Network coding with modular lattices

TL;DR

This paper extends the network coding error correction model from subspace lattices to arbitrary modular lattices, introducing new metrics, bounds, and methods for code analysis within these algebraic structures.

Contribution

It generalizes the existing subspace lattice model to modular lattices, providing new metrics, sphere size computations, and bounds for codes in these broader algebraic frameworks.

Findings

Introduces a generalized metric for modular lattices.

Provides methods to compute sphere sizes in modular lattices.

Establishes bounds like sphere packing, covering, and singleton for codes.

Abstract

In [1], K\"otter 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) \longmapsto dim(U + V) - dim(U \bigcap 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: (x, y) \longmapsto h(x \bigvee y) - h(x \bigwedge y), 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. [1] R. K\"otter, F.R. Kschischang: Coding for errors and erasures in random network coding, IEEE Trans. Inf. Theory, Vol. 54, No. 8, 2008