Arithmetic Field Circuit Complexity Bounds for Erasure Encoders

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 $\Omega \left(\tfrac{m\log n}{\log 1/p}\right)$ which is an asymptotically tight bound.Furthermore, a novel randomized arithmetic circuit with size $\mathcal{O}(m\log \log n)$ smaller than Raptor Codes with size $\mathcal{O}(m\log n)$ 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.