Construction of minimal tail-biting trellises

The subject of tail-biting trellis representation of block codes has attracted a lot of interest. One of the major open problems in this area is how to construct a minimal tail-biting trellis for a given linear code. We present a solution to this problem. Given a linear block code C, we show how to construct a linear tail-biting trellis for C that minimizes the product of vertex-space sizes among all possible linear tail-biting trellises for C. Our methods are based on the new concept of the characteristic matrix for a linear block code. The complexity of the resulting construction is O(n²) for a code of length n