New Constructions of Codes for Asymmetric Channels via Concatenation

We present new constructions of codes for asymmetric channels for both binary and nonbinary alphabets, based on methods of generalized code concatenation. For the binary asymmetric channel, our methods construct nonlinear single-error-correcting codes from ternary outer codes. We show that some of the Varshamov-Tenengol'ts-Constantin-Rao codes, a class of binary nonlinear codes for this channel, have a nice structure when viewed as ternary codes. In many cases, our ternary construction yields even better codes. For the nonbinary asymmetric channel, our methods construct linear codes for many lengths and distances which are superior to the linear codes of the same length capable of correcting the same number of symmetric errors

On asymmetric error-correcting codes

Historically, coding theory has dealt with binary codes correcting symmetric errors, in which errors are made in both 0 and 1 bits with equal likelihood. Within the past ten years, some study has been made of asymmetric codes, under the assumption that the only errors which occur are errors in which 1 becomes 0. This thesis continues this study. We first examine systematic asymmetric codes, binary codes for which information and check portions are in distinct bit fields. This is a new area of study in coding theory. We establish that systematic asymmetric codes can have higher information rates than systematic symmetric codes, but not too much higher. We also give a construction for building systematic codes from smaller ones, with necessary and sufficient conditions for the codes so built to be systematic asymmetric codes. Finally, we examine Constantin-Rao codes and their extension to multiple asymmetric error correction. We show that such codes are not systematic and describe conditions under which they are closed under complements. We also show that the multiple asymmetric error correcting codes can have higher information rates than their symmetric counterparts

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

Adaptively correcting quantum errors with entanglement

Contrary to the assumption that most quantum error-correcting codes (QECC) make, it is expected that phase errors are much more likely than bit errors in physical devices. By employing the entanglement-assisted stabilizer formalism, we develop a new kind of error-correcting protocol which can flexibly trade error correction abilities between the two types of errors, such that high error correction performance is achieved both in symmetric and in asymmetric situations. The characteristics of the QECCs can be optimized in an adaptive manner during information transmission. The proposed entanglement-assisted QECCs require only one ebit regardless of the degree of asymmetry at a given moment and can be decoded in polynomial time.Comment: 5 pages, final submission to ISIT 2011, Saint-Petersburg, Russi

Coding over Sets for DNA Storage

In this paper, we study error-correcting codes for the storage of data in synthetic deoxyribonucleic acid (DNA). We investigate a storage model where data is represented by an unordered set of $M$ sequences, each of length $L$. Errors within that model are losses of whole sequences and point errors inside the sequences, such as substitutions, insertions and deletions. We propose code constructions which can correct these errors with efficient encoders and decoders. By deriving upper bounds on the cardinalities of these codes using sphere packing arguments, we show that many of our codes are close to optimal.Comment: 5 page