Analog Error-Correcting Codes: Designs and Analysis
Abstract: A new type of analog error-correcting codes (Analog ECCs) has been proposed by Roth recently. The codes can correct errors of unlimited magnitudes even though the codeword is affected not only by such errors, but also by ubiquitous noise of limited magnitudes. The codes have the potential to accelerate the widely used vector-matrix multiplication in machine learning via their implementation in nanoscale analog circuits. Several Analog ECCs, which mainly focus on correcting or detecting a single unlimited-magnitude error, have been proposed. This paper explores the analysis and constructions of Analog ECCs in multiple ways. It presents a linear-programming based algorithm that computes the m-heights of Analog ECCs efficiently, which can be used to determine the error correction/detection capabilities of the codes. It then presents a family of Analog ECCs based on permutations, and proves that the time complexity for determining the m-heights of such codes can be further reduced substantially. The analysis forms a basis for the time-complexity tradeoff between the searching of codes and the verification of their performance. The paper then presents a number of newly discovered codes based on such a search and verification process, which achieve state-of-the-art performance.
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