  Basic properties of a characteristic matrix for a tail-biting convolutional
code are investigated. A tail-biting convolutional code can be regarded as a
linear block code. Since the corresponding scalar generator matrix Gt has a
kind of cyclic structure, an associated characteristic matrix also has a cyclic
structure, from which basic properties of a characteristic matrix are obtained.
Next, using the derived results, we discuss the possibility of trellis
reduction for a given tail-biting convolutional code. There are cases where we
can find a scalar generator matrix Gs equivalent to Gt based on a
characteristic matrix. In this case, if the polynomial generator matrix
corresponding to Gs has been reduced, or can be reduced by using appropriate
transformations, then trellis reduction for the original tail-biting
convolutional code is realized. In many cases, the polynomial generator matrix
corresponding to Gs has a monomial factor in some column and is reduced by
dividing the column by the factor. Note that this transformation corresponds to
cyclically shifting the associated code subsequence (a tail-biting path is
regarded as a code sequence) to the left. Thus if we allow partial cyclic
shifts of a tail-biting path, then trellis reduction is accomplished.
