On the Construction and Dimensionality of Linear Block Code Trellises

The trellis construction methods of Wolf [1], Massey [2], and Forey [3] for general linear block codes are briefly reviewed. An isomorphism between a. trellis constructed using Massey's method and one constructed using Wolf's method is derived. It is confirmed that Wolf's and Massey's trellis constructions also yield minimal trellises. Two simplified methods for minimal trellis construction ate presented, along with a. method to calculate the trellis dimensions that is art alternative to the methods of [2] and [3]. An improvement is found to a lower bound on the maximum trellis dimension due to Muder [4]. It is shown that when equivalent codes are constructed by permutations of the symbol positions the resulting trellis dimensions are fixed near either end, while in the central portion of the trellis the dimensions vary between an attainable upper bound and a lower bound. From the lower bound on the trellis dimensions in the central portion of the trellis it is seen that only codes (and their duals) that meet a certain condition on their minimum distances can possibly have a trellis with a relatively small number of states.