Algebraic Systems, Trellis Codes, and Rotational Invariance

Publisher Summary The major goal of coding theory is the design of error-correction codes, a type of code used to transmit information reliably over an unreliable medium. This chapter introduces notation and concepts related to error-correction coding and trellis codes. It addresses the structural aspects of trellis codes. The chapter illustrates diagramming techniques, the concepts of equivalence, minimality, and the finite input memory systems of trellis codes. It also focuses on the error-correcting functions of trellis codes. The techniques used for maximum likelihood decoding, calculation of free Euclidean distance, and heuristic code design are briefly reviewed in the chapter. Furthermore, the chapter reviews prior results and present several new results on the structure and behavior of finite group homatons (FGHs), focusing on their relevance to trellis codes. The chapter also explores the way to find new trellis codes using FGHs. It is important to note that the chapter focuses mainly on the structure of the encoder, rather than on the constellation.