On the trellis complexity of root lattices and their duals
Trellis complexity of root Lattices A(n), D-n, E-n, and their duals is investigated. Using N, the number of distinct paths in a trellis, as the measure of trellis complexity for Lattices, a trellis is called minimal if it minimizes N. It is proved that the previously discovered trellis diagrams of some of the above lattices (D-n, n odd, A(n), 4 less than or equal to n less than or equal to 9, A(4)*, A(5)*, A(6)*, A(9)* E-6, E-6*, and E-7*) are minimal. We also obtain minimal trellises fur A(7)* and A(8)*. It is known that the complexity N of any trellis of an n-dimensional lattice with coding gain gamma satisfies N greater than or equal to gamma(n/2). sere, this laser bound is improved for many of the root lattices and their duals. For A(n) and A(n)(*) lattices, we also propose simple constructions for low-complexity trellises in an arbitrary dimension ni and derive tight upper bounds on the complexity of the constructed trellises. In some dimensions, the constructed trellises are minimal, while for some other values of n they have lower complexity than previously known trellises.
trellis complexity root lattices lattices minimal trellises complexity bounds trellis construction