawi-Ho-Yao- Duursma ’14, Dau-Song-Yuen ’14 , Yan-Sprintson- Zelenko’ 14) • 𝑙 ≤ 4 (every 𝑜 ) • rows of 𝑯 partitioned into ≤ 3 groups, each has same “?” -pattern • rows have 𝑙 − 1 zeros & 2 different rows share ≤ 1 common zeros

Coding with Constraints: Review (Small field) MDS Case: 𝑒 𝑛𝑏𝑦 = 𝑜 − 𝑙 + 1 – Optimal codes exist in a few special cases (Halbawi-Ho-Yao- Duursma ’14, Dau-Song-Yuen ’14 , Yan-Sprintson- Zelenko’ 14) • 𝑙 ≤ 4 (every 𝑜 ) • rows of 𝑯 partitioned into ≤ 3 groups, each has same “?” -pattern • rows have 𝑙 − 1 zeros & 2 different rows share ≤ 1 common zeros General Case (Halbawi-Thill- Hassibi’15 ): 𝑒 𝑛𝑏𝑦 ≤ 𝑜 − 𝑙 + 1

Coding with Constraints: Review (Small field) MDS Case: 𝑒 𝑛𝑏𝑦 = 𝑜 − 𝑙 + 1 – Optimal codes exist in a few special cases (Halbawi-Ho-Yao- Duursma ’14, Dau-Song-Yuen ’14 , Yan-Sprintson- Zelenko’ 14) • 𝑙 ≤ 4 (every 𝑜 ) • rows of 𝑯 partitioned into ≤ 3 groups, each has same “?” -pattern • rows have 𝑙 − 1 zeros & 2 different rows share ≤ 1 common zeros General Case (Halbawi-Thill- Hassibi’15 ): 𝑒 𝑛𝑏𝑦 ≤ 𝑜 − 𝑙 + 1 – Optimal codes exist if there are ≥ 𝑒 𝑛𝑏𝑦 − 1 indices 𝑘 ’s where 𝐷 𝑘 = 𝑙

Coding with Constraints: Review (Small field) MDS Case: 𝑒 𝑛𝑏𝑦 = 𝑜 − 𝑙 + 1 – Optimal codes exist in a few special cases (Halbawi-Ho-Yao- Duursma ’14, Dau-Song-Yuen ’14 , Yan-Sprintson- Zelenko’ 14) • 𝑙 ≤ 4 (every 𝑜 ) • rows of 𝑯 partitioned into ≤ 3 groups, each has same “?” -pattern • rows have 𝑙 − 1 zeros & 2 different rows share ≤ 1 common zeros General Case (Halbawi-Thill- Hassibi’15 ): 𝑒 𝑛𝑏𝑦 ≤ 𝑜 − 𝑙 + 1 – Optimal codes exist if there are ≥ 𝑒 𝑛𝑏𝑦 − 1 indices 𝑘 ’s where 𝐷 𝑘 = 𝑙 – Optimal systematic codes always exists (smaller bound 𝑒 𝑡𝑧𝑡 ≤ 𝑒 𝑛𝑏𝑦 )

Coding with Constraints: Review (Small field) MDS Case: 𝑒 𝑛𝑏𝑦 = 𝑜 − 𝑙 + 1 – Optimal codes exist in a few special cases (Halbawi-Ho-Yao- Duursma ’14, Dau-Song-Yuen ’14 , Yan-Sprintson- Zelenko’ 14) • 𝑙 ≤ 4 (every 𝑜 ) • rows of 𝑯 partitioned into ≤ 3 groups, each has same “?” -pattern • rows have 𝑙 − 1 zeros & 2 different rows share ≤ 1 common zeros General Case (Halbawi-Thill- Hassibi’15 ): 𝑒 𝑛𝑏𝑦 ≤ 𝑜 − 𝑙 + 1 – Optimal codes exist if there are ≥ 𝑒 𝑛𝑏𝑦 − 1 indices 𝑘 ’s where 𝐷 𝑘 = 𝑙 – Optimal systematic codes always exists (smaller bound 𝑒 𝑡𝑧𝑡 ≤ 𝑒 𝑛𝑏𝑦 ) Common Technique: Reed-Solomon (sub-) code

Coding with Constraints: Review Common Technique: Reed-Solomon (sub-) code = 𝑑 1 (𝑦 1 , 𝑦 2 ) 𝑑 1 𝑦 1 = 𝑑 2 (𝑦 1 , 𝑦 3 ) 𝑑 2 𝑦 2 𝑑 3 = 𝑑 3 (𝑦 2 ) 𝑦 3 𝑑 4 = 𝑑 4 (𝑦 2 , 𝑦 3 ) 𝑑 5 = 𝑑 5 (𝑦 1 , 𝑦 3 ) ? ? 0 0 ? 𝑯 = ? 0 ? ? 0 0 ? 0 ? ?

Coding with Constraints: Review Common Technique: Reed-Solomon (sub-) code = 𝑑 1 (𝑦 1 , 𝑦 2 ) 𝑑 1 𝑦 1 = 𝑑 2 (𝑦 1 , 𝑦 3 ) 𝑑 2 𝑦 2 𝑑 3 = 𝑑 3 (𝑦 2 ) 𝑦 3 𝑑 4 = 𝑑 4 (𝑦 2 , 𝑦 3 ) 𝑑 5 = 𝑑 5 (𝑦 1 , 𝑦 3 ) 𝛽 1 𝛽 2 𝛽 3 𝛽 4 𝛽 5 ? ? 0 0 ? 𝑯 = ? 0 ? ? 0 0 ? 0 ? ?

Coding with Constraints: Review Common Technique: Reed-Solomon (sub-) code = 𝑑 1 (𝑦 1 , 𝑦 2 ) 𝑑 1 𝑦 1 = 𝑑 2 (𝑦 1 , 𝑦 3 ) 𝑑 2 𝑦 2 𝑑 3 = 𝑑 3 (𝑦 2 ) 𝑦 3 𝑑 4 = 𝑑 4 (𝑦 2 , 𝑦 3 ) 𝑑 5 = 𝑑 5 (𝑦 1 , 𝑦 3 ) 𝛽 1 𝛽 2 𝛽 3 𝛽 4 𝛽 5 𝑔 1 (𝛽 1 ) 𝑔 1 (𝛽 2 ) 0 0 𝑔 1 (𝛽 5 ) ? ? 0 0 ? 𝑔 2 (𝛽 1 ) 0 𝑔 2 (𝛽 3 ) 𝑔 2 (𝛽 4 ) 0 𝑯 = = ? 0 ? ? 0 0 ? 0 ? ? 0 𝑔 3 (𝛽 2 ) 0 𝑔 3 (𝛽 4 ) 𝑔 3 (𝛽 5 )

Coding with Constraints: Review Common Technique: Reed-Solomon (sub-) code = 𝑑 1 (𝑦 1 , 𝑦 2 ) 𝑑 1 𝑦 1 = 𝑑 2 (𝑦 1 , 𝑦 3 ) 𝑑 2 𝑦 2 𝑑 3 = 𝑑 3 (𝑦 2 ) 𝑦 3 𝑑 4 = 𝑑 4 (𝑦 2 , 𝑦 3 ) 𝑑 5 = 𝑑 5 (𝑦 1 , 𝑦 3 ) 𝛽 1 𝛽 2 𝛽 3 𝛽 4 𝛽 5 𝑔 1 (𝛽 1 ) 𝑔 1 (𝛽 2 ) 0 0 𝑔 1 (𝛽 5 ) ? ? 0 0 ? 𝑔 2 (𝛽 1 ) 0 𝑔 2 (𝛽 3 ) 𝑔 2 (𝛽 4 ) 0 𝑯 = = ? 0 ? ? 0 0 ? 0 ? ? 0 𝑔 3 (𝛽 2 ) 0 𝑔 3 (𝛽 4 ) 𝑔 3 (𝛽 5 ) Difficulty: G may not be full rank

Part II: Joint Design of Different MDS Codes (joint work with H. Kiah, W. Song, and C. Yuen)

MDS Codes for Distributed Storage 𝑑 1 MDS: tolerate any 𝑜 − 𝑙 node failures 𝑑 2 𝑦 1 𝑦 2 𝑜 coded 𝑙 data Encoded symbols symbols 𝑦 𝑙 Facebook: Reed-Solomon 𝑜 = 14, 𝑙 = 10 𝑑 𝑜 25

Question of Interest • If two (or more) independent DSS share some common data, can we jointly design the corresponding MDS codes to get a better overall failure protection?