### Abstract

We consider the communication problem over binary causal adversarial erasure channels. Such a channel maps n input bits to n output symbols in {0,1, Λ}, where Λ denotes erasure. The channel is causal if, for every i, the channel adversarially decides whether to erase the ith bit of its input based on inputs 1,..., i, before it observes bits i+1 to n. Such a channel is p-bounded if it can erase at most a p fraction of the input bits over the whole transmission duration. Causal channels provide a natural model for channels that obey basic physical restrictions but are otherwise unpredictable or highly variable. For a given erasure rate p, our goal is to understand the optimal rate (the "capacity") at which a randomized encoder/decoder can transmit reliably across all causal p- bounded erasure channels. In this paper, we introduce the causal erasure model and provide new upper bounds (impossibility results) and lower bounds (analyses of codes) on the achievable rate. Our bounds separate the achievable rate in the causal erasures setting from the rates achievable in two related models: random erasure channels (strictly weaker) and fully adversarial erasure channels (strictly stronger). Specifically, we show: A strict separation between random and causal erasures for all constant erasure rates p ∈ (0,1). In particular, we show that the capacity of causal erasure channels is 0 for p ≥ 1/2 (while it is nonzero for random erasures). A strict separation between causal and fully adversarial erasures for p ∈ (0, φ) where φ ≈ 0.348. For p ∈ [φ, 1 /2), we show codes for causal erasures that have higher rate than the best known constructions for fully adversarial channels. Our results contrast with existing results on correcting causal bit-flip errors (as opposed to erasures) [10, 9, 7, 4, 11], For the separations we provide, the analogous separations for bit-flip models are either not known at all or much weaker.

Original language | English (US) |
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Title of host publication | Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014 |

Publisher | Association for Computing Machinery |

Pages | 1844-1857 |

Number of pages | 14 |

ISBN (Print) | 9781611973389 |

State | Published - Jan 1 2014 |

Event | 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014 - Portland, OR, United States Duration: Jan 5 2014 → Jan 7 2014 |

### Publication series

Name | Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms |
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### Other

Other | 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014 |
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Country | United States |

City | Portland, OR |

Period | 1/5/14 → 1/7/14 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Software
- Mathematics(all)

### Cite this

*Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014*(pp. 1844-1857). (Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms). Association for Computing Machinery.

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*Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014.*Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms, Association for Computing Machinery, pp. 1844-1857, 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, OR, United States, 1/5/14.

**Causal erasure channels.** / Bassily, Raef; Smith, Adam Davison.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

TY - GEN

T1 - Causal erasure channels

AU - Bassily, Raef

AU - Smith, Adam Davison

PY - 2014/1/1

Y1 - 2014/1/1

N2 - We consider the communication problem over binary causal adversarial erasure channels. Such a channel maps n input bits to n output symbols in {0,1, Λ}, where Λ denotes erasure. The channel is causal if, for every i, the channel adversarially decides whether to erase the ith bit of its input based on inputs 1,..., i, before it observes bits i+1 to n. Such a channel is p-bounded if it can erase at most a p fraction of the input bits over the whole transmission duration. Causal channels provide a natural model for channels that obey basic physical restrictions but are otherwise unpredictable or highly variable. For a given erasure rate p, our goal is to understand the optimal rate (the "capacity") at which a randomized encoder/decoder can transmit reliably across all causal p- bounded erasure channels. In this paper, we introduce the causal erasure model and provide new upper bounds (impossibility results) and lower bounds (analyses of codes) on the achievable rate. Our bounds separate the achievable rate in the causal erasures setting from the rates achievable in two related models: random erasure channels (strictly weaker) and fully adversarial erasure channels (strictly stronger). Specifically, we show: A strict separation between random and causal erasures for all constant erasure rates p ∈ (0,1). In particular, we show that the capacity of causal erasure channels is 0 for p ≥ 1/2 (while it is nonzero for random erasures). A strict separation between causal and fully adversarial erasures for p ∈ (0, φ) where φ ≈ 0.348. For p ∈ [φ, 1 /2), we show codes for causal erasures that have higher rate than the best known constructions for fully adversarial channels. Our results contrast with existing results on correcting causal bit-flip errors (as opposed to erasures) [10, 9, 7, 4, 11], For the separations we provide, the analogous separations for bit-flip models are either not known at all or much weaker.

AB - We consider the communication problem over binary causal adversarial erasure channels. Such a channel maps n input bits to n output symbols in {0,1, Λ}, where Λ denotes erasure. The channel is causal if, for every i, the channel adversarially decides whether to erase the ith bit of its input based on inputs 1,..., i, before it observes bits i+1 to n. Such a channel is p-bounded if it can erase at most a p fraction of the input bits over the whole transmission duration. Causal channels provide a natural model for channels that obey basic physical restrictions but are otherwise unpredictable or highly variable. For a given erasure rate p, our goal is to understand the optimal rate (the "capacity") at which a randomized encoder/decoder can transmit reliably across all causal p- bounded erasure channels. In this paper, we introduce the causal erasure model and provide new upper bounds (impossibility results) and lower bounds (analyses of codes) on the achievable rate. Our bounds separate the achievable rate in the causal erasures setting from the rates achievable in two related models: random erasure channels (strictly weaker) and fully adversarial erasure channels (strictly stronger). Specifically, we show: A strict separation between random and causal erasures for all constant erasure rates p ∈ (0,1). In particular, we show that the capacity of causal erasure channels is 0 for p ≥ 1/2 (while it is nonzero for random erasures). A strict separation between causal and fully adversarial erasures for p ∈ (0, φ) where φ ≈ 0.348. For p ∈ [φ, 1 /2), we show codes for causal erasures that have higher rate than the best known constructions for fully adversarial channels. Our results contrast with existing results on correcting causal bit-flip errors (as opposed to erasures) [10, 9, 7, 4, 11], For the separations we provide, the analogous separations for bit-flip models are either not known at all or much weaker.

UR - http://www.scopus.com/inward/record.url?scp=84902089900&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84902089900&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84902089900

SN - 9781611973389

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 1844

EP - 1857

BT - Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014

PB - Association for Computing Machinery

ER -