On the trellis complexity of root lattices and their duals

Abstract

Trellis complexity of root lattices A/sub n/, D/sub n/, E/sub 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/sub n/, n odd, A/sub n/, 4/spl les/n/spl les/9, A/sub 4/*, A/sub 5/*, A/sub 6/*, A/sub 9/*, E/sub 6/, E/sub 6/*, and E/sub 7/*) are minimal. We also obtain minimal trellises for A/sub 7/* and A/sub 8/*. It is known that the complexity N of any trellis of an n-dimensional lattice with coding gain /spl gamma/ satisfies N/spl ges//spl gamma//sup n/2/. Here, this lower bound is improved for many of the root lattices and their duals. For A/sub n/ and A/sub n/* lattices, we also propose simple constructions for low-complexity trellises in an arbitrary dimension n, 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.