Linear tail-biting trellises, the square-root bound, and applications for Reed-Muller codes

Linear tail-biting trellises for block codes are considered. By introducing the notions of subtrellis, merging interval, and sub-tail-biting trellis. some, structural properties of linear tail-biting trellises are proved. It is shown that a linear tail-biting, trellis always has a certain simple structure, the parallel-merged-cosets structure. A necessary condition required from a Linear code in order to have a linear tail-biting trellis representation that achieves the square-root bound is presented. Finally, the above condition is used to show that for r greater than or equal to 2 and m greater than or equal to 4r - 1 or r greater than or equal to 4 and r + 3 less than or equal to m less than or equal to [(4r + 5)/3] the Reed-Muller code RM(r, m) under any bit order cannot be represented by a Linear tail-biting trellis whose state complexity is half of that of the minimal (conventional) trellis For the code under the standard bit order.