On Grid Codes

TL;DR

The paper extends grid-code theory by formulating grid-specific analogues of the Hamming and Gilbert–Varshamov bounds for codes in $\mathfrak{G}=\prod_{i=1}^n[0,m_i-1]$ under the Manhattan metric, using a local distance enumerator $p(t,a)$ to compute $r$-ball sizes and enabling explicit expressions for $\eta_r(\mathfrak{G})$ and $\gamma_r(\mathfrak{G})$. It develops a comprehensive framework relating Manhattan, Hamming, and Lee distances, and provides both theoretical bounds and practical SageMath implementations. In addition, the authors analyze grid codes that are cyclic subgroups of abelian groups, giving a method to compute the minimum Hamming distance and deriving lower bounds on the minimum Manhattan distance via the generator’s support, with illustrative examples. The work offers a cohesive toolkit for bounding code sizes and distances on multidimensional grids, with potential applications in crystallography, porous media modeling, and grid-structured digital data.

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.

Figures (5)

• Figure 1: The $2$-balls centered at $\mathscr{C}=\{00,41\}$ cover $\mathfrak{G}$.
• Figure 2: The $1$-balls centered at $\mathscr{C}'=\{01,20,41\}$ cover $\mathfrak{G}$.
• Figure 3: If $\mathfrak{G}=[0,3]^2$, the local distance enumerator polynomial of $a=(1,1)$ in $\mathfrak{G}$ is $p(t,a)=(1+2t+t^2)^{2}=\textbf{1}+\textbf{4}t+\textbf{6}t^2+\textbf{4}t^3+\textbf{1}t^4$. Hence $p(1,a)_{\leq 2}=\textbf{1}+\textbf{4}\cdot 1+\textbf{6}\cdot 1^2=11=|B_{2}(a)|$.
• Figure 4: The $39$ elements represented by the cerulean dots in $\mathfrak{G}$ coincide with the computation of $\eta_{8}([0,6]\times[0,6])$ given by Algorithm \ref{['algoeta']}.
• Figure :

Theorems & Definitions (27)

• Theorem 3.1: Hamming bound
• proof
• Example 3.2
• Remark 4.1
• Theorem 4.2: Gilbert–Varshamov bound
• proof
• Theorem 5.1
• proof
• Example 5.2
• Lemma 5.3