Linear Codes over Finite Rings

Linear codes over finite rings with identity have recently raised a great interest for their new role in algebraic coding theory and for their successful application in combined coding and modulation. Thus, in this paper we address the problems of constructing of new cyclic, BCH, alternant, Goppa and Srivastava codes over local finite commutative rings with identity. These constructions are very similar to the ones over finite fields and these constructions requires working on Galois extension rings, where some properties of the Galois extension fields are lost. Recent developments have contributed toward achieving the reliability required by todays high-speed digital systems, and the use of coding for error control has become an integral part in the design of modern communication systems and digital computers. Moreover, we mention that the investigation of codes over finite alphabets (for example, finite rings) which are less structured than finite fields may be more appropriate to use for computer-to-computer communication. This paper is organized as follows. In Section 2, we present a construction technique of cyclic codes over a commutative ring with identity. In Section 3, we present a construction technique of BCH and alternant codes over local finite commutative rings with identity. In Section 4, we describe a construction technique of Goppa and Srivastava codes over local finite commutative rings with identity.