Codes on Lattices for Random SAF Routing

Abstract: In this paper, a construction of constant weight codes based on the unique decomposition of elements in lattices is presented. The conditions for unique primary decomposition and unique irreducible decomposition in lattices are discussed and connections with the decomposition of ideals in Noetherian commutative rings established. In this context it is shown, drawing on the definitive works of Dilworth, Ward and others, that, as opposed to Noetherian commutative rings, the existence of unique irreducible decomposition in lattices does not guarantee unique primary decomposition. The source alphabet in our proposed construction is a set of uniquely decomposable elements constructed from a chosen subset of irreducible or primary elements of the appropriate lattice. The distance function between two lattice elements is based on the symmetric distance between sets of constituent elements. It is known that constructing such constant weight codes is equivalent to constructing a Johnson graph with appropriate parameters. Some bounds on the code sizes are also presented and a method to obtain codes of optimal size, utilizing the Johnson graph description of the codes, is discussed. As an application we show how these codes can be used for error and erasure correction in random networks employing store-and-forward (SAF) routing.