  Trellises are crucial graphical representations of codes. While conventional
trellises are well understood, the general theory of (tail-biting) trellises is
still under development. Iterative decoding concretely motivates such theory.
In this paper we first develop a new algebraic framework for a systematic
analysis of linear trellises which enables us to address open foundational
questions. In particular, we present a useful and powerful characterization of
linear trellis isomorphy. We also obtain a new proof of the Factorization
Theorem of Koetter/Vardy and point out unnoticed problems for the group case.
  Next, we apply our work to: describe all the elementary trellis
factorizations of linear trellises and consequently to determine all the
minimal linear trellises for a given code; prove that nonmergeable one-to-one
linear trellises are strikingly determined by the edge-label sequences of
certain closed paths; prove self-duality theorems for minimal linear trellises;
analyze quasi-cyclic linear trellises and consequently extend results on
reduced linear trellises to nonreduced ones. To achieve this, we also provide
new insight into mergeability and path connectivity properties of linear
trellises.
  Our classification results are important for iterative decoding as we show
that minimal linear trellises can yield different pseudocodewords even if they
have the same graph structure.
