Speaker
Description
Code Recovery using algebraic-geometric approaches becomes computationally expensive with the cardinality of the field and the complexity of the code structures. In response, we present a low-complexity algorithm that utilizes structures in algebraic-geometric codes over finite fields. The low-complexity algorithm recovers algebraic codes over finite fields locally, which we name as lrc algorithm. The lrc algorithm is derived based on a sparse matrix factorization of the inverse of an $(r+1) \times r$ locally recoverable matrix, whose elements are defined by the points on the surface in $\mathbb{P}^3$ over the finite field $\mathbb{F}_q$ having locality $r$. We will show that the lrc algorithm reduces the complexity from $\mathcal{O}(n^3)$ to $\mathcal{O}(n \log n)$ for the $n=2^s(s \geq 1) > r$ length codeword. Next, we utilize an extended lrc algorithm followed by a structured neural network (StNN) to globally recover codes over the surface $\mathbb{P}^3$ over the finite field $\mathbb{F}_q$. We discuss two StNNs, i.e., DFT-StNN and DCT-StNN, based on the factorization of the locally recoverable matrix to globally recover codes. Numerical simulations will be shown to compare the performance of the extended lrc algorithm in global recovery codes, with brute-force calculation, DFT-StNN, DCT-StNN, and a feedforward neural network for codewords with lengths from $n=6, 12, 27, 48, 96$ and $210$, having locality $2$ for points on the surface $\mathbb{P}^3$ over the finite field $\mathbb{F}_q$. Our empirical results demonstrate that the extended lrc algorithm achieves the lowest flops, the highest accuracy with an error order $10^{-16}$ compared to brute-force calculations, DFT-StNN, DCT-StNN, and the feedforward neural network, showing the existence of local to global codeword setting.