On the Theory of Linear Trellises

Trellis linearity, rst considered by McEliece in 1996, turns out to be crucial in the study of tail-biting trellises. In this chapter, basic structural properties of linear trellises are investigated. A rigorous deeni-tion of linearity is given for both conventional and tail-biting trellises. An algorithm that determines in polynomial time whether a given trellis is linear is then derived. The relationship between linear trellises and the trellis product construction is discussed. The key result of this chapter is that a trellis | either conventional or tail-biting | is linear if and only if it factors into a product of elementary trellises. Some consequences of this factorization of linear trellises are established.