Limited-magnitude Errors Research Articles

Snake-in-the-Box Codes for Rank Modulation

Motivated by the rank-modulation scheme with applications to flash memory, we consider Gray codes capable of detecting a single error, also known as snake-in-the-box codes. We study two error metrics: Kendall's τ-metric, which applies to charge-constrained errors, and the ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> -metric, which is useful in the case of limited-magnitude errors. In both cases, we construct snake-in-the-box codes with rate asymptotically tending to 1. We also provide efficient successor-calculation functions, as well as ranking and unranking functions. Finally, we also study bounds on the parameters of such codes.

Systematic, Single Limited Magnitude Error Correcting Codes for Flash Memories

A relatively new model of error correction is the limited magnitude error model. That is, it is assumed that the absolute difference between the sent and received symbols is bounded above by a certain value l. In this paper, we propose systematic codes for asymmetric limited magnitude channels that are able to correct a single error. We also show how this construction can be slightly modified to design codes that can correct a single symmetric error of limited magnitude. The designed codes achieve higher code rates than single error correcting codes previously given in the literature.

Codes for Asymmetric Limited-Magnitude Errors With Application to Multilevel Flash Memories

Several physical effects that limit the reliability and performance of multilevel flash memories induce errors that have low magnitudes and are dominantly asymmetric. This paper studies block codes for asymmetric limited-magnitude errors over q-ary channels. We propose code constructions and bounds for such channels when the number of errors is bounded by t and the error magnitudes are bounded by l. The constructions utilize known codes for symmetric errors, over small alphabets, to protect large-alphabet symbols from asymmetric limited-magnitude errors. The encoding and decoding of these codes are performed over the small alphabet whose size depends only on the maximum error magnitude and is independent of the alphabet size of the outer code. Moreover, the size of the codes is shown to exceed the sizes of known codes (for related error models), and asymptotic rate-optimality results are proved. Extensions of the construction are proposed to accommodate variations on the error model and to include systematic codes as a benefit to practical implementation.