Minimal trellis construction from convolutional ring encoders

The paper addresses minimality of encoders for basic convolutional codes over Zpr by using a recently developed concept of row reducedness for polynomial matrices over Zpr . It is known in the literature that the McMillan degree of a basic encoder is an upper bound for the minimum number of trellis states, but a general expression is missing. This open problem is solved in this paper. An expanded type of polynomial encoder is introduced, called “p-encoder”, whose rows are required to be a p-generator sequence. The latter property enables the working of fundamental linear algebraic properties, such as linear independence. It is shown that for any basic convolutional code a particular type of p-encoder can be constructed that is the ring analog of a canonical encoder from the field case. The open problem of constructing a minimal trellis representation of the code is then solved and the minimum number of trellis states is expressed in terms of an algebraic degree invariant of the code. In the literature this problem was only solved for the restrictive case where the code admits a basic row reduced encoder. The obtained results hold for the classical set-up of left compact support convolutional codes. It is also shown how they are extended to finite support convolutional codes.