The minimal generator matrix of a vector space

The atomic vectors of a finitely generated vector space C over a field F ar e characterized for C a subspace of the product vector space V = Pi(i=1)(n) V-i over F. For finite fields, the minimal trellis diagram for mixed-codes is determined, and this provides the L-section minimal trellis diagram for linear codes. As an example, an extremely simple yet comprehensive analysi s of the trellis structure of Reed-Muller codes is given. In particular, a trellis oriented generator matrix for the 2(l)-section minimal trellis diag ram of a I Reed-Muller code is presented.