Nonuniform Codes for Correcting Asymmetric Errors in Data Storage

Abstract

The construction of asymmetric error-correcting codes is a topic that was studied extensively, however; the existing approach for code construction assumes that every codeword should tolerate <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</i> asymmetric errors. Our main observation is that in contrast to symmetric errors, asymmetric errors are content dependent. For example, in Z-channels, the all-1 codeword is prone to have more errors than the all-0 codeword. This motivates us to develop nonuniform codes whose codewords can tolerate different numbers of asymmetric errors depending on their Hamming weights. The idea in a nonuniform codes' construction is to augment the redundancy in a content-dependent way and guarantee the worst case reliability while maximizing the code size. In this paper, we first study nonuniform codes for Z-channels, namely, they only suffer one type of errors, say 1→ 0. Specifically, we derive their upper bounds, analyze their asymptotic performances, and introduce two general constructions. Then, we extend the concept and results of nonuniform codes to general binary asymmetric channels, where the error probability for each bit from 0 to 1 is smaller than that from 1 to 0.