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

Coding with Constraints: Different Flavors

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

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Part I Coding with Constraints: A Quick Survey

Coding with Constraints: Definition

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𝑦1 𝑦2 𝑦𝑙

𝑑1 𝑑2 𝑑𝑜

ENCODED

𝑙 data symbols 𝑜 coded symbols CONVENTIONAL CODE 𝑑

𝑘 = 𝑑 𝑘(𝑦1, 𝑦2, … , 𝑦𝑙)

Coding with Constraints: Definition

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𝑦1 𝑦2 𝑦𝑙

𝑑1 𝑑2 𝑑𝑜

ENCODED

𝑙 data symbols 𝑜 coded symbols CONVENTIONAL CODE 𝑑

𝑘 = 𝑑 𝑘(𝑦1, 𝑦2, … , 𝑦𝑙)

CODE WITH CONSTRAINTS 𝑑

𝑘 = 𝑑 𝑘({𝑦𝑗: 𝑗 ∈ 𝐷 𝑘}), 𝐷 𝑘 ⊆ {1,2, … , 𝑙}

Coding with Constraints: Example

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𝑦1 𝑦2

𝑑1 𝑑2

𝑦3

𝑑3 𝑑4 𝑑5 = 𝑑1(𝑦1, 𝑦2) = 𝑑2(𝑦1, 𝑦3) = 𝑑3(𝑦2) = 𝑑4(𝑦2, 𝑦3) = 𝑑5(𝑦1, 𝑦3)

Coding with Constraints: Example

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𝑦1 𝑦2

𝑑1 𝑑2

𝑦3

𝑑3 𝑑4 𝑑5 = 𝑑1(𝑦1, 𝑦2) = 𝑑2(𝑦1, 𝑦3) = 𝑑3(𝑦2) = 𝑑4(𝑦2, 𝑦3) = 𝑑5(𝑦1, 𝑦3)

Linear code: 𝑑1, 𝑑2, 𝑑3, 𝑑4, 𝑑5 = 𝑦1, 𝑦2, 𝑦3 𝑯 where the generator matrix 𝑯 is 𝑯 = ? ? ? ? ? ? ? ? ?

Coding with Constraints: Main Problem

Given the constraints 𝑑

𝑘 = 𝑑 𝑘({𝑦𝑗: 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({𝑦𝑗: 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 𝑯 = ? ? ? ? ? ? ? ? ? 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 ≤ 𝑒𝑛𝑏𝑦 = 1 + min

∅≠𝐽⊆ 1,…,𝑙 ( ∪𝑗∈𝐽 𝑆𝑗 − |𝐽|)

where 𝑆𝑗 = {𝑘: 𝑗 ∈ 𝐷

𝑘}

Coding with Constraints: Upper Bound

Upper Bound (Halbawi-Thill-Hassibi’15, Song-Dau-Yuen’15) 𝑒 ≤ 𝑒𝑛𝑏𝑦 = 1 + min

∅≠𝐽⊆ 1,…,𝑙 ( ∪𝑗∈𝐽 𝑆𝑗 − |𝐽|)

where 𝑆𝑗 = {𝑘: 𝑗 ∈ 𝐷

𝑘}

Properties

– 𝑒𝑛𝑏𝑦 can be found in time poly(𝑜) – codes with 𝑒 = 𝑒𝑛𝑏𝑦 always exists over fields of size ≈ 𝑜 𝑒 − 1

Coding with Constraints: Upper Bound

Upper Bound (Halbawi-Thill-Hassibi’15, Song-Dau-Yuen’15) 𝑒 ≤ 𝑒𝑛𝑏𝑦 = 1 + min

∅≠𝐽⊆ 1,…,𝑙 ( ∪𝑗∈𝐽 𝑆𝑗 − |𝐽|)

where 𝑆𝑗 = {𝑘: 𝑗 ∈ 𝐷

𝑘}

Properties

– 𝑒𝑛𝑏𝑦 can be found in time poly(𝑜) – codes with 𝑒 = 𝑒𝑛𝑏𝑦 always exists over fields of size ≈ 𝑜 𝑒 − 1

Question of interest: how about fields of size poly(𝑜)?

Coding with Constraints: Review

(Small field) MDS Case: 𝑒𝑛𝑏𝑦 = 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: 𝑒𝑛𝑏𝑦 = 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: 𝑒𝑛𝑏𝑦 = 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: 𝑒𝑛𝑏𝑦 = 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: 𝑒𝑛𝑏𝑦 = 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): 𝑒𝑛𝑏𝑦 ≤ 𝑜 − 𝑙 + 1

Coding with Constraints: Review

(Small field) MDS Case: 𝑒𝑛𝑏𝑦 = 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): 𝑒𝑛𝑏𝑦 ≤ 𝑜 − 𝑙 + 1

– Optimal codes exist if there are ≥ 𝑒𝑛𝑏𝑦 − 1 indices 𝑘’s where 𝐷

𝑘 = 𝑙

Coding with Constraints: Review

(Small field) MDS Case: 𝑒𝑛𝑏𝑦 = 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): 𝑒𝑛𝑏𝑦 ≤ 𝑜 − 𝑙 + 1

– Optimal codes exist if there are ≥ 𝑒𝑛𝑏𝑦 − 1 indices 𝑘’s where 𝐷

𝑘 = 𝑙

– Optimal systematic codes always exists (smaller bound 𝑒𝑡𝑧𝑡 ≤ 𝑒𝑛𝑏𝑦)

Coding with Constraints: Review

(Small field) MDS Case: 𝑒𝑛𝑏𝑦 = 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): 𝑒𝑛𝑏𝑦 ≤ 𝑜 − 𝑙 + 1

– Optimal codes exist if there are ≥ 𝑒𝑛𝑏𝑦 − 1 indices 𝑘’s where 𝐷

𝑘 = 𝑙

– Optimal systematic codes always exists (smaller bound 𝑒𝑡𝑧𝑡 ≤ 𝑒𝑛𝑏𝑦)

Common Technique: Reed-Solomon (sub-) code

Coding with Constraints: Review

Common Technique: Reed-Solomon (sub-) code

𝑯 = ? ? ? ? ? ? ? ? ?

𝑦1 𝑦2

𝑑1 𝑑2

𝑦3

𝑑3 𝑑4 𝑑5 = 𝑑1(𝑦1, 𝑦2) = 𝑑2(𝑦1, 𝑦3) = 𝑑3(𝑦2) = 𝑑4(𝑦2, 𝑦3) = 𝑑5(𝑦1, 𝑦3)

Coding with Constraints: Review

Common Technique: Reed-Solomon (sub-) code

𝛽1 𝛽2 𝛽3 𝛽4 𝛽5 𝑯 = ? ? ? ? ? ? ? ? ?

𝑦1 𝑦2

𝑑1 𝑑2

𝑦3

𝑑3 𝑑4 𝑑5 = 𝑑1(𝑦1, 𝑦2) = 𝑑2(𝑦1, 𝑦3) = 𝑑3(𝑦2) = 𝑑4(𝑦2, 𝑦3) = 𝑑5(𝑦1, 𝑦3)

Coding with Constraints: Review

Common Technique: Reed-Solomon (sub-) code

𝛽1 𝛽2 𝛽3 𝛽4 𝛽5 𝑯 = ? ? ? ? ? ? ? ? ? = 𝑔

1(𝛽1)

𝑔

1(𝛽2)

𝑔

1(𝛽5)

𝑔

2(𝛽1)

𝑔

2(𝛽3)

𝑔

2(𝛽4)

𝑔

3(𝛽2)

𝑔

3(𝛽4)

𝑔

3(𝛽5) 𝑦1 𝑦2

𝑑1 𝑑2

𝑦3

𝑑3 𝑑4 𝑑5 = 𝑑1(𝑦1, 𝑦2) = 𝑑2(𝑦1, 𝑦3) = 𝑑3(𝑦2) = 𝑑4(𝑦2, 𝑦3) = 𝑑5(𝑦1, 𝑦3)

Coding with Constraints: Review

Common Technique: Reed-Solomon (sub-) code

𝛽1 𝛽2 𝛽3 𝛽4 𝛽5 𝑯 = ? ? ? ? ? ? ? ? ? = 𝑔

1(𝛽1)

𝑔

1(𝛽2)

𝑔

1(𝛽5)

𝑔

2(𝛽1)

𝑔

2(𝛽3)

𝑔

2(𝛽4)

𝑔

3(𝛽2)

𝑔

3(𝛽4)

𝑔

3(𝛽5)

Difficulty: G may not be full rank

𝑦1 𝑦2

𝑑1 𝑑2

𝑦3

𝑑3 𝑑4 𝑑5 = 𝑑1(𝑦1, 𝑦2) = 𝑑2(𝑦1, 𝑦3) = 𝑑3(𝑦2) = 𝑑4(𝑦2, 𝑦3) = 𝑑5(𝑦1, 𝑦3)

Part II: Joint Design of Different MDS Codes

(joint work with H. Kiah, W. Song, and C. Yuen)

MDS Codes for Distributed Storage

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𝑦1 𝑦2 𝑦𝑙

𝑑1 𝑑2 𝑑𝑜

Encoded

𝑙 data symbols 𝑜 coded symbols Facebook: Reed-Solomon 𝑜 = 14, 𝑙 = 10 MDS: tolerate any 𝑜 − 𝑙 node failures

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

• verall failure protection?

Question of Interest

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• If two (or more) independent DSS share some common data, can

we jointly design the corresponding MDS codes to get a better

• verall failure protection?

𝑦1 4,3 MDS Local Subcode 1 Local distance 𝑒1 = 2 𝑑2 𝑑3 𝑑4 𝑦2 𝑦3 = 𝑦1 + 𝑦2 + 𝑦3 = 𝑦1 + 2𝑦2 + 4𝑦3 = 𝑦1 + 3𝑦2 + 2𝑦3 = 𝑦1 + 4𝑦2 + 2𝑦3 𝑑1

Tolerate 1 node failure Code has minimum distance 𝑒: tolerate 𝑒 − 1 node failures

Question of Interest

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• If two (or more) independent DSS share some common data, can

we jointly design the corresponding MDS codes to get a better

• verall failure protection?

𝑦1 4,3 MDS Local Subcode 1 Local distance 𝑒1 = 2 𝑑2 𝑑3 𝑑4 𝑦2 𝑦3 𝑦2 Local Subcode 2 Local distance 𝑒2 = 3 𝑑6 𝑑7 𝑑8 𝑦3 𝑦4 𝑦5 𝑑9 𝑑10 = 𝑦1 + 𝑦2 + 𝑦3 = 𝑦1 + 2𝑦2 + 4𝑦3 = 𝑦1 + 3𝑦2 + 2𝑦3 = 𝑦1 + 4𝑦2 + 2𝑦3 6,4 MDS = 𝑦2 + 𝑦3 + 𝑦4 + 𝑦5 = 𝑦2 + 2𝑦3 + 4𝑦4 + 𝑦5 = 𝑦2 + 3𝑦3 + 2𝑦4 + 6𝑦5 = 𝑦2 + 4𝑦3 + 2𝑦4 + 𝑦5 = 𝑦2 + 5𝑦3 + 4𝑦4 + 6𝑦5 = 𝑦2 + 6𝑦3 + 𝑦4 + 6𝑦5 𝑑1 𝑑5

Tolerate 1 node failure Tolerate 2 node failures Code has minimum distance 𝑒: tolerate 𝑒 − 1 node failures

Question of Interest

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• If two (or more) independent DSS share some common data, can

we jointly design the corresponding MDS codes to get a better

• verall failure protection?

𝑦1 4,3 MDS Local Subcode 1 Local distance 𝑒1 = 2

Global minimum distance 𝑒 = 4

𝑑2 𝑑3 𝑑4 𝑦2 𝑦3 𝑦2 Local Subcode 2 Local distance 𝑒2 = 3 𝑑6 𝑑7 𝑑8 𝑦3 𝑦4 𝑦5 𝑑9 𝑑10 = 𝑦1 + 𝑦2 + 𝑦3 = 𝑦1 + 2𝑦2 + 4𝑦3 = 𝑦1 + 3𝑦2 + 2𝑦3 = 𝑦1 + 4𝑦2 + 2𝑦3 6,4 MDS = 𝑦2 + 𝑦3 + 𝑦4 + 𝑦5 = 𝑦2 + 2𝑦3 + 4𝑦4 + 𝑦5 = 𝑦2 + 3𝑦3 + 2𝑦4 + 6𝑦5 = 𝑦2 + 4𝑦3 + 2𝑦4 + 𝑦5 = 𝑦2 + 5𝑦3 + 4𝑦4 + 6𝑦5 = 𝑦2 + 6𝑦3 + 𝑦4 + 6𝑦5 𝑑1 𝑑5

Tolerate 1 node failure Tolerate 2 node failures Tolerate 3 node failures Code has minimum distance 𝑒: tolerate 𝑒 − 1 node failures

Upper Bound for Global Minimum Distance

For linear code 𝑑1, 𝑑2, … , 𝑑10 = 𝑦1, 𝑦2, … , 𝑦6 𝑯 where 𝑯 = ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

Upper Bound for Global Minimum Distance

For linear code 𝑑1, 𝑑2, … , 𝑑10 = 𝑦1, 𝑦2, … , 𝑦6 𝑯 where 𝑯 = ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Goal: replace “?”-entries with 𝐺

𝑟-elements so that

Upper Bound for Global Minimum Distance

For linear code 𝑑1, 𝑑2, … , 𝑑10 = 𝑦1, 𝑦2, … , 𝑦6 𝑯 where 𝑯 = ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Goal: replace “?”-entries with 𝐺

𝑟-elements so that

– two local subcodes are MDS (additional requirement) [4,3]-MDS [6,4]-MDS

Upper Bound for Global Minimum Distance

For linear code 𝑑1, 𝑑2, … , 𝑑10 = 𝑦1, 𝑦2, … , 𝑦6 𝑯 where 𝑯 = ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Goal: replace “?”-entries with 𝐺

𝑟-elements so that

– two local subcodes are MDS (additional requirement) – the global code has optimal distance (same as coding with constraints) [4,3]-MDS [6,4]-MDS

Upper Bound for Global Minimum Distance

For linear code 𝑑1, 𝑑2, … , 𝑑10 = 𝑦1, 𝑦2, … , 𝑦6 𝑯 where 𝑯 = ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Goal: replace “?”-entries with 𝐺

𝑟-elements so that

– two local subcodes are MDS (additional requirement) – the global code has optimal distance (same as coding with constraints)

Same cut-set bound apply & codes over large fields achieve this bound 𝑒 ≤ 𝑒𝑛𝑏𝑦 = 1 + min

∅≠𝐽⊆ 1,…,𝑙 ( ∪𝑗∈𝐽 𝑆𝑗 − |𝐽|)

[4,3]-MDS [6,4]-MDS

Upper Bound for Global Minimum Distance: Two Local Subcodes

• 𝑒 ≤ 𝑒𝑛𝑏𝑦 = 1 + 𝑢 + min 𝑜1 − 𝑙1, 𝑜2 − 𝑙2
• 𝑢 = #{common 𝑦𝑗}

Upper Bound for Global Minimum Distance: Two Local Subcodes

36 𝑦1 4,3 MDS Local Subcode 1 Local distance 𝑒1 = 2

Global minimum distance 𝑒 = 4

𝑑2 𝑑3 𝑑4 𝑦2 𝑦3 𝑦2 Local Subcode 2 Local distance 𝑒2 = 3 𝑑6 𝑑7 𝑑8 𝑦3 𝑦4 𝑦5 𝑑9 𝑑10 = 𝑦1 + 𝑦2 + 𝑦3 = 𝑦1 + 2𝑦2 + 4𝑦3 = 𝑦1 + 3𝑦2 + 2𝑦3 = 𝑦1 + 4𝑦2 + 2𝑦3 6,4 MDS = 𝑦2 + 𝑦3 + 𝑦4 + 𝑦5 = 𝑦2 + 2𝑦3 + 4𝑦4 + 𝑦5 = 𝑦2 + 3𝑦3 + 2𝑦4 + 6𝑦5 = 𝑦2 + 4𝑦3 + 2𝑦4 + 𝑦5 = 𝑦2 + 5𝑦3 + 4𝑦4 + 6𝑦5 = 𝑦2 + 6𝑦3 + 𝑦4 + 6𝑦5 𝑑1 𝑑5

In this example: 𝑒 ≤ 1 + 2 + min 4 − 3, 6 − 4 = 4 -> optimal code here 𝑜1 = 4, 𝑙1= 3 𝑜2 = 6, 𝑙2= 4 𝑢 = 𝑦2, 𝑦3 = 2

• 𝑒 ≤ 𝑒𝑛𝑏𝑦 = 1 + 𝑢 + min 𝑜1 − 𝑙1, 𝑜2 − 𝑙2
• 𝑢 = #{common 𝑦𝑗}

Generator Matrix Representation

37

1st code: 𝑯𝟐 = 1 1 1 1 1 2 3 4 1 4 2 2 =

𝑽 𝑩

𝑦1 4,3 MDS Local Subcode 1 Local distance 𝑒1 = 2 𝑑2 𝑑3 𝑑4 𝑦2 𝑦3 = 𝑦1 + 𝑦2 + 𝑦3 = 𝑦1 + 2𝑦2 + 4𝑦3 = 𝑦1 + 3𝑦2 + 2𝑦3 = 𝑦1 + 4𝑦2 + 2𝑦3 𝑑1

Generator Matrix Representation

38

1st code: 𝑯𝟐 = 1 1 1 1 1 2 3 4 1 4 2 3 =

𝑽 𝑩

2nd code: 𝑯𝟑 = 1 1 1 1 1 1 1 2 3 4 5 6 1 4 2 2 4 1 1 1 6 1 6 6 =

𝑪 𝑾

𝑦1 4,3 MDS Local Subcode 1 Local distance 𝑒1 = 2 𝑑2 𝑑3 𝑑4 𝑦2 𝑦3 𝑦2 Local Subcode 2 Local distance 𝑒2 = 3 𝑑6 𝑑7 𝑑8 𝑦3 𝑦4 𝑦5 𝑑9 𝑑10 = 𝑦1 + 𝑦2 + 𝑦3 = 𝑦1 + 2𝑦2 + 4𝑦3 = 𝑦1 + 3𝑦2 + 2𝑦3 = 𝑦1 + 4𝑦2 + 2𝑦3 6,4 MDS = 𝑦2 + 𝑦3 + 𝑦4 + 𝑦5 = 𝑦2 + 2𝑦3 + 4𝑦4 + 𝑦5 = 𝑦2 + 3𝑦3 + 2𝑦4 + 6𝑦5 = 𝑦2 + 4𝑦3 + 2𝑦4 + 𝑦5 = 𝑦2 + 5𝑦3 + 4𝑦4 + 6𝑦5 = 𝑦2 + 6𝑦3 + 𝑦4 + 6𝑦5 𝑑1 𝑑5

Generator Matrix Representation

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1st code: 𝑯𝟐 = 1 1 1 1 1 2 3 4 1 4 2 3 =

𝑽 𝑩

2nd code: 𝑯𝟑 = 1 1 1 1 1 1 1 2 3 4 5 6 1 4 2 2 4 1 1 1 6 1 6 6 =

𝑪 𝑾

𝑦1 4,3 MDS Local Subcode 1 Local distance 𝑒1 = 2

Global minimum distance 𝑒 = 4

𝑑2 𝑑3 𝑑4 𝑦2 𝑦3 𝑦2 Local Subcode 2 Local distance 𝑒2 = 3 𝑑6 𝑑7 𝑑8 𝑦3 𝑦4 𝑦5 𝑑9 𝑑10 = 𝑦1 + 𝑦2 + 𝑦3 = 𝑦1 + 2𝑦2 + 4𝑦3 = 𝑦1 + 3𝑦2 + 2𝑦3 = 𝑦1 + 4𝑦2 + 2𝑦3 6,4 MDS = 𝑦2 + 𝑦3 + 𝑦4 + 𝑦5 = 𝑦2 + 2𝑦3 + 4𝑦4 + 𝑦5 = 𝑦2 + 3𝑦3 + 2𝑦4 + 6𝑦5 = 𝑦2 + 4𝑦3 + 2𝑦4 + 𝑦5 = 𝑦2 + 5𝑦3 + 4𝑦4 + 6𝑦5 = 𝑦2 + 6𝑦3 + 𝑦4 + 6𝑦5 𝑑1 𝑑5

Global code: 𝑯 = 𝑽 𝑷 𝑩 𝑪 𝑷 𝑾

Easy Case: Two Codes Have Few Common Data

• If few common data, i.e.

𝑢 ≤ 1 + max 𝑜2 − 𝑙2, 𝑜1 − 𝑙1 using two Vandermonde matrices as 𝐻1, 𝐻2 : optimal minimum distance

• Finite field size required: |𝐺

𝑟| ≥ max{𝑜1, 𝑜2}

Easy Case: Two Codes Have Few Common Data

• If few common data, i.e.

𝑢 ≤ 1 + max 𝑜2 − 𝑙2, 𝑜1 − 𝑙1 using two Vandermonde matrices as 𝐻1, 𝐻2 : optimal minimum distance

• Finite field size required: |𝐺

𝑟| ≥ max{𝑜1, 𝑜2}

More specifically, in this case, if

• 𝑯𝟐 =

𝑽 𝑩 is a nested MDS code: 𝑯𝟐, 𝑽 are generator matrices of

MDS codes

• 𝑯𝟑 =

𝑪 𝑾 is a nested MDS code: 𝑯𝟑, 𝑾 are generator matrices of

MDS codes then the global code achieves the optimal minimum distance (attains the upper bound)

Harder Case: Two Codes Have the Same Redundancy

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• If same redundancy, i.e.

𝑜1 − 𝑙1 = 𝑜2 − 𝑙2 we construct codes that have optimal global minimum distance

• Finite field size required: 𝑟 > 𝑜 = 𝑜1 + 𝑜2

The construction uses the BCH bound

Two Codes Have the Same Redundancy: BCH Bound

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• 𝐺

𝑟: finite field of 𝑟 elements

• 𝜕: primitive element of 𝐺

𝑟, i.e. 𝐺 𝑟 = {0,1, 𝜕, 𝜕2, 𝜕3, … }

• Identify a vector 𝑑 = 𝑑1, … , 𝑑𝑜 ∈ 𝐺

𝑟 𝑜 with the polynomial

𝑑 x = 𝑑1 + 𝑑2x + 𝑑3x2 + ⋯ + 𝑑𝑜x𝑜−1 BCH Bound: If every coded vector 𝒅 satisfies 𝑑 𝜕𝑗 = 0, for every 𝑗 = 0,1, … , 𝜀 − 1 i.e, they all have 𝜀 consecutive powers of 𝜕 as roots, then the code has minimum distance 𝑒 ≥ 𝜀 + 1

Two Codes Have the Same Redundancy: Construction

We construct the generator matrix of the optimal code. Example: 𝐺

13

𝑯 = 𝑽 𝑷 𝑩 𝑪 𝑷 𝑾 =

Two Codes Have the Same Redundancy: Construction

We construct the generator matrix of the optimal code. Example: 𝐺

13

𝑯 = 𝑽 𝑷 𝑩 𝑪 𝑷 𝑾 =

5 1 6 1 5 1 6 1 5 11 10 9 4 12 1 7 5 1 5 1 6 1 5 1 6 1 5 1 6 1 𝑜1 = 5 𝑜1 = 6 𝑙2 = 5 𝑙1 = 4

Two Codes Have the Same Redundancy: Construction

We construct the generator matrix of the optimal code. Example: 𝐺

13

𝑯 = 𝑽 𝑷 𝑩 𝑪 𝑷 𝑾 =

5 1 6 1 5 1 6 1 5 11 10 9 4 12 1 7 5 1 5 1 6 1 5 1 6 1 5 1 6 1 Rows of 𝐻1 has root 1 = 𝜕0

• > distance 2=1+1

𝑜1 = 5 𝑜1 = 6 𝑙2 = 5 𝑙1 = 4

Two Codes Have the Same Redundancy: Construction

We construct the generator matrix of the optimal code. Example: 𝐺

13

𝑯 = 𝑽 𝑷 𝑩 𝑪 𝑷 𝑾 =

5 1 6 1 5 1 6 1 5 11 10 9 4 12 1 7 5 1 5 1 6 1 5 1 6 1 5 1 6 1 Rows of 𝐻1 has root 1 = 𝜕0

• > distance 2=1+1

Rows of 𝐻2 has root 1 = 𝜕0

• > distance 2=1+1

𝑜1 = 5 𝑜1 = 6 𝑙2 = 5 𝑙1 = 4

Two Codes Have the Same Redundancy: Construction

We construct the generator matrix of the optimal code. Example: 𝐺

13

𝑯 = 𝑽 𝑷 𝑩 𝑪 𝑷 𝑾 =

5 1 6 1 5 1 6 1 5 11 10 9 4 12 1 7 5 1 5 1 6 1 5 1 6 1 5 1 6 1 Rows of 𝐻1 has root 1 = 𝜕0

• > distance 2=1+1

Rows of 𝐻2 has root 1 = 𝜕0

• > distance 2=1+1

Rows of 𝑯 has roots 𝜕0, 𝜕, 𝜕2

• > distance 4=3+1

𝑜1 = 5 𝑜1 = 6 𝑙2 = 5 𝑙1 = 4

Two Codes Have the Same Redundancy: Construction

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Summary of this construction

• Rows: treated as polynomial having certain roots
• Solving systems of linear equations to determine rows
• BCH bound → global code & local codes have desired distances

5 1 6 1 5 1 6 1 5 11 10 9 4 12 1 7 5 1 5 1 6 1 5 1 6 1 5 1 6 1

Conclusions

What we have done

• Introduce a new coding problem: how to jointly design 2 (or more)

MDS codes to have better overall failure tolerance

• Construct optimal codes for two cases

– There are few common data – Two codes have the same amount of redundancy Open Questions – Codes over small field size for 2 local codes: 𝑜1 − 𝑙1 ≠ 𝑜2 − 𝑙2 – Codes over small field size for more than two local codes