A Singleton Bound for Lattice Schemes

TL;DR

This paper establishes a Singleton bound for lattice schemes, generalizing known bounds for binary and subspace codes, and introduces a new upper bound for non-constant dimension codes, highlighting the influence of modular structure.

Contribution

It derives a unified Singleton bound for lattice schemes, showing the impact of modular structure and providing a tight new upper bound for non-constant dimension codes.

Findings

Singleton bound for lattice schemes derived

Bound is tight for certain code parameters

New upper bound for non-constant dimension codes

Abstract

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. It is shown that the modular structure affects the strength of the Singleton bound. 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.