On codes that correct asymmetric errors with graded magnitude distribution

Abstract

In multi-level flash memories, the dominant cell errors are asymmetric with limited-magnitude. With such an error model in mind, Cassuto et al. recently developed bounds and constructions for codes correcting t asymmetric errors with magnitude no more than ℓ. However, a more refined model of these memory devices reflects the fact that typically only a small number of errors have large magnitude while the remainder are of smaller magnitude. In this work, we study such an error model, in which at most t <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> errors of maximum magnitude ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> and at most t <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> errors of maximum magnitude ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> , with ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> <; ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> , can occur. We adapt the analysis and code construction of Cassuto, et al. for the refined error model and assess the relative efficiency of the new codes. We then consider in more detail specific constructions for the case where t <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> = t <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> = 1, ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> = 1, and ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> >; 1.