Abstract
One of the most important and challenging problems in coding theory is to construct codes with good parameters. There are various methods to construct codes with the best possible parameters. A promising and fruitful approach has been to focus on the class of quasi-twisted (QT) codes which includes constacyclic codes as a special case. This class of codes is known to contain many codes with good parameters. Although constacyclic codes and QT codes have been the subject of numerous studies and computer searches over the last few decades, we have been able to discover a new fundamental result about the structure of constacyclic codes which was instrumental in our comprehensive search method for new QT codes over GF(7). We have been able to find 41 QT codes with better parameters than the previously best known linear codes. Furthermore, we derived a number of additional new codes via Construction X as well as standard constructions, such as shortening and puncturing.
Highlights
One of the goals in coding theory is to construct linear codes with optimal parameters
A record of parameters and constructions for existing BKLCs is maintained in the online database of best known linear codes [15], which is a major source of reference for coding theory researchers
We first review some of the basic facts about constacyclic and QT codes, describe our search method and present the new codes we obtained
Summary
One of the goals in coding theory is to construct linear codes with optimal parameters. There exist theoretical upper bounds on the value of the minimum distance for a code with a given set of parameters; those codes with known constructions that come as close as possible to this upper bound are referred to as best known linear codes (BKLCs). A record of parameters and constructions for existing BKLCs is maintained in the online database of best known linear codes [15], which is a major source of reference for coding theory researchers. Many generalizations of cyclic codes have been introduced over the years Among these generalizations is the class of quasi-twisted (QT) codes that turned out to be very promising in constructing codes with best known parameters. We first review some of the basic facts about constacyclic and QT codes, describe our search method and present the new codes we obtained
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