| In this dissertation maximum distance separable convolutional codes are introduced and studied.; In particular, the Singleton Bound on the minimum distance of block codes is generalized to an upper bound on the free distance of convolutional codes. The convolutional codes attaining this bound will be called maximum distance separable convolutional codes, or shortly, MDS codes. A general construction of a rate k/n, degree δ, MDS convolutional code will be provided, starting with a large Reed Solomon block code.; Following the same direction, strongly MDS convolutional codes are defined in the case of rate 1/2. These are codes having optimal column distances. Properties of these codes are given and a concrete construction is provided. This construction has the advantage that the field required has the fewest elements among all other existing constructions. Finally, a decoding algorithm for these codes is given, along with several properties that improve on the time of decoding.; The input-state-output representation of a convolutional code is also introduced. A construction of rate 1/n MDS convolutional codes is developed and a theoretical algebraic geometric proof of the existence of a general k/n MDS convolutional code is given using this representation. This proof sets the MDS convolutional codes into the framework of algebraic geometry, which generalizes the algebraic geometric setting for the Reed Solomon block codes.; Finally the class of binary unit memory convolutional codes that are MDS is studied. Some algebraic constructions of convolutional codes over small fields are given. |
