Coding with Constraints: Different Flavors

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

2 Part I Coding with Constraints: A Quick Survey

3 Coding with Constraints: Definition c 1 x 1 c 2 k data symbols x 2 ENCODED n coded symbols x k c n CONVENTIONAL CODE c j = c j (x 1, x 2,, x k ) 3

4 Coding with Constraints: Definition c 1 x 1 c 2 k data symbols x 2 ENCODED n coded symbols x k c n CONVENTIONAL CODE c j = c j (x 1, x 2,, x k ) CODE WITH CONSTRAINTS c j = c j ({x i : i C j }), C j {1,2,, k} 4

5 Coding with Constraints: Example x 1 c 1 c 2 = c 1 (x 1, x 2 ) = c 2 (x 1, x 3 ) x 2 c 3 = c 3 (x 2 ) x 3 c 4 c 5 = c 4 (x 2, x 3 ) = c 5 (x 1, x 3 ) 5

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

7 Coding with Constraints: Main Problem Given the constraints c j = c j ({x i : i C j }), C j {1,2,, k} how to construct codes that achieve the optimal minimum distance over small field size q poly n

8 Coding with Constraints: Main Problem Given the constraints c j = c j ({x i : i C j }), C j {1,2,, k} how to construct codes that achieve the optimal minimum distance over small field size q poly n Linear case: given G =?? 0 0?? 0?? 0 0? 0?? how to replace? -entries by elements of F q (q poly(n)) so that G generate a code with optimal distance

9 Coding with Constraints: Upper Bound Upper Bound (Halbawi-Thill-Hassibi 15, Song-Dau-Yuen 15) d d max = 1 + min ( i I R i I ) I 1,,k where R i = {j: i C j }

10 Coding with Constraints: Upper Bound Upper Bound (Halbawi-Thill-Hassibi 15, Song-Dau-Yuen 15) d d max = 1 + min ( i I R i I ) I 1,,k where R i = {j: i C j } Properties d max can be found in time poly(n) codes with d = d max always exists over fields of size n d 1

11 Coding with Constraints: Upper Bound Upper Bound (Halbawi-Thill-Hassibi 15, Song-Dau-Yuen 15) d d max = 1 + min ( i I R i I ) I 1,,k where R i = {j: i C j } Properties d max can be found in time poly(n) codes with d = d max always exists over fields of size n d 1 Question of interest: how about fields of size poly(n)?

12 Coding with Constraints: Review (Small field) MDS Case: d max = n k + 1 Optimal codes exist in a few special cases (Halbawi-Ho-Yao- Duursma 14, Dau-Song-Yuen 14, Yan-Sprintson-Zelenko 14)

13 Coding with Constraints: Review (Small field) MDS Case: d max = n k + 1 Optimal codes exist in a few special cases (Halbawi-Ho-Yao- Duursma 14, Dau-Song-Yuen 14, Yan-Sprintson-Zelenko 14) k 4 (every n)

14 Coding with Constraints: Review (Small field) MDS Case: d max = n k + 1 Optimal codes exist in a few special cases (Halbawi-Ho-Yao- Duursma 14, Dau-Song-Yuen 14, Yan-Sprintson-Zelenko 14) k 4 (every n) rows of G partitioned into 3 groups, each has same? -pattern

15 Coding with Constraints: Review (Small field) MDS Case: d max = n k + 1 Optimal codes exist in a few special cases (Halbawi-Ho-Yao- Duursma 14, Dau-Song-Yuen 14, Yan-Sprintson-Zelenko 14) k 4 (every n) rows of G partitioned into 3 groups, each has same? -pattern rows have k 1 zeros & 2 different rows share 1 common zeros

16 Coding with Constraints: Review (Small field) MDS Case: d max = n k + 1 Optimal codes exist in a few special cases (Halbawi-Ho-Yao- Duursma 14, Dau-Song-Yuen 14, Yan-Sprintson-Zelenko 14) k 4 (every n) rows of G partitioned into 3 groups, each has same? -pattern rows have k 1 zeros & 2 different rows share 1 common zeros General Case (Halbawi-Thill-Hassibi 15): d max n k + 1

17 Coding with Constraints: Review (Small field) MDS Case: d max = n k + 1 Optimal codes exist in a few special cases (Halbawi-Ho-Yao- Duursma 14, Dau-Song-Yuen 14, Yan-Sprintson-Zelenko 14) k 4 (every n) rows of G partitioned into 3 groups, each has same? -pattern rows have k 1 zeros & 2 different rows share 1 common zeros General Case (Halbawi-Thill-Hassibi 15): d max n k + 1 Optimal codes exist if there are d max 1 indices j s where C j = k

18 Coding with Constraints: Review (Small field) MDS Case: d max = n k + 1 Optimal codes exist in a few special cases (Halbawi-Ho-Yao- Duursma 14, Dau-Song-Yuen 14, Yan-Sprintson-Zelenko 14) k 4 (every n) rows of G partitioned into 3 groups, each has same? -pattern rows have k 1 zeros & 2 different rows share 1 common zeros General Case (Halbawi-Thill-Hassibi 15): d max n k + 1 Optimal codes exist if there are d max 1 indices j s where C j = k Optimal systematic codes always exists (smaller bound d sys d max )

19 Coding with Constraints: Review (Small field) MDS Case: d max = n k + 1 Optimal codes exist in a few special cases (Halbawi-Ho-Yao- Duursma 14, Dau-Song-Yuen 14, Yan-Sprintson-Zelenko 14) k 4 (every n) rows of G partitioned into 3 groups, each has same? -pattern rows have k 1 zeros & 2 different rows share 1 common zeros General Case (Halbawi-Thill-Hassibi 15): d max n k + 1 Optimal codes exist if there are d max 1 indices j s where C j = k Optimal systematic codes always exists (smaller bound d sys d max ) Common Technique: Reed-Solomon (sub-) code

20 Coding with Constraints: Review Common Technique: Reed-Solomon (sub-) code x 1 c 1 c 2 = c 1 (x 1, x 2 ) = c 2 (x 1, x 3 ) x 2 c 3 = c 3 (x 2 ) x 3 c 4 c 5 = c 4 (x 2, x 3 ) = c 5 (x 1, x 3 ) G =?? 0 0?? 0?? 0 0? 0??

21 Coding with Constraints: Review Common Technique: Reed-Solomon (sub-) code x 1 c 1 c 2 = c 1 (x 1, x 2 ) = c 2 (x 1, x 3 ) x 2 c 3 = c 3 (x 2 ) x 3 c 4 c 5 = c 4 (x 2, x 3 ) = c 5 (x 1, x 3 ) G = α 1 α 2 α 3 α 4 α 5?? 0 0?? 0?? 0 0? 0??

22 Coding with Constraints: Review Common Technique: Reed-Solomon (sub-) code x 1 c 1 c 2 = c 1 (x 1, x 2 ) = c 2 (x 1, x 3 ) x 2 c 3 = c 3 (x 2 ) x 3 c 4 c 5 = c 4 (x 2, x 3 ) = c 5 (x 1, x 3 ) G = α 1 α 2 α 3 α 4 α 5?? 0 0?? 0?? 0 0? 0?? = f 1 (α 1 ) f 1 (α 2 ) 0 0 f 1 (α 5 ) f 2 (α 1 ) 0 f 2 (α 3 ) f 2 (α 4 ) 0 0 f 3 (α 2 ) 0 f 3 (α 4 ) f 3 (α 5 )

23 Coding with Constraints: Review Common Technique: Reed-Solomon (sub-) code x 1 c 1 c 2 = c 1 (x 1, x 2 ) = c 2 (x 1, x 3 ) x 2 c 3 = c 3 (x 2 ) x 3 c 4 c 5 = c 4 (x 2, x 3 ) = c 5 (x 1, x 3 ) G = α 1 α 2 α 3 α 4 α 5?? 0 0?? 0?? 0 0? 0?? = f 1 (α 1 ) f 1 (α 2 ) 0 0 f 1 (α 5 ) f 2 (α 1 ) 0 f 2 (α 3 ) f 2 (α 4 ) 0 0 f 3 (α 2 ) 0 f 3 (α 4 ) f 3 (α 5 ) Difficulty: G may not be full rank

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

25 MDS Codes for Distributed Storage x 1 c 1 c 2 MDS: tolerate any n k node failures k data symbols x 2 Encoded n coded symbols x k c n Facebook: Reed-Solomon n = 14, k = 10 25

26 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?

27 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? Local Subcode 1 Tolerate 1 node failure x 1 x 2 4,3 MDS c 1 c 2 c 3 = x 1 + x 2 + x 3 = x 1 + 2x 2 + 4x 3 = x 1 + 3x 2 + 2x 3 x 3 Local distance d 1 = 2 c 4 = x 1 + 4x 2 + 2x 3 Code has minimum distance d: tolerate d 1 node failures 27

28 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? Local Subcode 1 Tolerate 1 node failure x 1 x 2 4,3 MDS c 1 c 2 c 3 = x 1 + x 2 + x 3 = x 1 + 2x 2 + 4x 3 = x 1 + 3x 2 + 2x 3 x 3 Local distance d 1 = 2 c 4 = x 1 + 4x 2 + 2x 3 Local Subcode 2 c 5 = x 2 + x 3 + x 4 + x 5 x 2 c 6 = x 2 + 2x 3 + 4x 4 + x 5 Tolerate 2 node failures x 3 x 4 x 5 6,4 MDS Local distance d 2 = 3 c 7 c 8 c 9 c 10 = x 2 + 3x 3 + 2x 4 + 6x 5 = x 2 + 4x 3 + 2x 4 + x 5 = x 2 + 5x 3 + 4x 4 + 6x 5 = x 2 + 6x 3 + x 4 + 6x 5 Code has minimum distance d: tolerate d 1 node failures 28

29 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? Local Subcode 1 Tolerate 3 node failures Tolerate 1 node failure Tolerate 2 node failures x 1 x 2 x 3 4,3 MDS Local distance d 1 = 2 Local Subcode 2 x 2 x 3 x 4 x 5 6,4 MDS Local distance d 2 = 3 c 1 c 2 c 3 c 4 c 5 c 6 c 7 c 8 c 9 c 10 = x 1 + x 2 + x 3 = x 1 + 2x 2 + 4x 3 = x 1 + 3x 2 + 2x 3 = x 1 + 4x 2 + 2x 3 = x 2 + x 3 + x 4 + x 5 = x 2 + 2x 3 + 4x 4 + x 5 = x 2 + 3x 3 + 2x 4 + 6x 5 = x 2 + 4x 3 + 2x 4 + x 5 = x 2 + 5x 3 + 4x 4 + 6x 5 = x 2 + 6x 3 + x 4 + 6x 5 Global minimum distance d = 4 Code has minimum distance d: tolerate d 1 node failures 29

30 Upper Bound for Global Minimum Distance For linear code where G = c 1, c 2,, c 10 = x 1, x 2,, x 6 G???? ???????????????????? ?????? ??????

31 Upper Bound for Global Minimum Distance For linear code where G = c 1, c 2,, c 10 = x 1, x 2,, x 6 G???? ???????????????????? ?????? ?????? Goal: replace? -entries with F q -elements so that

32 Upper Bound for Global Minimum Distance For linear code where [4,3]-MDS G = c 1, c 2,, c 10 = x 1, x 2,, x 6 G???? ???????????????????? ?????? ?????? [6,4]-MDS Goal: replace? -entries with F q -elements so that two local subcodes are MDS (additional requirement)

33 Upper Bound for Global Minimum Distance For linear code where [4,3]-MDS G = c 1, c 2,, c 10 = x 1, x 2,, x 6 G???? ???????????????????? ?????? ?????? [6,4]-MDS Goal: replace? -entries with F q -elements so that two local subcodes are MDS (additional requirement) the global code has optimal distance (same as coding with constraints)

34 Upper Bound for Global Minimum Distance For linear code where [4,3]-MDS G = c 1, c 2,, c 10 = x 1, x 2,, x 6 G???? ???????????????????? ?????? ?????? [6,4]-MDS Goal: replace? -entries with F q -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 d d max = 1 + min ( i I R i I ) I 1,,k

35 Upper Bound for Global Minimum Distance: Two Local Subcodes d d max = 1 + t + min n 1 k 1, n 2 k 2 t = #{common x i }

36 Upper Bound for Global Minimum Distance: Two Local Subcodes d d max = 1 + t + min n 1 k 1, n 2 k 2 t = #{common x i } Local Subcode 1 x 1 c 1 = x 1 + x 2 + x 3 n 1 = 4, k 1 = 3 t = x 2, x 3 = 2 n 2 = 6, k 2 = 4 x 2 x 3 4,3 MDS Local distance d 1 = 2 Local Subcode 2 x 2 x 3 x 4 x 5 6,4 MDS Local distance d 2 = 3 c 2 c 3 c 4 c 5 c 6 c 7 c 8 c 9 c 10 = x 1 + 2x 2 + 4x 3 = x 1 + 3x 2 + 2x 3 = x 1 + 4x 2 + 2x 3 = x 2 + x 3 + x 4 + x 5 = x 2 + 2x 3 + 4x 4 + x 5 = x 2 + 3x 3 + 2x 4 + 6x 5 = x 2 + 4x 3 + 2x 4 + x 5 = x 2 + 5x 3 + 4x 4 + 6x 5 = x 2 + 6x 3 + x 4 + 6x 5 Global minimum distance d = 4 In this example: d min 4 3, 6 4 = 4 -> optimal code here 36

37 Generator Matrix Representation 1 st code: G 1 = = U A Local Subcode 1 x 1 x 2 4,3 MDS c 1 c 2 c 3 = x 1 + x 2 + x 3 = x 1 + 2x 2 + 4x 3 = x 1 + 3x 2 + 2x 3 x 3 Local distance d 1 = 2 c 4 = x 1 + 4x 2 + 2x 3 37

38 Generator Matrix Representation 1 st code: G 1 = 2 nd code: G 2 = = U A = B V Local Subcode 1 x 1 x 2 x 3 4,3 MDS Local distance d 1 = 2 Local Subcode 2 x 2 c 1 c 2 c 3 c 4 c 5 c 6 = x 1 + x 2 + x 3 = x 1 + 2x 2 + 4x 3 = x 1 + 3x 2 + 2x 3 = x 1 + 4x 2 + 2x 3 = x 2 + x 3 + x 4 + x 5 = x 2 + 2x 3 + 4x 4 + x 5 x 3 x 4 6,4 MDS c 7 c 8 = x 2 + 3x 3 + 2x 4 + 6x 5 = x 2 + 4x 3 + 2x 4 + x 5 x 5 Local distance d 2 = 3 c 9 c 10 = x 2 + 5x 3 + 4x 4 + 6x 5 = x 2 + 6x 3 + x 4 + 6x 5 38

39 Generator Matrix Representation 1 st code: G 1 = 2 nd code: G 2 = Global code: G = = U A U A O O B V = B V Local Subcode 1 x 1 x 2 x 3 4,3 MDS Local distance d 1 = 2 Local Subcode 2 x 2 x 3 x 4 x 5 6,4 MDS Local distance d 2 = 3 c 1 c 2 c 3 c 4 c 5 c 6 c 7 c 8 c 9 c 10 = x 1 + x 2 + x 3 = x 1 + 2x 2 + 4x 3 = x 1 + 3x 2 + 2x 3 = x 1 + 4x 2 + 2x 3 = x 2 + x 3 + x 4 + x 5 = x 2 + 2x 3 + 4x 4 + x 5 = x 2 + 3x 3 + 2x 4 + 6x 5 = x 2 + 4x 3 + 2x 4 + x 5 = x 2 + 5x 3 + 4x 4 + 6x 5 = x 2 + 6x 3 + x 4 + 6x 5 Global minimum distance d = 4 39

40 Easy Case: Two Codes Have Few Common Data - If few common data, i.e. t 1 + max n 2 k 2, n 1 k 1 using two Vandermonde matrices as G 1, G 2 : optimal minimum distance - Finite field size required: F q max{n 1, n 2 }

41 Easy Case: Two Codes Have Few Common Data - If few common data, i.e. t 1 + max n 2 k 2, n 1 k 1 using two Vandermonde matrices as G 1, G 2 : optimal minimum distance - Finite field size required: F q max{n 1, n 2 } More specifically, in this case, if G 1 = U A is a nested MDS code: G 1, U are generator matrices of MDS codes G 2 = B V is a nested MDS code: G 2, V are generator matrices of MDS codes then the global code achieves the optimal minimum distance (attains the upper bound)

42 Harder Case: Two Codes Have the Same Redundancy - If same redundancy, i.e. n 1 k 1 = n 2 k 2 we construct codes that have optimal global minimum distance - Finite field size required: q > n = n 1 + n 2 The construction uses the BCH bound 42

43 Two Codes Have the Same Redundancy: BCH Bound F q : finite field of q elements ω: primitive element of F q, i.e. F q = {0,1, ω, ω 2, ω 3, } Identify a vector c = c 1,, c n Fn q with the polynomial c x = c 1 + c 2 x + c 3 x c n x n 1 BCH Bound: If every coded vector c satisfies c ω i = 0, for every i = 0,1,, δ 1 i.e, they all have δ consecutive powers of ω as roots, then the code has minimum distance d δ

44 Two Codes Have the Same Redundancy: Construction We construct the generator matrix of the optimal code. Example: F 13 U O G = A B = O V

45 Two Codes Have the Same Redundancy: Construction We construct the generator matrix of the optimal code. Example: F 13 U O G = A B = O V n 1 = 5 k 1 = n 1 = 6 k 2 = 5

46 Two Codes Have the Same Redundancy: Construction We construct the generator matrix of the optimal code. Example: F 13 U O G = A B = O V n 1 = 5 k 1 = Rows of G 1 has root 1 = ω 0 -> distance 2= k 2 = 5 n 1 = 6

47 Two Codes Have the Same Redundancy: Construction We construct the generator matrix of the optimal code. Example: F 13 U O G = A B = O V n 1 = 5 k 1 = Rows of G 1 has root 1 = ω 0 -> distance 2= k 2 = 5 n 1 = 6 Rows of G 2 has root 1 = ω 0 -> distance 2=1+1

48 Two Codes Have the Same Redundancy: Construction We construct the generator matrix of the optimal code. Example: F 13 U O G = A B = O V n 1 = 5 Rows of G has roots ω 0, ω, ω 2 -> distance 4=3+1 k 1 = Rows of G 1 has root 1 = ω 0 -> distance 2= k 2 = 5 n 1 = 6 Rows of G 2 has root 1 = ω 0 -> distance 2=1+1

49 Two Codes Have the Same Redundancy: Construction 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

50 What we have done Conclusions 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: n 1 k 1 n 2 k 2 Codes over small field size for more than two local codes

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