On Error Correcting Codes and Their Applications

Abstract

To expand the use of codes and provide a form of error-correction, it is useful to extend the use of binary streams into another representation. In this work, we have used different monoid rings for the construction of a new family of error correcting codes having better error correction capability. Initially we have constructed binary cyclic codes using monoid rings instead of polynomial ring. For an n length binary cyclic code, three di¤erent binary cyclic codes of length an; bn and abn are obtain. These codes are interleaved codes capable of correcting burst of errors alongwith random error correction. The BCH codes form a class of parameterized error-correcting codes which have been the subject of interest. Instead of primitive BCH codes we have showed the existence of non- primitive BCH codes of length bn over the elds F2, F4 and nite rings Z2m along with their applications. The value of b is investigated for which the existence of the non-primitive BCH code Cbn is assured. It is noticed that the code Cn is embedded in the code Cbn. Therefore, the data transmitted by the code Cn can also be transmitted by the code Cbn. The BCH codes Cbn have better error correction capability whereas the BCH code Cn has better code rate.