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