Constructions of Codes with the Locality Property Alexander Barg - PowerPoint PPT Presentation

Constructions of Codes with the Locality Property Alexander Barg University of Maryland DIMACS Workshop “Network Coding: The Next 15 years” A. Barg (UMD) LRC codes 1 / 30

Acknowledgment Based on joint works with Itzhak Tamo Alexey Frolov Serge Vl˘ adut ¸ Sreechakra Goparaju Robert Calderbank A. Barg (UMD) LRC codes 2 / 30

Definitions Locally recoverable codes Table of codewords ������ ������ ������ ������ a The code C ⊂ F n is locally recoverable ������ ������ with locality r if every symbol can be b recovered by accessing some other r symbols in the encoding (recovery set of coordinate i ) i J i A. Barg (UMD) LRC codes 3 / 30

Definitions ( n , k , r ) LRC code Definition (LRC codes) Code C has locality r if for every i ∈ [ n ] there exists a subset J i ⊂ [ n ] \ i , | J i | ≤ r and a function φ i such that for every codeword c ∈ C c i = φ i ( { c j , j ∈ J i } ) J. Han and L. Lastras-Montano, ISIT 2007; C. Huang, M. Chen, and J. Li, Symp. Networks App. 2007; F . Oggier and A. Datta ’10; P . Gopalan, C. Huang, H. Simitci, and S. Yekhanin, IEEE Trans. Inf. Theory, Nov. 2012. Linear index codes are duals of linear DS codes on graphs (Mazumdar ’14; Shanmugam-Dimakis ’14) A. Barg (UMD) LRC codes 4 / 30

Definitions ( n , k , r ) LRC code Definition (LRC codes) Code C has locality r if for every i ∈ [ n ] there exists a subset J i ⊂ [ n ] \ i , | J i | ≤ r and a function φ i such that for every codeword c ∈ C c i = φ i ( { c j , j ∈ J i } ) Examples: Repetition codes, Single parity-check codes [ n , r , n − r + 1 ] RS code Early constructions: Prasanth, Kamath, Lalitha, Kumar , ISIT 2012 Silberstein, Rawat, Koyluoglu Vishwanath , ISIT 2013 Tamo, Papailiopoulos, Dimakis , ISIT 2013 A. Barg (UMD) LRC codes 5 / 30

Outline RS-like LRC codes Bounds on LRC codes LRC codes on curves Cyclic LRC codes A. Barg (UMD) LRC codes 6 / 30

RS-like LRC codes RS codes and Evaluation codes Given a polynomial f ∈ F q [ x ] and a set A = { P 1 , . . . , P n } ⊂ F q define the map ev A : f �→ ( f ( P i ) , i = 1 , . . . , n ) Example: Let q = 8 , f ( x ) = 1 + α x + α x 2 f ( x ) �→ ( 1 , α 4 , α 6 , α 4 , α, α, α 6 ) Evaluation code C ( A ) Let V = { f ∈ F q [ x ] } be a set of polynomials, dim ( V ) = k C : V → F n q f �→ ev A ( f ) = ( f ( P i ) , i = 1 , . . . , n ) A. Barg (UMD) LRC codes 7 / 30

RS-like LRC codes Reed-Solomon codes A. Barg (UMD) LRC codes 8 / 30

RS-like LRC codes Reed-Solomon codes � � � � � � � � � � � � � � A. Barg (UMD) LRC codes 9 / 30

RS-like LRC codes Reed-Solomon codes � � � � � � � � � � � � � � � A. Barg (UMD) LRC codes 10 / 30

RS-like LRC codes Evaluation codes with locality � � � � � � � � � � � � � � � A. Barg (UMD) LRC codes 11 / 30

RS-like LRC codes Construction of ( n , k , r ) LRC codes: Example Parameters: n = 9 , k = 4 , r = 2 , q = 13 ; Set of points: A = { P 1 , . . . , P 9 } ⊂ F 13 A = { A 1 = ( 1 , 3 , 9 ) , A 2 = ( 2 , 6 , 5 ) , A 3 = ( 4 , 12 , 10 ) } Set of functions: P = { f a ( x ) = a 0 + a 1 x + a 3 x 3 + a 4 x 4 } Code construction: ev A : f a �→ ( f ( P i ) , i = 1 , . . . 9 ) E.g., a = ( 1111 ) then f a ( x ) = 1 + x + x 3 + x 4 c := ev A ( f a ) = ( 4 , 8 , 7 | 1 , 11 , 2 | 0 , 0 , 0 ) � �� � � �� � � �� � A 1 A 2 A 3 f a ( x ) | A 1 = a 0 + a 3 + ( a 1 + a 4 ) x = 2 + 2 x f a ( x ) | A 2 = a 0 + 8 a 3 + ( a 1 + 8 a 4 ) x A. Barg (UMD) LRC codes 12 / 30

RS-like LRC codes Construction of ( n , k , r ) LRC codes A = ( P 1 , . . . , P n ) ⊂ F q A = A 1 ∪ A 2 ∪ · · · ∪ A n r + 1 r + 1 (above g ( x ) = x 3 ) n Basis of functions: Take g ( x ) constant on A i , i = 1 , . . . , ⟨ g ( x ) j x i , i = 0 , . . . , r − 1 ; j = 0 , . . . , k ⟩ V = r − 1 ; dim ( V ) = k   k r − 1 r − 1   ∑ ∑ a ij g ( x ) j x i V =  f a ( x ) =  i = 0 j = 0 We obtain a family of optimal r -LRC codes Erasure recovery by polynomial interpolation over r points. I. Tamo and A.B. , IEEE Trans. Inf. Theory, Aug. 2014. A. Barg (UMD) LRC codes 13 / 30

RS-like LRC codes Extensions Codes with multiple disjoint recovery sets for every coordinate Codes that recover locally from ρ ≥ 2 erasures: The local codes are [ r + ρ − 1 , r , ρ ] MDS Systematic encoding A. Barg (UMD) LRC codes 14 / 30

Bounds Finite-length bounds Let C ⊂ F n q be an r -LRC code, |C| = q k , distance d ⌈ k ⌉ d ≤ n − k − + 2 r ( P . Gopalan e.a. 2012) k ≤ min s ≥ 1 { sr + k q ( n − s ( r + 1 ) , d ) } ( V. Cadambe and A. Mazumdar , 2013-15) Bounds for multiple recovery sets (work with I. Tamo, 2014) A. Barg (UMD) LRC codes 15 / 30

Bounds Asymptotic bounds R 0.7 0.6 S i 0.5 n g l Plotkin bound e t o n 0.4 b o u n d 0.3 0.2 LP bound 0.1 GV bound 1.0 ∆ 0.2 0.4 0.6 0.8 Binary codes, r = 3; n → ∞ R q ( r , δ ) > 0 , 0 ≤ δ < ( q − 1 ) / q r q − 1 R q ( r , 0 ) = r + 1 , R q ( r , δ ) = 0 , ≤ δ ≤ 1 q A. Barg (UMD) LRC codes 16 / 30

Codes on curves Geometric view of LRC codes A = { 1 , . . . , 9 } ⊂ F 13 A = A 1 ∪ A 2 ∪ A 3 A 1 = ( 1 , 3 , 9 ) A 2 = ( 2 , 6 , 5 ) A 3 = ( 4 , 12 , 10 ) g : A → F 13 x �→ x 3 − 1 g : F 13 → { 0 , 7 , 8 } ⊂ F 13 | g − 1 ( y ) | = r + 1 A. Barg (UMD) LRC codes 17 / 30

Codes on curves LRC codes on curves Consider the set of pairs ( x , y ) ∈ F 9 that satisfy the equation x 3 + x = y 4 α 7 • • • • α 6 • α 5 • • • • α 4 • • • • x α 3 • • • • α 2 • α • • • • 1 • • • • 0 • 0 1 α α 2 α 3 α 4 α 5 α 6 α 7 y Affine points of the Hermitian curve X over F 9 ; α 2 = α + 1 A. Barg (UMD) LRC codes 18 / 30

Codes on curves Hermitian codes P 1 g : X → ( x , y ) �→ y Space of functions V := ⟨ 1 , y , y 2 , x , xy , xy 2 ⟩ A= { Affine points of the Hermitian curve over F 9 } ; n = 27 , k = 6 C : V → F n 9 E.g., message ( 1 , α, α 2 , α 3 , α 4 , α 5 ) F ( x , y ) = 1 + α y + α 2 y 2 + α 3 x + α 4 xy + α 5 xy 2 F ( 0 , 0 ) = 1 etc. A. Barg (UMD) LRC codes 19 / 30

Codes on curves LRC codes on curves α 7 α 7 α 5 α 0 α 6 α 2 α 5 α 6 α 4 α 2 0 α 4 α 7 α 3 α 5 α 5 x α 3 α 3 α 7 α α α 2 α 3 α 0 0 0 0 α 6 α 4 1 1 0 0 1 α α 2 α 3 α 4 α 5 α 6 α 7 0 1 y A. Barg (UMD) LRC codes 20 / 30

Codes on curves Hermitian LRC codes α 7 α 7 α 5 α 0 α 6 α 2 α 5 α 6 α 4 α 2 0 α 4 α 7 α 3 α 5 α 5 x α 3 α 3 α 7 α α α 2 α 3 X α 0 0 0 0 α 6 α 4 1 1 0 0 1 α α 2 α 3 α 4 α 5 α 6 α 7 0 1 y Let P = ( α, 1 ) be the erased location. A. Barg (UMD) LRC codes 21 / 30

Codes on curves Local recovery with Hermitian codes α 7 α 7 α 5 α 0 α 6 α 2 α 5 α 6 α 4 α 2 0 α 4 α 7 α 3 α 5 α 5 x α 3 α 3 α 7 α α α 2 α 3 α ? 0 0 0 α 6 α 4 1 1 0 0 1 α α 2 α 3 α 4 α 5 α 6 α 7 0 1 y Let P = ( α, 1 ) be the erased location. Recovery set I P = { ( α 4 , 1 ) , ( α 3 , 1 ) } Find f ( x ) : f ( α 4 ) = α 7 , f ( α 3 ) = α 3 f ( x ) = α x − α 2 ⇒ f ( α ) = 0 = F ( P ) A. Barg (UMD) LRC codes 22 / 30

Codes on curves Hermitian codes q = q 2 0 , q 0 prime power X : x q 0 + x = y q 0 + 1 0 = q 3 / 2 points in F q X has q 3 Let g : X → Y = P 1 , g ( P ) = g ( x , y ) := y We obtain a family of q -ary codes of length n = q 3 0 , k = ( t + 1 )( q 0 − 1 ) , d ≥ n − tq 0 − ( q 0 − 2 )( q 0 + 1 ) with locality r = q 0 − 1 . It is also possible to take g ( P ) = x (projection on x ); we obtain LRC codes with locality q 0 A. Barg (UMD) LRC codes 23 / 30

Codes on curves General construction Map of curves X , Y smooth projective absolutely irreducible curves over k g : X → Y rational separable map of degree r + 1 Lift the points of Y S = { P 1 , . . . , P s } ⊂ Y ( k ) . Partition of points: A := g − 1 ( S ) = { P ij , i = 0 , . . . , r , j = 1 , . . . , s } ⊆ X ( k ) such that g ( P ij ) = P j for all i , j Basis of the function space: Q ∞ = π − 1 ( ∞ ) , where π : Y → P 1 k { f 1 , . . . , f m } span L ( tQ ∞ ) , t ≥ 1 { f j x i , i = 0 , . . . , r − 1 ; j = 1 , . . . , m } Construct LRC codes Evaluation codes constructed on the set A are LRC codes with locality r A. Barg (UMD) LRC codes 24 / 30

Asymptotically good codes Asymptotically good sequences of codes Let q = q 2 0 , where q 0 is a prime power. Take Garcia-Stichtenoth towers of curves: x 0 := 1 ; X 1 := P 1 , k ( X 1 ) = k ( x 1 ); l − 1 , x l − 1 := z l − 1 X l : z q 0 + z l = x q 0 + 1 ∈ k ( X l − 1 ) (if l ≥ 3 ) l x l − 2 There exist families of q -ary LRC codes with locality r whose rate and relative dis- tance satisfy r ( 3 ) r = √ q − 1 R ≥ 1 − δ − √ q + 1 , r + 1 1 − δ − 2 √ q ( ) r = √ q r R ≥ , r + 1 q − 1 ∗ ) Recall the TVZ bound without locality: R ≥ 1 − δ − 1 √ q − 1 A. Barg (UMD) LRC codes 25 / 30

Asymptotically good codes LRC codes on curves better than the GV bound R 0.8 � 0.6 LRC codes 0.4 on curves LRC GV bound 0.2 1.0 ∆ 0.2 0.4 0.6 0.8 A. Barg (UMD) LRC codes 26 / 30