Layered Error-Correction Codes for Real-Time Streaming over Erasure Channels

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1 Layered Error-Correction Codes for Real-Time Streaming over Erasure Channels Ashish Khisti University of Toronto Joint Work: Ahmed Badr (Toronto), Wai-Tian Tan (Cisco), John Apostolopoulos (Cisco). Sequential and Adaptive Information Theory Workshop McGill University Nov 7, 2013

2 Real-Time Communication System Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

3 Real-Time Communication System Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

4 Real-Time Communication System Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

5 Real-Time Communication System Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

6 Real-Time Communication System Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

7 Real-Time Communication System Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

8 Real-Time Communication System Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

9 Real-Time Communication System Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

10 Real-Time Communication System Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

11 Real-Time Communication System Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

12 Real-Time Communication System Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

13 Real-Time Communication System Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

14 Motivating Example k n s 0 s 1 s 2 s 3 s 4 s 5 s 6 s 7 x i p 0 p 1 p 2 p 3 p 4 p 5 p 6 p 7 p i = s i H 0 + s i 1 H s i M H M, Erasure Codes: H i F k n k q Random Linear Codes Strongly-MDS Codes (Gabidulin 88, Gluesing-Luerssen 06 ) Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

15 Motivating Example k n s 0 s 1 s 2 s 3 s 4 s 5 s 6 s 7 x i p 0 p 1 p 2 p 3 p 4 p 5 p 6 p 7 p i = s i H 0 + s i 1 H s i M H M, Erasure Codes: H i F k n k q Random Linear Codes Strongly-MDS Codes (Gabidulin 88, Gluesing-Luerssen 06 ) Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

16 Motivating Example k n s 0 s 1 s 2 s 3 s 4 s 5 s 6 s 7 x i p 0 p 1 p 2 p 3 p 4 p 5 p 6 p 7 p i = s i H 0 + s i 1 H s i M H M, Erasure Codes: H i F k n k q Random Linear Codes Strongly-MDS Codes (Gabidulin 88, Gluesing-Luerssen 06 ) Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

17 Motivating Example k n s 0 s 1 s 2 s 3 s 4 s 5 s 6 s 7 x i p 0 p 1 p 2 p 3 p 4 p 5 p 6 p 7 p i = s i H 0 + s i 1 H s i M H M, Erasure Codes: H i F k n k q Random Linear Codes Strongly-MDS Codes (Gabidulin 88, Gluesing-Luerssen 06 ) Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

18 Motivating Example k n s 0 s 1 s 2 s 3 s 4 s 5 s 6 s 7 x i p 0 p 1 p 2 p 3 p 4 p 5 p 6 p 7 p i = s i H 0 + s i 1 H s i M H M, Erasure Codes: H i F k n k q Random Linear Codes Strongly-MDS Codes (Gabidulin 88, Gluesing-Luerssen 06 ) Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

19 Motivating Example k n s 0 s 1 s 2 s 3 s 4 s 5 s 6 s 7 x i p 0 p 1 p 2 p 3 p 4 p 5 p 6 p 7 p i = s i H 0 + s i 1 H s i M H M, Erasure Codes: H i F k n k q Random Linear Codes Strongly-MDS Codes (Gabidulin 88, Gluesing-Luerssen 06 ) Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

20 Motivating Example k n s 0 s 1 s 2 s 3 s 4 s 5 s 6 s 7 x i p 0 p 1 p 2 p 3 p 4 p 5 p 6 p 7 p i = s i H 0 + s i 1 H s i M H M, Erasure Codes: Recover s 0, s 1, s 2, s 3 H i F k n k q Random Linear Codes Strongly-MDS Codes (Gabidulin 88, Gluesing-Luerssen 06 ) Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

21 Motivating Example k n s 0 s 1 s 2 s 3 s 4 s 5 s 6 s 7 x i p 0 p 1 p 2 p 3 p 4 p 5 p 6 p 7 p i = s i H 0 + s i 1 H s i M H M, Erasure Codes: Recover s 0, s 1, s 2, s 3 H i F k n k q Random Linear Codes Strongly-MDS Codes (Gabidulin 88, Gluesing-Luerssen 06 ) Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

22 Motivating Example k n s 0 s 1 s 2 s 3 s 4 s 5 s 6 s 7 x i p 0 p 1 p 2 p 3 p 4 p 5 p 6 p 7 p i = s i H 0 + s i 1 H s i M H M, Erasure Codes: Recover s 0, s 1, s 2, s 3 H i F k n k q Random Linear Codes Strongly-MDS Codes (Gabidulin 88, Gluesing-Luerssen 06 ) p 4 H 4 H 3 H 2 H 1 s 0 p 5 p 6 = H 5 H 4 H 3 H 2 s 1 0 H 5 H 4 H 3 s 2 p H 5 H 4 s 3 }{{} full rank Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

23 Motivating Example- II B = 4, T = 8 Rate 1/2 Baseline Erasure Codes, T = 7 v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 p 0 p 1 p 2 p 3 p 4 p 5 p 6 p 7 v 8 v 9 v 10 v 11 p 8 p 9 p 10 p 11 x i v 0,v 1,v 2,v 3 Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

24 Motivating Example- II B = 4, T = 8 Rate 1/2 Baseline Erasure Codes, T = 7 v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 p 0 p 1 p 2 p 3 p 4 p 5 p 6 p 7 v 8 v 9 v 10 v 11 p 8 p 9 p 10 p 11 x i Rate 1/2 Repetition Code, T = 8 v 0,v 1,v 2,v 3 u 0 u 1 u 2 u 3 u 4 u 5 u 6 u 7 u -8 u -7 u -6 u -5 u -4 u -3 u -2 u -1 u 8 u 9 u 10 u 11 u 0 u 1 u 2 u 3 x i u 0 u 1 u 2 u 3 Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

25 Motivating Example - II B = 4, T = 8 v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 10 v 11 p 0 p 1 p 2 p 3 p 4 p 5 p 6 p 7 p 8 p 9 p 10 p 11 v u u 0 u 1 u 2 u 3 u 4 u 5 u 6 u 7 u 8 u 9 u 10 u 11 u -8 u -7 u -6 u -5 u -4 u -3 u -2 u -1 u 0 u 1 u 2 u 3 u u Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

26 Motivating Example - II B = 4, T = 8 u 0 v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 10 v 11 p 0 p 1 p 2 p 3 p 4 p 5 p 6 p 7 p 8 p 9 p 10 u -8 u 1 u -7 u 2 u -6 u 3 u -5 u 4 u -4 u 5 u -3 u 6 u -2 u 7 u -1 u 8 u 0 u 9 u 1 u 10 u 2 u 11 p 11 u 3 u v u u R= u+v 3u+v = 1 2 Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

27 Motivating Example - II B = 4, T = 8 s i u 0 u 1 u 2 u 3 u 4 u 5 u 6 u 7 u 8 u 9 u 10 u 11 v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 10 v 11 p 0 p 1 p 2 p 3 p 4 p 5 p 6 p 7 p 8 p 9 p 10 p 11 +u -8 +u -7 +u -6 +u -5 +u -3 +u -2 +u -1 +u 0 +u 1 +u 2 +u 3 +u -4 u v u x i R= u+v 2 u+v = 2 3 Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

28 Motivating Example - II B = 4, T = 8 s i u 0 u 1 u 2 u 3 u 4 u 5 u 6 u 7 u 8 u 9 u 10 u 11 v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 10 v 11 p 0 p 1 p 2 p 3 p 4 p 5 p 6 p 7 p 8 p 9 p 10 p 11 +u -8 +u -7 +u -6 +u -5 +u -3 +u -2 +u -1 +u 0 +u 1 +u 2 +u 3 +u -4 u v u x i R= u+v 2 u+v = 2 3 Encoding: 1 Split each source symbol into 2 groups s i = (u i, v i ) 2 Apply Erasure code to the v i stream generating p i parities 3 Repeat the u i symbols with a shift of T 4 Combine the repeated u i s with the p i s Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

29 Motivating Example - II B = 4, T = 8 s i u 0 u 1 u 2 u 3 u 4 u 5 u 6 u 7 u 8 u 9 u 10 u 11 v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 10 v 11 p 0 p 1 p 2 p 3 p 4 p 5 p 6 p 7 p 8 p 9 p 10 p 11 +u -8 +u -7 +u -6 +u -5 +u -3 +u -2 +u -1 +u 0 +u 1 +u 2 +u 3 +u -4 u v u x i R= u+v 2 u+v = 2 3 v 0,v 1,v 2,v 3 Encoding: 1 Split each source symbol into 2 groups s i = (u i, v i ) 2 Apply Erasure code to the v i stream generating p i parities 3 Repeat the u i symbols with a shift of T 4 Combine the repeated u i s with the p i s Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

30 Motivating Example - II B = 4, T = 8 s i u 0 u 1 u 2 u 3 u 4 u 5 u 6 u 7 u 8 u 9 u 10 u 11 v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 10 v 11 p 0 p 1 p 2 p 3 p 4 p 5 p 6 p 7 p 8 p 9 p 10 p 11 +u -8 +u -7 +u -6 +u -5 +u -3 +u -2 +u -1 +u 0 +u 1 +u 2 +u 3 +u -4 u v u x i R= u v 2 u v v 0,v 1,v 2,v 3 u 0 u 1 u 2 u 3 Encoding: 1 Split each source symbol into 2 groups s i = (u i, v i ) 2 Apply Erasure code to the v i stream generating p i parities 3 Repeat the u i symbols with a shift of T 4 Combine the repeated u i s with the p i s Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

31 Motivating Example - II B = 4, T = 8 s i u 0 u 1 u 2 u 3 u 4 u 5 u 6 u 7 u 8 u 9 u 10 u 11 v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 10 v 11 p 0 p 1 p 2 p 3 p 4 p 5 p 6 p 7 p 8 p 9 p 10 p 11 +u -8 +u -7 +u -6 +u -5 +u -3 +u -2 +u -1 +u 0 +u 1 +u 2 +u 3 +u -4 u v u x i R= u+v 2 u+v = 2 3 v 0,v 1,v 2,v 3 u 0 u 1 u 2 u 3 Encoding: s 0 s 1 s 2 s 3 1 Split each source symbol into 2 groups s i = (u i, v i ) 2 Apply Erasure code to the v i stream generating p i parities 3 Repeat the u i symbols with a shift of T 4 Combine the repeated u i s with the p i s Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

32 Motivating Example - II B = 4, T = 8 s i v u 0 p v 0 v u 1 p v 1 v u 2 p v 2 v u 3 p v 3 v u 4 p v 4 v u 5 p v 5 v u 6 p v 6 v u 7 7 p v 7 7 v u 8 p v 8 v u 9 p v 9 v u 10 p v 10 v u 11 p v 11 p 0 p 1 p 2 p 3 p 4 p 5 p 6 p 7 p 8 p 9 p 10 p 11 +u -8 +u -7 +u -6 +u -5 +u -4 +u -3 +u -2 +u -1 +u 0 +u 1 +u 2 +u 3 u -8 u -7 u -6 u -5 +u u -4-4 u -3 u 0 u 1 R= u u+v u v 2 u 3 u 4 u 5 u 6 u u+v 1 2 v u v 0,v 1,v 2,v 3 u 0 u 1 u 2 u 2 = u -8 u -7 u -6 u -5 u -4 u -3 u -2 u -1 u 0 u 1 u 2 Encoding: u -2 u -1 u 0 u 1 u 2 u 3 u 3 s 0 s 1 s 2 s 3 uv vu u u u u x i 1 Split each source symbol into 2 groups s i = (u i, v i ) 2 Apply Erasure code to the v i stream generating p i parities 3 Repeat the u i symbols with a shift of T 4 Combine the repeated u i s with the p i s 5 Maximally Short Codes (Martinian and Sundberg 04, Martinian and Trott 07) Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

33 Key Questions Can we construct streaming codes for realistic channels? Bad State Good State Bad State Gilbert-Elliott Model How much performance gains can we obtain? What are the fundamental metrics for low-delay error correction codes? Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

34 Problem Setup System Model Source Model : i.i.d. sequence s[t] p s ( )= Unif{(F q ) k } Streaming Encoder: x[t] = f t (s[1],..., s[t]), x[t] (F q ) n Erasure Channel Delay-Constrained Decoder: ŝ[t] = g t (y[1],..., y[t + T ]) Rate R = k n Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

35 Problem Setup System Model Source Model : i.i.d. sequence s[t] p s ( )= Unif{(F q ) k } Streaming Encoder: x[t] = f t (s[1],..., s[t]), x[t] (F q ) n Erasure Channel Delay-Constrained Decoder: ŝ[t] = g t (y[1],..., y[t + T ]) Rate R = k n (N,B,W) = (2,3,6) W = W 6= 6 N = N 2= 2 Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

36 Problem Setup System Model Source Model : i.i.d. sequence s[t] p s ( )= Unif{(F q ) k } Streaming Encoder: x[t] = f t (s[1],..., s[t]), x[t] (F q ) n Erasure Channel Delay-Constrained Decoder: ŝ[t] = g t (y[1],..., y[t + T ]) Rate R = k n (N,B,W) = (2,3,6) W = W 6= W 6= 6 N = N 2= N 2= 2 Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

37 Problem Setup System Model Source Model : i.i.d. sequence s[t] p s ( )= Unif{(F q ) k } Streaming Encoder: x[t] = f t (s[1],..., s[t]), x[t] (F q ) n Erasure Channel Delay-Constrained Decoder: ŝ[t] = g t (y[1],..., y[t + T ]) Rate R = k n (N,B,W) = (2,3,6) W = W 6= W 6= W 6= 6 N = N 2= N 2= B 2= 3 Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

38 Problem Setup System Model Source Model : i.i.d. sequence s[t] p s ( )= Unif{(F q ) k } Streaming Encoder: x[t] = f t (s[1],..., s[t]), x[t] (F q ) n Erasure Channel Delay-Constrained Decoder: ŝ[t] = g t (y[1],..., y[t + T ]) Rate R = k n (N,B,W) = (2,3,6) W = W 6= W 6= W 6= W 6= 6 N = N 2= N 2= B 2= B 3= 3 Capacity: C(N, B, W, T ) Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

39 Streaming Codes Capacity Badr-Khisti-Tan-Apostolopoulos (2013) Theorem Consider the C(N, B, W ) channel, with W B + 1, and let the delay be T. Upper-Bound For any rate R code, we have: ( R 1 R ) B + N min(w, T + 1) Lower-Bound: There exists a rate R code that satisfies: ( ) R B + N min(w, T + 1) 1. 1 R The gap between the upper and lower bound is 1 unit of delay. Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

40 Streaming Codes - Burst Erasure Channels C(N = 1, B, W T + 1) = T T +B s i u 0 u 1 u 2 u 3 u 4 u 5 u 6 u 7 u 8 u 9 u 10 u 11 v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 10 v 11 p 0 p 4 +u -8 +u -7 +u -6 +u -5 +u -3 +u -2 +u -1 +u 0 +u 1 +u 2 +u 3 p 1 p 2 p 3 p 5 p 6 p 7 p 8 p 9 p 10 p 11 +u -4 B T-B B x i B T-B...v 3 v 0 Assume s i F T q. Let v i F T q B, u i, p i F B q Recovery of {v i } by time T 1 Number of Unknowns: B symbols over Fq T B Number of Equations: T B symbols over F B q Recovery of u i at time i + T Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

41 Streaming Codes - Burst Erasure Channels C(N = 1, B, W T + 1) = T T +B s i u 0 u 1 u 2 u 3 u 4 u 5 u 6 u 7 u 8 u 9 u 10 u 11 v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 10 v 11 p 0 p 4 +u -4 +u -8 +u -7 +u -6 +u -5 +u -3 +u -2 +u -1 +u 0 +u 1 +u 2 +u 3 p 1 p 2 p 3 p 5 p 6 p 7 p 8 p 9 p 10 p 11 B T-B B x i B T-B v 0...v 3 u 0 u 1 u 2 u 3 Assume s i F T q. Let v i F T q B, u i, p i F B q Recovery of {v i } by time T 1 Number of Unknowns: B symbols over Fq T B Number of Equations: T B symbols over F B q Recovery of u i at time i + T Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

42 Streaming Codes - Burst Erasure Channels C(N = 1, B, W T + 1) = T T +B s i u 0 u 1 u 2 u 3 u 4 u 5 u 6 u 7 u 8 u 9 u 10 u 11 v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 10 v 11 p 0 p 4 +u -4 +u -8 +u -7 +u -6 +u -5 +u -3 +u -2 +u -1 +u 0 +u 1 +u 2 +u 3 p 1 p 2 p 3 p 5 p 6 p 7 p 8 p 9 p 10 p 11 B T-B B x i B T-B v 0...v 3 u 0 u 1 u 2 u 3 Assume s i F T q. Let v i F T q B, u i, p i F B q Recovery of {v i } by time T 1 Number of Unknowns: B symbols over Fq T B Number of Equations: T B symbols over F B q Recovery of u i at time i + T Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

43 Streaming Codes - Isolated Erasures C(N 2, B, W) T = 8 u 0 u 1 u 2 u 3 u 4 u 5 u 6 u 7 u 8 u 9 u 10 u 11 v 0 v 1 v 2 v 3 v 4 +u -4 v 5 v 6 v 7 v 8 v 9 v 10 v 11 p 0 p 1 p 2 p 3 p 4 p 5 p 6 p 7 p 8 p 9 p 10 p 11 +u -8 +u -7 +u -6 +u -5 +u -3 +u -2 +u -1 +u 0 +u 1 +u 2 +u 3 u v u x i Erasures at time t = 0 and t = 8 u 0 cannot be recovered due to a repetition code Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

44 Proposed Approach: Layering C(N 2, B, W) u 0 u 1 u 2 u 3 u 4 u 5 u 6 u 7 u 8 u 9 u 10 u 11 v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 10 v 11 p 0 p 1 p 2 p 3 p 4 p 5 p 6 p 7 p 8 p 9 p 10 p 11 u +u -8 +u -7 +u -6 +u -5 +u -4 +u -3 +u -2 +u -1 +u 0 +u 1 +u 2 +u 3 q 1 q 2 q 3 q 4 q 5 q 6 q 7 q 8 q 9 q 10 q 11 k q 0 u v x i Layered Code Design Burst-Erasure Streaming Code C 1 : (u i, v i, p i + u i T ) Erasure Code: q i = M t=1 u i t H u t, q i F k q C 2 : (u i, v i, p i + u i T, q i ) R = T T + B + k, k = N T N + 1 B Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

45 Proposed Approach: Layering C(N 2, B, W) s i u 0 u 1 u 2 u 3 u 4 u 5 u 6 u 7 u 8 u 9 u 10 u 11 v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 10 v 11 p 0 p 1 p 2 p 3 p 4 p 5 p 6 p 7 p 8 p 9 p 10 p 11 u +u -8 +u -7 +u -6 +u -5 +u -4 +u -3 +u -2 +u -1 +u 0 +u 1 +u 2 +u 3 q 1 q 2 q 3 q 4 q 5 q 6 q 7 q 8 q 9 q 10 q 11 k q 0 u v x i Layered Code Design Burst-Erasure Streaming Code C 1 : (u i, v i, p i + u i T ) Erasure Code: q i = M t=1 u i t H u t, q i F k q C 2 : (u i, v i, p i + u i T, q i ) R = T T + B + k, k = N T N + 1 B Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

46 Proposed Approach: Layering C(N 2, B, W) s i u 0 u 1 u 2 u 3 u 4 u 5 u 6 u 7 u 8 u 9 u 10 u 11 v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 10 v 11 p 0 p 1 p 2 p 3 p 4 p 5 p 6 p 7 p 8 p 9 p 10 p 11 u +u -8 +u -7 +u -6 +u -5 +u -4 +u -3 +u -2 +u -1 +u 0 +u 1 +u 2 +u 3 q 1 q 2 q 3 q 4 q 5 q 6 q 7 q 8 q 9 q 10 q 11 k q 0 u v x i Layered Code Design Burst-Erasure Streaming Code C 1 : (u i, v i, p i + u i T ) Erasure Code: q i = M t=1 u i t H u t, q i F k q C 2 : (u i, v i, p i + u i T, q i ) R = T T + B + k, k = N T N + 1 B Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

47 Proposed Approach: Layering C(N 2, B, W) s i u 0 u 1 u 2 u 3 u 4 u 5 u 6 u 7 u 8 u 9 u 10 u 11 v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 10 v 11 p 0 p 1 p 2 p 3 p 4 p 5 p 6 p 7 p 8 p 9 p 10 p 11 u +u -8 +u -7 +u -6 +u -5 +u -4 +u -3 +u -2 +u -1 +u 0 +u 1 +u 2 +u 3 q 1 q 2 q 3 q 4 q 5 q 6 q 7 q 8 q 9 q 10 q 11 k q 0 u v x i Layered Code Design v 0 v 2 Burst-Erasure Streaming Code C 1 : (u i, v i, p i + u i T ) Erasure Code: q i = M t=1 u i t H u t, q i F k q C 2 : (u i, v i, p i + u i T, q i ) R = T T + B + k, k = N T N + 1 B Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

48 Proposed Approach: Layering C(N 2, B, W) s i u 0 u 1 u 2 u 3 u 4 u 5 u 6 u 7 u 8 u 9 u 10 u 11 u v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 10 v 11 v p 0 p 4 +u -4 p 1 p 2 p 3 p 5 p 6 p 7 p 8 p 9 p 10 p 11 u +u -8 +u -7 +u -6 +u -5 +u -3 +u -2 +u -1 +u 0 +u 1 +u 2 +u 3 q 1 q 2 q 3 q 4 q 5 q 6 q 7 q 8 q 9 q 10 q 11 k q 0 x i Layered Code Design v 0 v 2 u 0 u 2 Burst-Erasure Streaming Code C 1 : (u i, v i, p i + u i T ) Erasure Code: q i = M t=1 u i t H u t, q i F k q C 2 : (u i, v i, p i + u i T, q i ) R = T T + B + k, k = N T N + 1 B Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

49 Upper Bound W T + 1 Periodic Erasure Channel with P = T + 1 N + B. B erasures in each burst. Guard Interval = T + 1 N. Link: B T + 1 N B T + 1 N B T + 1 N Show that any low-delay code recovers every symbol on this erasure channel. R T + 1 N T + 1 N + B Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

50 Simulation Results Gilbert-Eliott Channel (α, β) = (5 10 4, 0.5), T = 12 and R = 12/23 Gilbert Elliott Channel Good State: Pr(loss) = ε Bad State: Pr(loss) = 1 Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

51 Simulation Results Gilbert-Eliott Channel (α, β) = (5 10 4, 0.5), T = 12 and R = 12/23 Loss Probability Gilbert Channel (, ) = (5x10 4,0.5) T = 12, Rate = 12/ Uncoded Maximally Short Code (N,B) = (1,11) Strongly MDS Code (N,B) = (6,6) MiDAS Code (N,B) = (2,9) x 10 3 Code N B Code N B Strongly MDS 6 6 MiDAS 2 9 Burst-Erasure 1 11 Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

52 Simulation Results-II Fritchman Channel (α, β) = (1e 5, 0.5) and T = 40 and R = 40/79, 9 states α Good E 1 E 2 E 3 E N-1 E N 1- α α = 1e 5 β = Histogram of Burst Lengths for 9 States Fritchman Channel (, ) = (1E 5,0.5) Analytical Actual Probability of Occurence Burst Length Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

53 Simulation Results-II Fritchman Channel (α, β) = (1e 5, 0.5) and T = 40 and R = 40/79, 9 states Uncoded RLC SCo E RLC (Shift = 32) E RLC (Shift = 36) 9 States Fritchman Channel (, ) = (1e 005,0.5), T = 40 Loss Probability x 10 3 Code N B Code N B Strongly MDS MiDAS-I 8 31 Burst Erasure 1 39 MiDAS-II 4 35 Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

54 Simulation Results - III Gilbert-Eliott Channel (α, β) = (5 10 5, 0.2), T = 50 and R Gilbert Channel (, ) = (5x10 5,0.2), Simulation Length = 10 8, T = 50, Rate = 50/ MS (N,B) = (1,33) S BEC (N,B) = (20,20) MIDAS (N,B) = (4,30) PRC (N,B) = (4,25) Loss Probability x 10 3 Code N B Code N B Strongly MDS MiDAS 4 30 Burst-Erasure 1 33 PRC 4 25 Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

55 Streaming Codes Burst plus Isolated Erasures Badr-Khisti-Tan-Apostolopoulos (2013) When do Earlier Constructions Fail? s i u 0 u 1 u 2 u 3 u 4 u 5 u 6 u 7 u 8 u 9 u 10 u 11 v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 10 v 11 p 0 p 4 +u -8 +u -7 +u -6 +u -5 +u -3 +u -2 +u -1 +u 0 +u 1 +u 2 +u 3 p 1 p 2 p 3 p 5 p 6 p 7 p 8 p 9 p 10 p 11 +u -4 B T-B B x i v 0...v 2??? Burst spans t [0, 2] Isolated Erasure i [7, 10] Interference from v i during the recovery of u 0,..., u 2 Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

56 Streaming Codes Burst plus Isolated Erasures Badr-Khisti-Tan-Apostolopoulos (2013) Layered Construction for Partial Recovery s i u 0 u 1 u 2 u 3 u 4 u 5 u 6 u 7 u 8 u 9 u 10 u 11 u v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 10 v 11 v p 0 p 4 p 1 p 2 p 3 p 5 p 6 p 7 p 8 p 9 p 10 p 11 u +u -8 +u -7 +u -6 +u -5 +u -4 +u -3 +u -2 +u -1 +u 0 +u 1 +u 2 +u 3 r u s0 0 rs1 1 rs2 2 r 3 r 4 r 5 r 6 r 7 r 8 r 9 r 10 r 11 r x i r i = M t=1 v i t H v t Shift in Repetition Code: p i + u i (T ) Isolated Loss: v ( + 1)r Burst Loss: v(b + 1) (T B)(u + r) Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

57 Streaming Codes Burst plus Isolated Erasures Badr-Khisti-Tan-Apostolopoulos (2013) Layered Construction for Partial Recovery s i u 0 u 1 u 2 u 3 u 4 u 5 u 6 u 7 u 8 u 9 u 10 u 11 u v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 10 v 11 v p 0 p 4 p 1 p 2 p 3 p 5 p 6 p 7 p 8 p 9 p 10 p 11 u +u -8 +u -7 +u -6 +u -5 +u -4 +u -3 +u -2 +u -1 +u 0 +u 1 +u 2 +u 3 r u s0 0 rs1 1 rs2 2 r 3 r 4 r 5 r 6 r 7 r 8 r 9 r 10 r 11 r x i r i = M t=1 v i t H v t Shift in Repetition Code: p i + u i (T ) Isolated Loss: v ( + 1)r Burst Loss: v(b + 1) (T B)(u + r) Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

58 Streaming Codes Burst plus Isolated Erasures Badr-Khisti-Tan-Apostolopoulos (2013) Layered Construction for Partial Recovery s i T=8 u 0 u 1 u 2 u 3 u 4 u 5 u 6 u 7 u 8 u 9 u 10 u 11 u v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 10 v 11 v p 0 p 1 p 2 p 3 +u u u -4-6 p 4 p 5 p 6 p 7 p 8 p 9 p 10 p 11 u +u u -2 +u -1 +u 0 +u 1 +u 2 +u 3 +u 4 +u 5 r x i rs0 0 r u s1 1 rs2 2 r 3 r 4 r 5 r 6 r 7 r 8 r 9 r 10 r 11 T eff =6 =2 v 6 r i = M t=1 v i t H v t Shift in Repetition Code: p i + u i (T ) Isolated Loss: v ( + 1)r Burst Loss: v(b + 1) (T B)(u + r) Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

59 Streaming Codes Burst plus Isolated Erasures Badr-Khisti-Tan-Apostolopoulos (2013) Layered Construction for Partial Recovery s i u 0 u 1 u 2 u 3 u 4 u 5 u 6 u 7 u 8 u 9 u 10 u 11 u v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 10 v 11 v p 0 T=8 p 4 +u -4-2 p 1 p 2 p 3 p 5 p 6 p 7 p 8 p 9 p 10 p 11 u +u u u u u u u u 0 2 +u 1 3 +u 2 4 +u 3 5 r x i r s0 0 rs1 1 r u s2 2 r 3 r 4 r 5 r 6 r 7 r 8 r 9 r 10 r 11 T eff =6 =2 v 6 v 0 v 1 v 2 r i = M t=1 v i t H v t Shift in Repetition Code: p i + u i (T ) Isolated Loss: v ( + 1)r Burst Loss: v(b + 1) (T B)(u + r) Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

60 Streaming Codes Burst plus Isolated Erasures Badr-Khisti-Tan-Apostolopoulos (2013) s i T=8 u 0 u 1 u 2 u 3 u 4 u 5 u 6 u 7 u 8 u 9 u 10 u 11 u v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 10 v 11 v p 0 p 4 p 1 p 2 p 3 p 5 p 6 p 7 p 8 p 9 p 10 p 11 u +u u u u u -2 +u -1 +u 0 +u 1 +u 2 +u 3 +u 4 +u 5 r u s0 0 rs1 1 rs2 2 r 3 r 4 r 5 r 6 r 7 r 8 r 9 r 10 r 11 r T eff =6 =2 v 0 v 1 v 2 x i Code-Params v = (T + 1)( B 1) u =(B + 1)(T + 1) ( B 1) r = ( B 1) Rate R = u + v 2u + v + r (T ) + (B + 1) = (T ) + (B + 1)(T + 2) = T + 1 T B Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

61 Streaming Codes Burst plus Isolated Erasures Badr-Khisti-Tan-Apostolopoulos (2013) s i T=8 u 0 u 1 u 2 u 3 u 4 u 5 u 6 u 7 u 8 u 9 u 10 u 11 u v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 10 v 11 v p 0 p 4 p 1 p 2 p 3 p 5 p 6 p 7 p 8 p 9 p 10 p 11 u +u u u u u -2 +u -1 +u 0 +u 1 +u 2 +u 3 +u 4 +u 5 r u s0 0 rs1 1 rs2 2 r 3 r 4 r 5 r 6 r 7 r 8 r 9 r 10 r 11 r T eff =6 =2 v 0 v 1 v 2 x i Layering Principle Start with a code for C(N, B, W ) Add additional parity check symbols for more sophisticated patterns. Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

62 Unequal Source/Channel Rates Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

63 Unequal Source/Channel Rates Source model: i.i.d. process with s[i] uniform over F k q Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

64 Unequal Source/Channel Rates Source model: i.i.d. process with s[i] uniform over F k q Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

65 Unequal Source/Channel Rates Source model: i.i.d. process with s[i] uniform over F k q Streaming encoder:x[i, j] = f i,j(s[0], s[1],, s[i]) F n q Macro-packet: X[i, :] = [x[i, 1]... x[i, M]] Rate: R = H(s) n M = k n M Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

66 Unequal Source/Channel Rates Source model: i.i.d. process with s[i] uniform over F k q Streaming encoder:x[i, j] = f i,j(s[0], s[1],, s[i]) F n q Macro-packet: X[i, :] = [x[i, 1]... x[i, M]] Rate: R = H(s) n M = k n M Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

67 Unequal Source/Channel Rates Source model: i.i.d. process with s[i] uniform over F k q Streaming encoder:x[i, j] = f i,j(s[0], s[1],, s[i]) F n q Macro-packet: X[i, :] = [x[i, 1]... x[i, M]] Rate: R = H(s) n M = k n M Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

68 Unequal Source/Channel Rates Source model: i.i.d. process with s[i] uniform over F k q Streaming encoder:x[i, j] = f i,j(s[0], s[1],, s[i]) F n q Macro-packet: X[i, :] = [x[i, 1]... x[i, M]] Rate: R = H(s) n M = k n M Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

69 Unequal Source/Channel Rates Source model: i.i.d. process with s[i] uniform over F k q Streaming encoder:x[i, j] = f i,j(s[0], s[1],, s[i]) F n q Macro-packet: X[i, :] = [x[i, 1]... x[i, M]] Rate: R = H(s) = k n M n M Packet erasure channel: erasure burst of maximum B channel packets Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

70 Unequal Source/Channel Rates Source model: i.i.d. process with s[i] uniform over F k q Streaming encoder:x[i, j] = f i,j(s[0], s[1],, s[i]) F n q Macro-packet: X[i, :] = [x[i, 1]... x[i, M]] Rate: R = H(s) = k n M n M Packet erasure channel: erasure burst of maximum B channel packets Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

71 Unequal Source/Channel Rates Source model: i.i.d. process with s[i] uniform over F k q Streaming encoder:x[i, j] = f i,j(s[0], s[1],, s[i]) F n q Macro-packet: X[i, :] = [x[i, 1]... x[i, M]] Rate: R = H(s) n M = k n M Packet erasure channel: erasure burst of maximum B channel packets Delay-constrained decoder: s[i] recovered by macro-packet i + T Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

72 Unequal Source/Channel Rates Source model: i.i.d. process with s[i] uniform over F k q Streaming encoder:x[i, j] = f i,j(s[0], s[1],, s[i]) F n q Macro-packet: X[i, :] = [x[i, 1]... x[i, M]] Rate: R = H(s) n M = k n M Packet erasure channel: erasure burst of maximum B channel packets Delay-constrained decoder: s[i] recovered by macro-packet i + T Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

73 Unequal Source/Channel Rates Source-splitting based scheme Source Stream Split Source s 1 s 2 s 3 s 4 s 5 s 6 s 11 s 12 s 13 s 21 s 22 s 23 s 31 s 32 s 33 s 41 s 42 s 43 s 51 s 52 s 53 s 61 Channel Stream x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12 x 13 x 14 x 15 x 16 Received Stream y 1 y 2 y 3 y 4 y 5 y 6 y 7 y 8 y 9 y 10 y 11 y 12 y 13 y 14 y 15 y 16 T'=6 s 11 s 12 s 13 s 21 s 22 s 23 s 31 s 32 s 33 Split s[i] into sub-symbols (s[i, 1], s[i, 2],..., s[i, M]). Apply a streaming code for M = 1. Decoding Delay T = M T. R = MT MT +B Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

74 Unequal Source/Channel Rates Patil-Badr-Khisti-Tan M = 20, T = 5 Optimal Codes SCo Codes MDP Codes Rate (R) Burst Length (B) Source Splitting: R = MT MT +B Strongly MDS: R = 1 B M(T +1) Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

75 Unequal Source/Channel Rates Patil-Badr-Khisti-Tan M = 20, T = 5 Optimal Codes SCo Codes MDP Codes Rate (R) Burst Length (B) Source Splitting: R = MT MT +B Strongly MDS: R = 1 B M(T +1) Capacity: Let B = bm + B, where B {0,..., M 1} { T T +b C =, 0 B b T +b M M(T +b+1) B b, M(T +b+1) T +b M B M 1 Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

76 Optimal Code Construction I Patil-Badr-Khisti-Tan 2013 Key Idea: 1 Apply code for M = 1 on complete source-packet to generate the entire macro-packet. 2 Map/reshape the generated macro-packet into M individual channel-packets. Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

77 Optimal Code Construction II Patil-Badr-Khisti-Tan Source splitting split s[i] into into two groups s[i] = (s 1 [i],..., s k [i]) = (u 1 [i],..., u ku [i], v }{{} 1 [i],..., v kv [i]) }{{} u vec[i] v vec[i] Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

78 Optimal Code Construction III Patil-Badr-Khisti-Tan Parity generation layer 1: (k v + k u, k v, T ) Strongly-MDS code applied to v vec [ ] generating p vec [i] layer 2: repetition code on u vec with a shift of T Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

79 Optimal Code Construction IV Patil-Badr-Khisti-Tan 2013 Overall combined parity: q vec [i] = p vec [i] + u vec [i T ] Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

80 Optimal Code Construction V Patil-Badr-Khisti-Tan Mapping of macro-packet to individual channel-packets Map the generated macro-packet into M individual channel-packets } Rate of the code= ku+kv 2k u+k v Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

81 Optimal Code Construction VI Patil-Badr-Khisti-Tan 2013 Overall macro-packet structure: Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

82 Decoding Analysis Patil-Badr-Khisti-Tan 2013 Key Fact: The worst case burst pattern starts at the beginning of the macro-packet. Let B = bm + B. Two cases depending on whether B (1 R)M: Burst only erases symbols from u vec[i + b] Recovery of v symbols: (k u + k v)b = k ut Optimal spitting ratio: k u kv = b T b R = T T +b Burst erases symbols from v vec[i + b] Recovery of v symbols: (k u + k v)b + (B n k u) = k ut Optimal spitting ratio: k u B kv = M(T +b+1) 2B R = M(T +b+1) B M(T +b+1) Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

83 Capacity Result Theorem For the streaming setup considered, with any M, T and B, the streaming capacity C is given by the following expression: T T +b, B b T +bm, T b, M(T +b+1) B C = M(T +b+1), B > b T +bm, T > b, M B M, B > M 2, T = b, 0, T < b. where the constants b and B are defined via B = bm + B, B {0, 1,..., M 1}, b N 0. Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

84 Robust Extension for M = 1 Append an additional layer parity checks containing Strongly-MDS code on u u 0 u 1 u 2 u 3 u 4 u 5 u 6 u 7 u 8 u 9 u 10 u 11 v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 10 v 11 p 0 p 1 p 2 p 3 p 4 p 5 p 6 p 7 p 8 p 9 p 10 p 11 u +u -8 +u -7 +u -6 +u -5 +u -4 +u -3 +u -2 +u -1 +u 0 +u 1 +u 2 +u 3 q 1 q 2 q 3 q 4 q 5 q 6 q 7 q 8 q 9 q 10 q 11 k q 0 u v x i R = u+v 2u+v+k. Very close to being optimal for k = N T N+1 B Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

85 Robust Extension for any M Many possibilities for the placement of Strongly-MDS parities for u! Approach I: Repetition code replaced by Strongly-MDS code for u Rate of the code unchanged. N = r is achievable. Approach II Append additional layer of Strongly-MDS code for u R = ku+kv 2k u+k v+k s. k s chosen according to given N. } } Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

86 Related Works Burst Erasure Channel: Maximally Short Codes (Block Code + Interleaving), Martinian and Sundberg (IT-2004), Martinian and Trott (ISIT-2007) Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

87 Related Works Burst Erasure Channel: Maximally Short Codes (Block Code + Interleaving), Martinian and Sundberg (IT-2004), Martinian and Trott (ISIT-2007) Other Variations of Streaming Codes Burst and isolated loss patterns (Badr-Khisti-Tan-Apostolopoulos, ISIT 2013) Unequal Source Channel Rates (Patil-Badr-Khisti-Tan Asilomar 2013) Multicast Extension (Khisti-Singh 2009, Badr-Lui-Khisti Allerton 2010) Parallel Channels (Lui-Badr-Khisti CWIT 2011) Multi-Source Streaming Codes (Lui Thesis, 2011) Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

88 Related Works Burst Erasure Channel: Maximally Short Codes (Block Code + Interleaving), Martinian and Sundberg (IT-2004), Martinian and Trott (ISIT-2007) Other Variations of Streaming Codes Burst and isolated loss patterns (Badr-Khisti-Tan-Apostolopoulos, ISIT 2013) Unequal Source Channel Rates (Patil-Badr-Khisti-Tan Asilomar 2013) Multicast Extension (Khisti-Singh 2009, Badr-Lui-Khisti Allerton 2010) Parallel Channels (Lui-Badr-Khisti CWIT 2011) Multi-Source Streaming Codes (Lui Thesis, 2011) Connections between Network Coding and Real-Time Streaming Codes (Tekin-Ho-Yao-Jaggi ITA-2012, Leong and Ho ISIT-2013) Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

89 Related Works Burst Erasure Channel: Maximally Short Codes (Block Code + Interleaving), Martinian and Sundberg (IT-2004), Martinian and Trott (ISIT-2007) Other Variations of Streaming Codes Burst and isolated loss patterns (Badr-Khisti-Tan-Apostolopoulos, ISIT 2013) Unequal Source Channel Rates (Patil-Badr-Khisti-Tan Asilomar 2013) Multicast Extension (Khisti-Singh 2009, Badr-Lui-Khisti Allerton 2010) Parallel Channels (Lui-Badr-Khisti CWIT 2011) Multi-Source Streaming Codes (Lui Thesis, 2011) Connections between Network Coding and Real-Time Streaming Codes (Tekin-Ho-Yao-Jaggi ITA-2012, Leong and Ho ISIT-2013) Tree Codes: Schulman (IT 1996), Sahai (2001), Martinian and Wornell (Allerton 2004), Sukhavasi and Hassibi (2011) Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

90 Distance and Span Properties Append an additional layer parity checks containing Strongly-MDS code on u u 0 u 1 u 2 u 3 u 4 u 5 u 6 u 7 u 8 u 9 u 10 u 11 v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 10 v 11 p 0 p 1 p 2 p 3 p 4 p 5 p 6 p 7 p 8 p 9 p 10 p 11 u +u -8 +u -7 +u -6 +u -5 +u -4 +u -3 +u -2 +u -1 +u 0 +u 1 +u 2 +u 3 q 1 q 2 q 3 q 4 q 5 q 6 q 7 q 8 q 9 q 10 q 11 k q 0 u v x i R = u+v 2u+v+k. Very close to being optimal for k = N T N+1 B MiDAS (Near) Maximum Distance And Span tradeoff Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

91 Distance and Span Properties Consider (n, k, m) Convolutional code: x i = m j=0 s i jg j States Trellis Diagram Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

92 Distance and Span Properties Consider (n, k, m) Convolutional code: x i = m j=0 s i jg j States Trellis Diagram Free Distance Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

93 Distance and Span Properties Consider (n, k, m) Convolutional code: x i = m j=0 s i jg j States Column Distance in [0,3] Column Distance: d T G 0 G 1... G T [ ] 0 G 0... G T 1 d T = min wt s0... s T [s 0,...,s T ]..... s G 0 Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

94 Distance and Span Properties Consider (n, k, m) Convolutional code: x i = m j=0 s i jg j States Column Distance in [0,3] Column Distance: d T G 0 G 1... G T [ ] 0 G 0... G T 1 d T = min wt s0... s T [s 0,...,s T ]..... s G 0 Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

95 Distance and Span Properties Consider (n, k, m) Convolutional code: x i = m j=0 s i jg j States Trellis Column Diagram Distance Span in [0,3] Free in [0,3] Distance Column Span: c T G 0 G 1... G T [ ] 0 G 0... G T 1 c T = min span s0... s T [s 0,...,s T ]..... s G 0 Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

96 Column-Distance & Column Span Tradeoff Badr-Khisti-Tan-Apostolopoulos (2013) Theorem Consider a C(N, B, W ) channel with delay T and W T + 1. A streaming code is feasible over this channel if and only if it satisfies: d T N + 1 and c T B + 1 Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

97 Column-Distance & Column Span Tradeoff Badr-Khisti-Tan-Apostolopoulos (2013) Theorem Consider a C(N, B, W ) channel with delay T and W T + 1. A streaming code is feasible over this channel if and only if it satisfies: d T N + 1 and c T B + 1 Theorem For any rate R convolutional code and any T 0 the Column-Distance d T and Column-Span c T satisfy the following: ( R 1 R ) c T + d T T R There exists a rate R code (MiDAS Code) over a sufficiently large field that satisfies: ( ) R c T + d T T R 1 R Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

98 Multicast (Low-Delay) Codes Burst Erasure Broadcast Channel Src. Stream Encoder X[i] B 1 Decoder 1 Delay = T 1 B 2 Decoder 2 Delay = T 2 Motivation B 1 < B 2 Receiver 1 : Good Channel State Receiver 2: Weaker Channel State Delay adapts to Channel State Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

99 Channel-Adaptive Delay Example: B 1 = 2, T 1 = 4, B 2 = 3 and T 2 = 5. B 1 B 2 x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12 x 13 x 14 Output s 0 s 1 s 2 s 3 s 4,s 5 s 6 s 7 s 8 T 1 T 2 A burst of length B 1 results in a delay of T 1. A burst of length B 2 results in a delay of T 2. Stretch-Factor: s = T 2 B 1 T 1 B 1 Tradeoff between s and Pr(loss). Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

100 Diversity Embedded Streaming Codes: DE-SCo Badr-Khisti-Martinian 2011 Burst Erasure Broadcast Channel Src. Stream Encoder X[i] B 1 Decoder 1 Delay = T 1 B 2 Decoder 2 Delay = T 2 Theorem There exists a {(B 1, T 1 ), (B 2, T 2 )} DE-SCo construction of rate R = T 1 T 1 +B 1 provided T 2 T 2 B 2 B 1 T 1 +B 1. This code has polynomial-time encoding and decoding complexity. Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

101 Multicast (Low Delay) Capacity Burst Erasure Broadcast Channel Src. Stream Encoder X[i] B 1 Decoder 1 Delay = T 1 B 2 Decoder 2 Delay = T 2 Capacity Function Capacity function C(T 1, T 2, B 1, B 2 ) Single User Upper Bound: C min ( T1 T 1 +B 1, Concatenation Lower Bound: C 1 1+ B 1 T 1 + B 2 T 2 ) T 2 T 2 +B 2 Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

102 Multicast (Low Delay) Capacity Badr-Khisti-Lui 2011 Assume w.l.o.g. B 2 B 1 T 2 T B 2 2 T1 B1 B1 Region A Capacity T2 T B 2 2 B C B 1 + B 2 E (B 1, B 2 ) DE-SCo B 2 D T 2 =T 1 +B 2 T 2 =T 1 SCo A T 1 B C D T1 T B T1 T B E?? T2 B1 T B B 2 2 Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

103 Other Extensions Multiple Erasure Bursts (Li-Khisti-Girod 2011) - Interleaved Low-Delay Codes Multiple Links (Lui-Badr-Khisti 2011) - Layered coding for burst erasure channels Multiple Source Streams with Different Decoding Delays (Lui 2011) - Embedded Codes Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42

104 Conclusions Error Correction Codes for Real-Time Streaming Deterministic Channel Models C(N, B, W ) Tradeoff between achievable N and B MiDAS Constructions Column-Distance and Column-Span Tradeoff Partial Recovery Codes for Burst + Isolated Erasures Unequal Source-Channel Rates Sequential and Adaptive Information Theory Workshop McGill University Nov 7, / 42