## Abstract

In this paper, the computational circuit complexity of encoding erasure codes over a (finite) field mathbb{F} with arithmetic circuits is investigated. An erasure encoder computes m code symbols from mathbb{F} given the n input symbols from mathbb{F} such that after erasing some code symbols the input can be reconstructed. Linear erasure encoders compute symbols by linear transformations of the input symbols.Two erasure channel models are considered: a worst case channel where a perfect erasure code can reconstruct the input from any n of m output symbols; a probabilistic channel model where each output symbol is erased with probability p and the reconstruction succeeds with high probability.The circuit complexity by the depth and size of fan-in bounded arithmetic circuits over F is measured. It is shown that the minimum circuit depth for perfect encoders is log n + log left(1 - tfrac{n - 1}{m}right). For the probabilistic model an encoder circuit needs depth Ω(log log n - log log 1/p) to succeed with at least some constant probability. For linear encoders it is shown that the number of all positive entries in the encoder matrix in the perfect case is n(m - n + 1) and for the probabilistic setting the number of entries is in Omega left(tfrac{mlog n}{log 1/p}right) which is an asymptotically tight bound.Furthermore, a novel randomized arithmetic circuit with size mathcal{O}(mlog log n) smaller than Raptor Codes with size mathcal{O}(mlog n) is introduced. The circuit depth of the probabilistic erasure encoder as mathcal{O}(log log n), which is depth-optimal like Raptor codes is also presented.

Original language | English |
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Title of host publication | 11th International Conference on Communications, Circuits and Systems, ICCCAS 2022 |

Publisher | Institute of Electrical and Electronics Engineers Inc. |

Pages | 19-28 |

Number of pages | 10 |

ISBN (Electronic) | 9781665469890 |

DOIs | |

State | Published - 2022 |

Externally published | Yes |

Event | 11th International Conference on Communications, Circuits and Systems, ICCCAS 2022 - Singapore, Singapore Duration: 13 May 2022 → 15 May 2022 |

### Publication series

Name | 11th International Conference on Communications, Circuits and Systems, ICCCAS 2022 |
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### Conference

Conference | 11th International Conference on Communications, Circuits and Systems, ICCCAS 2022 |
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Country/Territory | Singapore |

City | Singapore |

Period | 13/05/22 → 15/05/22 |

### Bibliographical note

Publisher Copyright:© 2022 IEEE.

## Keywords

- complexity theory
- fields (algebraic)
- forward error correction
- graph theory
- random codes