A Singleton Bound for Lattice Schemes

The binary coding theory and subspace codes for random network coding exhibit similar structures. The method used to obtain a Singleton bound for subspace codes mimic the technique used in obtaining the Singleton bound for binary codes. This motivates the question of whether there is an abstract framework that captures these similarities. As a first step towards answering this question, we use the lattice framework proposed in [1]. A lattice is a partially ordered set in which any two elements have a least upper bound and a greatest lower bound. A ‘lattice scheme’ is defined as a subset of a lattice. In this paper, we derive a Singleton bound for lattice schemes and obtain Singleton bounds known for binary codes and subspace codes as special cases. The lattice framework gives additional insights into the behaviour of Singleton bound for subspace codes. We also obtain a new upper bound on the code size for non-constant dimension codes. The plots of this bound along with plots of the code sizes of known non-constant dimension codes in the literature reveal that our bound is tight for certain parameters of the code.