Quasi-cyclic structure of Reed-Muller codes and their smallest regular trellis diagram

The largest quasi-cyclic subcode of the Reed-Muller code R(r,m), invariant under the shift T^2m-l is determined. This code, denoted QCR(r,m,l), is presented through its module decomposition into cyclic submodules. The smallest regular trellis diagram (SRTD) is defined for block codes. This trellis and its construction algorithm are given for the class of cyclic-form codes. Using the cyclic-form structure of QCR(r,m,l), the 2^l-section SRTD of this code is determined. The eight-section SRTD is given for the Reed-Muller codes R(r,m). The quasi-cyclic subcodes of R(r,m) with regular 2^l-section minimal trellis diagrams are presented