Arithmetic Field Circuit Complexity Bounds for Erasure Encoders

Andreas Jakoby, Christian Schindelhauer, Sneha Mohanty, Sven Kohler

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

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 languageEnglish
Title of host publication11th International Conference on Communications, Circuits and Systems, ICCCAS 2022
PublisherInstitute of Electrical and Electronics Engineers Inc.
Pages19-28
Number of pages10
ISBN (Electronic)9781665469890
DOIs
StatePublished - 2022
Externally publishedYes
Event11th International Conference on Communications, Circuits and Systems, ICCCAS 2022 - Singapore, Singapore
Duration: 13 May 202215 May 2022

Publication series

Name11th International Conference on Communications, Circuits and Systems, ICCCAS 2022

Conference

Conference11th International Conference on Communications, Circuits and Systems, ICCCAS 2022
Country/TerritorySingapore
CitySingapore
Period13/05/2215/05/22

Bibliographical note

Publisher Copyright:
© 2022 IEEE.

Keywords

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

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