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We construct error-correcting write-once memory (WOM) codes that guarantee correction of any specified number of errors in $q$ -level memories. The constructions use suitably designed short $q$ -ary WOM codes and concatenate them with outer error-correcting codes over different alphabets using suitably designed mappings. With a new storage-efficiency measure, we call EC-rate and show that for common error types the codes save redundancy and implementation complexity over straightforward concatenation. In addition to constructions for guaranteed error correction, we extend the error-correcting WOM scheme to binary multi-level coding for random errors, and toward soft-decision decoding provide an efficient way to extract reliability information without using higher-precision readout. [ABSTRACT FROM AUTHOR]

We propose a new structure of a self-checking combinational device where, based on the properties of parity and Berger codes, as well as a code with the detection of all double errors in information vectors, the problem of detecting all single faults of logical elements can be solved without transforming the structure of the source device. The properties of binary codes with the detection of all double errors that can be used in constructing the proposed structure are considered. We give an example of constructing a new structure. [ABSTRACT FROM AUTHOR]

We consider codes over fixed alphabets against worst case symbol deletions. For any fixed k \ge 2 , we construct a family of codes over alphabet of size k with positive rate, which allow efficient recovery from a worst case deletion fraction approaching 1-(2/(k+\sqrt k)) . In particular, for binary codes, we are able to recover a fraction of deletions approaching 1/(\sqrt 2 +1)=\sqrt 2-1 \approx 0.414 . Previously, even non-constructively, the largest deletion fraction known to be correctable with positive rate was 1-\Theta (1/\sqrt k) , and around 0.17 for the binary case. Our result pins down the largest fraction of correctable deletions for $k$ -ary codes as 1-\Theta (1/k)$ , since 1-1/k$ is an upper bound even for the simpler model of erasures where the locations of the missing symbols are known. Closing the gap between and 1/2 for the limit of worst case deletions correctable by binary codes remains a tantalizing open question. [ABSTRACT FROM PUBLISHER]

Improved lower and upper bounds on the size and the rate of grain-correcting codes are presented. The lower bound is Gilbert–Varshamov-like combined with a construction by Gabrys et al., and it improves on the previously best known lower bounds on the asymptotic rate of \lceil \tau n \rceil -grain-correcting codes of length n on the interval [0,0.0668] . One of the two newly presented upper bounds improves on the best known upper bounds on the asymptotic rate of \lceil \tau n \rceil -grain-correcting codes of length n on the interval \tau \in (0,{1}/{8}] and meets the lower bound of {1}/{2} for \tau \geq 1/8 . Moreover, in a nonasymptotic regime, both upper bounds improve on the previously best known results on the largest size of t -grain-correcting codes of length $n$ , for certain values of n$ and t$ . Constructions of 1-grain-correcting codes based on a partitioning technique are presented for lengths up to 18. Finally, a lower bound of on the minimum redundancy of $\infty $ -grain-detecting codes of length $n$ is presented. [ABSTRACT FROM PUBLISHER]

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. [ABSTRACT FROM AUTHOR]