Structural analysis of error-correcting codes for discrete channels that involve combinations of three basic error types

TLDR: It can be said that unique decodability is decidable also in the presence of errors in the class of bounded error effects channels (BEE channels), which are used to model, for instance, scattered errors and bursts of errors of any combination of the three error types.

Abstract: A nonprobabilistic mathematical model is introduced of discrete channels that involve the error types substitution, insertion, and deletion. The model is based on the novelty that errors can be expressed as strings over an alphabet of basic error symbols. Some general conditions on errors are defined which bound the error effects on messages, obtaining thus the class of bounded error effects channels (BEE channels). These channels can be used to model, for instance, scattered errors and bursts of errors of any combination of the three error types. A general notion of error-correcting code is defined and a characterization of the error-correcting codes for a given BEE channel is obtained. Then, an algorithm is presented that tests, for a given finite code (not necessarily of fixed length) and a given description of a BEE channel, whether the code is error-correcting for the channel defined by the given description. This result can be considered as an extension of the well-known theorem of Sardinas and Patterson (1953) for testing the unique decodability of a given finite code. In this sense, it can be said that unique decodability is decidable also in the presence of errors.