Efficiently decodable codes for the binary deletion channel

Transcription

1 Efficiently decodable codes for the binary deletion channel Venkatesan Guruswami Ray Li * (rayyli@stanford.edu) Carnegie Mellon University August 18, 2017 V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

2 Outline for section 1 1 Introduction 2 Binary deletion channel background 3 Construction ingredients 4 Construction 5 Open questions V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

3 Deletion channel m = 01 Alice V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

4 Deletion channel m = 01 Alice Bob V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

5 Deletion channel m = 01 Alice Channel Bob V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

6 Deletion channel 01 1 m = 01 Alice Channel Bob V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

7 Deletion channel 01 1 m = 01 Alice Channel Bob m = 11 m = 10 m = 01 V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

8 Deletion channel 01 1 m = 01 Alice Channel Bob m = 11 m = 10 m = 01 Error type Before After Substitution Erasure ? Deletion V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

9 Error correcting codes m = 01 Alice (Enc) Bob (Dec) V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

10 Error correcting codes m = 01 Alice (Enc) Channel Bob (Dec) V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

11 Error correcting codes m = 01 Alice (Enc) Channel Bob (Dec) V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

12 Error correcting codes m = 01 Alice (Enc) Channel Bob (Dec) m = 01 V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

13 Error correcting codes m = 01 Alice (Enc) Channel Bob (Dec) m = 01 Tradeoff between redundancy and robustness of the code. Efficient construction, encoding, decoding. V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

14 Error correcting codes m = 01 Alice (Enc) Channel Bob (Dec) m = 01 Tradeoff between redundancy and robustness of the code. Efficient construction, encoding, decoding. This work V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

15 Error correcting codes m = 01 Alice (Enc) Channel Bob (Dec) m = 01 Tradeoff between redundancy and robustness of the code. Efficient construction, encoding, decoding. This work Note: Order of redundancy matters for deletions. m mmm fails for 1 deletion ( , ). V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

16 Error correcting codes: Notation m = 01 Alice (Enc) c = s = Channel Bob (Dec) m = 01 Alphabet: Σ (e.g. {0, 1}) (Block) length: n, m, N Codeword: c Σ n V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

17 Error correcting codes: Notation m = 01 Alice (Enc) c = s = Channel Bob (Dec) m = 01 Alphabet: Σ (e.g. {0, 1}) (Block) length: n, m, N Codeword: c Σ n Rate R = log(#messages) N (0, 1). R is proportion of non-redundant symbols Want families of codes (implicitly N ) V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

18 Binary deletion channel m = 01 Alice (Enc) c = s = Channel Bob (Dec) m = 01 Adversarial: # deletions fixed (pn), decoding 100% success Random: i.i.d deletions, decoding success w.h.p. V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

19 Binary deletion channel m = 01 Alice (Enc) c = s = Channel Bob (Dec) m = 01 Adversarial: # deletions fixed (pn), decoding 100% success Random: i.i.d deletions, decoding success w.h.p. This work Definition For p (0, 1), the binary deletion channel with deletion probability p (BDC p ) deletes bits of binary strings independently w.p. p. V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

20 Outline for section 2 1 Introduction 2 Binary deletion channel background 3 Construction ingredients 4 Construction 5 Open questions V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

21 Capacity Definition Capacity C BDC (p) = sup{r : exists rate R code family against BDC p }. Question. What is the capacity of a binary deletion channel with deletion probability p? V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

22 Capacity Definition Capacity C BDC (p) = sup{r : exists rate R code family against BDC p }. Question. What is the capacity of a binary deletion channel with deletion probability p? Binary Symmetric Channel (BSC): Well understood Binary Erasure Channel (BEC): Well understood Binary Deletion Channel (BDC): Don t know capacity Error type Before After Substitution Erasure ? Deletion V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

23 Capacity Definition Capacity C BDC (p) = sup{r : exists rate R code family against BDC p }. Question. What is the capacity of a binary deletion channel with deletion probability p? Binary Symmetric Channel (BSC): Well understood Binary Erasure Channel (BEC): Well understood Binary Deletion Channel (BDC): Don t know capacity Error type Before After Substitution Erasure ? Deletion Aside: Adversarial deletions, unknown if capacity is 0 for p [ 2 1, 1 2 ). V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

24 Existing bounds on BDC p capacity Definition Capacity C BDC (p) = sup{r : exists rate R code family against BDC p }. Recall: p [0, 1], H(p) = p log 1 p + (1 p) log 1 1 p. V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

25 Existing bounds on BDC p capacity Definition Capacity C BDC (p) = sup{r : exists rate R code family against BDC p }. Recall: p [0, 1], H(p) = p log 1 p Lower bounds 1 2H(p) (adversarial bound) 1 H(p) [Gallager 61, Zigangirov 69] (1 p)/9 [Drinea, Mitzenmacher 06] Upper bounds + (1 p) log 1 1 p. 1 H(p) as p 0 [Kalai, Mitzenmacher, Sudan 10] 1 p (BEC Capacity).4143(1 p) if p 0.65 [Rahmati, Duman 15] Numerical bounds [Fertonani, Duman 10] a Capacity understood as p 0 (1 H(p)), p 1 (α(1 p)). V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

26 Existing bounds on BDC p capacity Definition Capacity C BDC (p) = sup{r : exists rate R code family against BDC p }. Recall: p [0, 1], H(p) = p log 1 p Lower bounds 1 2H(p) (adversarial bound) 1 H(p) [Gallager 61, Zigangirov 69] (1 p)/9 [Drinea, Mitzenmacher 06] + (1 p) log 1 1 p. a Capacity understood as p 0 (1 H(p)), p 1 (α(1 p)). Upper bounds 1 H(p) as p 0 [Kalai, Mitzenmacher, Sudan 10] 1 p (BEC Capacity).4143(1 p) if p 0.65 [Rahmati, Duman 15] Numerical bounds [Fertonani, Duman 10] (1 p)/9 result is non-constructive V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

27 Existing bounds on BDC p capacity Lower bounds 1 H(p) [Gallager 61, Zigangirov 69] (1 p)/9 [Drinea, Mitzenmacher 06] Upper bounds.4143(1 p) if p 0.65 [Rahmati, Duman 15] Numerical bounds [Fertonani, Duman 10] *Plot made with Mathematica V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

28 New algorithmic result Theorem (Guruswami, Li 17) Let p (0, 1). There is a constant α > 0 and an explicit a family of binary codes that has rate α(1 p), is constructible, encodable, decodable in poly time on BDC p. V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

29 New algorithmic result Theorem (Guruswami, Li 17) Let p (0, 1). There is a constant α > 0 and an explicit a family of binary codes that has rate α(1 p), (α = ) is constructible, encodable, decodable in poly time on BDC p. V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

30 New algorithmic result Theorem (Guruswami, Li 17) Let p (0, 1). There is a constant α > 0 and an explicit a family of binary codes that has rate α(1 p), (α = ) is constructible, encodable, decodable in poly time on BDC p. Throughout the talk, think of p close to 1. V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

31 Outline for section 3 1 Introduction 2 Binary deletion channel background 3 Construction ingredients Concatenated codes Haeupler-Shahrasbi code 4 Construction 5 Open questions V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

32 Ingredient 1: Concatenated codes Example Enc out (m) = aabc C out Enc in ( ) : a 0011, b 1100, c 0101 Enc(m) = C m a a b c C out C V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

33 Ingredient 1: Concatenated codes C out : alphabet K, length n C in : alphabet k, length m C: alphabet k, length N = mn Message encoded into c = σ1 σ 2... σ n C out Each σi encoded into codeword of C in R = R out R in m σ 1 σ 2... σ n C out C in (σ 1 ) C in (σ 2 )... C in (σ n ) C V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

34 Ingredient 2: What do we want in an outer code? m σ 1 σ 2... σ n C out C in (σ 1 ) C in (σ 2 )... C in (σ n ) C Rate close to 1 Tolerate constant fraction of adversarial insertions and deletions Fast construction, encoding, decoding V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

35 Ingredient 2: Haeupler-Shahrasbi code (STOC 17) m σ 1 σ 2... σ n C out C in (σ 1 ) C in (σ 2 )... C in (σ n ) C Σ = poly(1/ɛ), Rate ɛ, Corrects 0.001n adversarial insertions and deletions Construction time: poly(n) Encoding time: O(n) Decoding time: O(n 2 ) V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

36 Outline for section 4 1 Introduction 2 Binary deletion channel background 3 Construction ingredients 4 Construction First attempts Our construction Remarks on construction 5 Open questions V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

37 Reminder: New algorithmic result Theorem (Guruswami, Li 17) Let p (0, 1). There is a constant α > 0 and an explicit a family of binary codes that has rate α(1 p), is constructible, encodable, decodable in polynomial time on BDC p. V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

38 Attempt 1: Vanilla concatenation C out is Haeupler-Shahrasbi code. C in {0, 1} m robust against BDC p m σ 1 σ 2... σ n C in (σ 1 ) C in (σ 2 )... C in (σ n ) Rate 0.999R in Outer code decodes in O(n 2 ) time V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

39 Attempt 1: Vanilla concatenation C out is Haeupler-Shahrasbi code. C in {0, 1} m robust against BDC p (m = poly(1/ɛ)) m σ 1 σ 2... σ n C in (σ 1 ) C in (σ 2 )... C in (σ n ) Rate 0.999R in Outer code decodes in O(n 2 ) time Inner code decodes in constant time V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

40 Attempt 1: Vanilla concatenation C out is Haeupler-Shahrasbi code. C in {0, 1} m robust against BDC p (m = poly(1/ɛ)) m σ 1 σ 2... σ n C in (σ 1 ) C in (σ 2 )... C in (σ n ) Rate 0.999R in Outer code decodes in O(n 2 ) time Inner code decodes in constant time Issue: where do inner codewords start? V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

41 Attempt 1: Vanilla concatenation C out is Haeupler-Shahrasbi code. C in {0, 1} m robust against BDC p (m = poly(1/ɛ)) m σ 1 σ 2... σ n C in (σ 1 ) C in (σ 2 )... C in (σ n ) Rate 0.999R in Outer code decodes in O(n 2 ) time Inner code decodes in constant time Issue: where do inner codewords start? V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

42 Attempt 1: Vanilla concatenation C out is Haeupler-Shahrasbi code. C in {0, 1} m robust against BDC p (m = poly(1/ɛ)) m σ 1 σ 2... σ n C in (σ 1 ) C in (σ 2 )... C in (σ n ) Rate 0.999R in Outer code decodes in O(n 2 ) time Inner code decodes in constant time Issue: where do inner codewords start? V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

43 Attempt 2: Concatenation with buffers C out is Haeupler-Shahrasbi code. C in {0, 1} m robust against BDC p (m = poly(1/ɛ)) m σ 1 C in (σ 1 ) m Decoding σ 2 C in (σ 2 ) m m σ n Identify decoding buffers as long runs of 0s C in (σ n ) Inner decode the decoding windows in between the decoding buffers V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

44 Attempt 2: Concatenation with buffers C out is Haeupler-Shahrasbi code. C in {0, 1} m robust against BDC p (m = poly(1/ɛ)) m σ 1 C in (σ 1 ) m Decoding σ 2 C in (σ 2 ) m m σ n Identify decoding buffers as long runs of 0s C in (σ n ) Inner decode the decoding windows in between the decoding buffers Bits of inner codewords not deleted according to BDC p Cannot use DM06 R = (1 p)/9 code as a black box inner code V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

45 Concatenation with buffers and duplication σ 1 C B in (σ 1) Bm σ 2 C B in (σ 2) m Bm Bm σ n C B in (σ n) Outer code. Haeupler-Shahrasbi code Inner code. Word have runs of length 1 and 2 only, start and end with 1, chosen greedily, correct against δn adversarial deletions Buffer m 0s between adjacent inner codewords. Duplication. Duplicate each bit B(p) times (B(p) = 60/(1 p)) 1 B 0 B 1 2B 0 2B 1 2B 0 B 1 B 0 B 1 B 0 B 1 2B 0 2B 1 B V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

46 Concatenation with buffers and duplication σ 1 C B in (σ 1) Bm σ 2 C B in (σ 2) m Bm Bm σ n C B in (σ n) Outer code. Haeupler-Shahrasbi code Inner code. Word have runs of length 1 and 2 only, start and end with 1, chosen greedily, correct against δn adversarial deletions. δ 0.008, R in Buffer m 0s between adjacent inner codewords. Duplication. Duplicate each bit B(p) times (B(p) = 60/(1 p)) 1 B 0 B 1 2B 0 2B 1 2B 0 B 1 B 0 B 1 B 0 B 1 2B 0 2B 1 B V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

47 Concatenation with buffers and duplication σ 1 C B in (σ 1) Bm σ 2 C B in (σ 2) m Bm Bm σ n C B in (σ n) Outer code. Haeupler-Shahrasbi code Inner code. Word have runs of length 1 and 2 only, start and end with 1, chosen greedily, correct against δn adversarial deletions. δ 0.008, R in Buffer m 0s between adjacent inner codewords. Duplication. Duplicate each bit B(p) times (B(p) = 60/(1 p)) 1 B 0 B 1 2B 0 2B 1 2B 0 B 1 B 0 B 1 B 0 B 1 2B 0 2B 1 B Rate (1 p)/60 = α(1 p). V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

48 Concatenation with buffers and duplication σ 1 C B in (σ 1) Bm σ 2 C B in (σ 2) m Bm Bm σ n C B in (σ n) Outer code. Haeupler-Shahrasbi code Inner code. Runs of length 1 and 2. Start and end with Buffer. ηm 0s between inner codewords. Duplication. B = 60/(1 p). E[ BDC p (1 B ) ] = 60, E[ BDC p (1 2B ) ] = 120 Idea: Decode 1 α as 1 for α 86, and 11 for α > 86 V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

49 Decoding algorithm aaa m σ 1 σ 2 σ n C 60 in (σ 1) m Decoding algorithm. C 60 in (σ 2) m m C 60 in (σ n) Runs of at least 0.03m zeros (decoding buffers) divide word into decoding windows V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

50 Decoding algorithm aaa m σ 1 σ 2 σ n C 60 in (σ 1) m Decoding algorithm. C 60 in (σ 2) m m C 60 in (σ n) Runs of at least 0.03m zeros (decoding buffers) divide word into decoding windows V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

51 Decoding algorithm aaa m σ 1 σ 2 σ n C 60 in (σ 1) m Decoding algorithm. C 60 in (σ 2) m m C 60 in (σ n) Runs of at least 0.03m zeros (decoding buffers) divide word into decoding windows Deduplicate the runs: b α as b for α 86, and bb for α > 86 V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

52 Decoding algorithm aaa m σ 1 C in (σ 1 ) m σ 2 Decoding algorithm. C in (σ 2 ) m m σ n C in (σ n ) Runs of at least 0.03m zeros (decoding buffers) divide word into decoding windows Deduplicate the runs: b α as b for α 86, and bb for α > 86 V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

53 Decoding algorithm aaa m σ 1 C in (σ 1 ) m σ 2 Decoding algorithm. C in (σ 2 ) m m σ n C in (σ n ) Runs of at least 0.03m zeros (decoding buffers) divide word into decoding windows Deduplicate the runs: b α as b for α 86, and bb for α > 86 For each decoding window recover outer symbol σ from Dec in V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

54 Decoding algorithm aaa m σ 1 C in (σ 1 ) m σ 2 Decoding algorithm. C in (σ 2 ) m m σ n C in (σ n ) Runs of at least 0.03m zeros (decoding buffers) divide word into decoding windows Deduplicate the runs: b α as b for α 86, and bb for α > 86 For each decoding window recover outer symbol σ from Dec in V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

55 Decoding algorithm aaa m σ 1 C in (σ 1 ) m σ 2 Decoding algorithm. C in (σ 2 ) m m σ n C in (σ n ) Runs of at least 0.03m zeros (decoding buffers) divide word into decoding windows Deduplicate the runs: b α as b for α 86, and bb for α > 86 For each decoding window recover outer symbol σ from Dec in Use Dec out to decode σ 1 σ 2... σ n into message m. V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

56 Decoding algorithm aaa m σ 1 C in (σ 1 ) m σ 2 Decoding algorithm. C in (σ 2 ) m m σ n C in (σ n ) Runs of at least 0.03m zeros (decoding buffers) divide word into decoding windows Deduplicate the runs: b α as b for α 86, and bb for α > 86 For each decoding window recover outer symbol σ from Dec in Use Dec out to decode σ 1 σ 2... σ n into message m. V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

57 Decoding algorithm aaa m σ 1 C in (σ 1 ) m σ 2 Decoding algorithm. C in (σ 2 ) m m σ n C in (σ n ) Runs of at least 0.03m zeros (decoding buffers) divide word into decoding windows O(n) Deduplicate the runs: b α as b for α 86, and bb for α > 86 n O(1) For each decoding window recover outer symbol σ from Dec in n O(1) Use Dec out to decode σ 1 σ 2... σ n into message m. O(n 2 ) Total runtime: O(n 2 ) V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

58 What could go wrong? Deleted buffer 2 deletions, 1 insertion m σ 1 σ 1 σ 2 σ n 1 C 60 in (σ 1) m Decoding algorithm. C 60 in (σ 2) m m C 60 in (σ n) Runs of at least 0.03m zeros (decoding buffers) divide word into decoding windows Deduplicate the runs: b α as b for α 86, and bb for α > 86 For each decoding window recover outer symbol σ from Dec in Use Dec out to decode σ 1 σ 2... σ n into message m. V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

59 What could go wrong? Created buffer 2 insertions, 1 deletion m σ 1 σ 2 σ 3 σ n+1 x0 0.06m y m Decoding algorithm. C 60 in (σ 2) m m C 60 in (σ n) Runs of at least 0.03m zeros (decoding buffers) divide word into decoding windows Deduplicate the runs: b α as b for α 86, and bb for α > 86 For each decoding window recover outer symbol σ from Dec in Use Dec out to decode σ 1 σ 2... σ n into message m. V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

60 What could go wrong? Corrupted decoding window 1 insertion, 1 deletion m σ 1 x 0 σ m C 60 Decoding algorithm. in (σ 2) m m σ n C 60 in (σ n) Runs of at least 0.03m zeros (decoding buffers) divide word into decoding windows Deduplicate the runs: b α as b for α 86, and bb for α > 86 For each decoding window recover outer symbol σ from Dec in Use Dec out to decode σ 1 σ 2... σ n into message m. V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

61 What could go wrong? Corrupted decoding window 1 insertion, 1 deletion m σ 1 x 0 σ m C 60 Decoding algorithm. in (σ 2) m m σ n C 60 in (σ n) Runs of at least 0.03m zeros (decoding buffers) divide word into decoding windows Deduplicate the runs: b α as b for α 86, and bb for α > 86 For each decoding window recover outer symbol σ from Dec in Use Dec out to decode σ 1 σ 2... σ n into message m. Total number of ins/dels in σ 1... σ n is < 0.001n w.h.p V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

62 Remark: Rate improvement Rate is 1 p 1 p 110. Can improve to 60 if duplication is Poisson No easy way to use 1 p 9 directly as a black box inner code (else rate is 1 p 9 ) V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

63 Remark: Alternative outer codes Reed Solomon code (encoding (i, α i ) into the inner code) Similar rate, Worse runtime, Inner code is log n length High rate binary code efficiently decodable against insertions and deletions [Guruswami, Li 16] Worse rate and runtime Outer code Error type Inner len Rate Error frac Decoding HS 17 Ins/del c 1 ɛ Ω(ɛ) O(N 2 ) GL 16 Ins/del c 1 ɛ Ω(ɛ 5 ) poly(n) RS Erase/Sub Ω(log n) 1 ɛ Ω(ɛ) O(N 3 ) V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

64 Outline for section 5 1 Introduction 2 Binary deletion channel background 3 Construction ingredients 4 Construction 5 Open questions V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

65 Open questions Capacity of the binary deletion channel Efficiently decodable codes for BDC with rate α(1 p) for larger α, perhaps α 1/9 Efficiently decodable codes for BDC with rate 1 O(H(p)) for p 0 Capacity of random channels applying insertions and deletions V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31

66 Thank you! Research supported in part by NSF grant CCF V. Guruswami and R. Li (CMU) Efficiently decodable codes for the BDC August 18, / 31