Dimension/length profiles and trellis complexity of linear block codes

This semi-tutorial paper discusses the connections between the dimension/length profile (DLP) of a linear code, which is essentially the same as its " generalized Hamming weight hierarchy " 111, and the complexity of its minimal trellis diagram. These connections are close andtdeep. DLP duality is closely related to trellis duality. The DLP of a code gives tight bounds on its state and branch complexity profiles under any coordinate ordering; these bounds can often be met. A maximum distance separable (MDS) code is characterized by a certain extrema1 DLP, from which the main properties of MDS codes are easily derived. The simplicity and generality of these interrelationships are emphasized.