On Grid Codes

TL;DR

This paper explores the combinatorial properties of grid codes using generating functions, providing explicit bounds and relations between different distance metrics, with applications demonstrated through SageMath implementations.

Contribution

It introduces explicit formulas for grid sphere sizes, extends classical bounds to grid codes, and relates Hamming, Manhattan, and Lee distances in abelian groups.

Findings

Explicit formulas for minimum and maximum size of r-balls in grids.

Extended Hamming and Gilbert-Varshamov bounds to grid codes.

Lower bounds for Manhattan distances based on Hamming and Lee distances.

Abstract

Generating functions for the size of a $r$-sphere, with respect to the Manhattan distance in an $n$-dimensional grid, are used to provide explicit formulas for the minimum and maximum size of an $r$-ball centered at a point of the grid. This allows us to offer versions of the Hamming and Gilbert-Varshamov bounds for codes in these grids. Relations between the Hamming, Manhattan, and Lee distances defined in an abelian group $G$ are studied. A formula for the minimum Hamming distance of codes that are cyclic subgroups of $G$ is presented. Furthermore, several lower bounds for the minimum Manhattan distance of these codes based on their minimum Hamming and Lee distances are established. Examples illustrating the main results are presented, including several SageMath implementations.