4 research outputs found
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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
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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
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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