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/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.
展开▼
机译：研究了根格A / sub n /，D / sub n /，E / sub n /及其对偶的网格复杂度。使用N（网格中不同路径的数量）作为网格的网格复杂度的度量，如果网格将N最小化，则称为最小网格。证明了先前发现的上述某些网格的网格图（D / sub n /，n奇数，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 / *和E / sub 7 / *）最小。我们还获得了A / sub 7 / *和A / sub 8 / *的最小网格。已知具有编码增益/ spl gamma /的n维晶格的任何网格的复杂度N满足N / spl ges // spl gamma // sup n / 2 /。在这里，对于许多根晶格及其对偶，此下限得到了改善。对于A / sub n /和A / sub n / *晶格，我们还针对任意维数n的低复杂度网格提出了简单的构造，并得出了所构造网格的复杂性的严格上限。在某些维度上，构造的网格最小，而对于其他一些n值，它们的复杂度比以前已知的网格低。
展开▼
