New Single Asymmetric Error-Correcting Codes

New single asymmetric error-correcting codes are proposed. These codes are better than existing codes when the code length n is greater than 10, except for n = 12 and n = 15 . In many cases one can construct a code C containing at least [2^n/n] codewords. It is known that a code with |C| >= [2^n/(n + 1)] can be easily obtained. It should be noted that the proposed codes for n = 12 and n = 15 are also the best known codes that can be explicitly constructed, since the best of the existing codes for these values of n are based on combinatorial arguments. Useful partitions of binary vectors are also presented

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

Limited magnitude error control codes

A relatively new model of error control is the limited magnitude error over high radix channels. In this error model, the error magnitude does not exceed a certain limit known beforehand. In this dissertation, we study systematic error control codes for common channels under the assumption that the maximum error magnitude is known a priori. Optimal codes correcting all asymmetric and symmetric errors are given. Further, as it is often the case that we only need to correct a small number of errors, codes that can correct a single error over asymmetric and symmetric channels are also proposed. The designed codes achieve higher code rates than single error correcting codes previously given in the literature. From the error detection point of view, we study both all and t error detecting codes for asymmetric/unidirectional channels and design close-to-optimal codes. Finally, we show how the all asymmetric error correcting codes proposed in this dissertation can be used to detect all symmetric errors

New bounds and constructions for error control codes

The bulk of the theory on error control codes has been developed under the fault assumption of random (symmetric) errors, where 1 → 0 and 0 → 1 errors are equally likely. In the past few years, several applications have emerged in which the observed errors are highly asymmetric. This has prompted the study of codes that offer a combination of symmetric and asymmetric error control capabilities. This research is a part of this ongoing study. The main results of the research are listed below. 1. New upper bounds on t-unordered codes. Exact bounds are established in some cases. 2. A new method for constructing constant weight distance four codes that gives the best known bounds in several cases. 3. A new method for constructing single asymmetric error correcting codes. The method establishes several new lower bounds. 4. A construction for symmetric error correcting code. The code is suited for a photon channel and other highly asymmetric channels because it has far fewer 1's than 0's. The code uses one extra bit of redundancy over the BCH code in almost all cases, and it is relatively easy to encode and decode. 5. A new construction for systematic double asymmetric error correcting code. The resulting code is easier to decode than the BCH code and is optimal in several cases. The code has fewer 1's than 0's. 6. A new construction for double symmetric error correcting linear code. The resulting code is easier to decode than the BCH code and is optimal in several cases. 7. A new construction for linear codes. The construction yields best known codes in many cases

Unidirectional error correcting/detecting codes

An extensive theory of symmetric error control coding has been developed in the last few decades. The recently developed VLSI circuits, ROM, and RAM memories have given an impetus to the extension of error control coding to include asymmetric and unidirectional types of error control. The maximal numbers of unidirectional errors which can be detected by systematic codes using r checkbits are investigated. They are found for codes with k, the number of information bits, being equal to 2[superscript r] and 2[superscript r] + 1. The importance of their characteristic in unidirectional error detection is discussed. A new method of constructing a systematic t-error correcting/all-unidirectional error detecting(t-EC/AUED) code, which uses fewer checkbits than any of the previous methods, is developed. It is constructed by appending t + 1 check symbols to a systematic t-error correcting and (t+l)-error detecting code. Its decoding algorithm is developed. A bound on the number of checkbits for a systematic t-EC/AUED code is also discussed. Bose-Rao codes, which are the best known single error correcting/all-unidirectional error detecting(SEC/AUED) codes, are completely analyzed. The maximal Bose-Rao codes for a fixed weight and for all weights are found. Of course, the base group and the group element which make the Bose-Rao code maximal are found, too. The bounds on the size of SEC/AUED codes are discussed. Nonsystematic single error correcting/d-unidirectional error detecting codes are constructed. Three methods for constructing the systematic t-error correcting/d-unidirectional error detecting(t-EC/d-UED) codes are developed. From these, simple and efficient t-EC/(t+2)-UED codes are derived. The decoding algorithm for one of these methods, which can be applied to the other two methods with slight modification, is described. A lower bound on the number of checkbits for a systematic t-EC/d-UED code is derived. Finally, future research efforts are proposed