Network coding with modular lattices

In [13], Kötter and Kschischang presented a new model for error correcting codes in network coding. The alphabet in this model is the subspace lattice of a given vector space, a code is a subset of this lattice and the used metric on this alphabet is the map d ∶ (U,V ) ↦ dim(U + V ) − dim(U ∩ V ). In this paper we generalize this model to arbitrary modular lattices, i.e. we consider codes, which are subsets of modular lattices. The used metric in this general case is the map d ∶ (u, v) ↦ h(u ∨ v) − h(u ∧ v), where h is the height function of the lattice. We apply this model to submodule lattices. Moreover, we show a method to compute the size of spheres in certain modular lattices and present a sphere packing bound, a sphere covering bound, and a singleton bound for codes, which are subsets of modular lattices. 2010 Mathematics Subject Classification: 06C05, 68P30, 94B65, 05A15, 20K27.