Difference-set cyclic codes

Codes exist which are capable of correcting large numbers of random errors. Such codes are rarely used in practical data transmission systems, however, because the equipment necessary to realize their capabilities — that is, to actually correct the errors — is usually prohibitively complex and expensive. The problem of finding simply implemented decoding algorithms or, equivalently, codes which can be decoded simply with existing methods, is perhaps the outstanding unsolved problem in coding theory today. In this paper, a new class of random-error-correcting cyclic codes is defined. These codes have two very desirable features: the binary members of the class are nearly as powerful as the best-known codes in the range of interest, and they can be decoded with the simplest known decoding algorithm. Unfortunately there are relatively few codes with useful parameters in this class, despite the fact that the class is infinite.