Characteristic Matrices and Trellis Reduction for Tail-Biting Convolutional Codes

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 G 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 G′ equivalent to G based on a characteristic matrix. In this case, if the polynomial generator matrix corresponding to G′ 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 G′ 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.