On Zero Error Capacity of Nearest Neighbor Error Channels with Multilevel Alphabet

TL;DR

This paper investigates the zero error capacity of nearest neighbor error channels with multilevel alphabets, deriving bounds and a formula by leveraging perfect Lee codes and linear programming techniques.

Contribution

It provides the first comprehensive bounds and a concise formula for the zero error capacity of NNE channels with multilevel alphabets.

Findings

Derived a lower bound based on perfect Lee codes.

Established an upper bound via linear programming.

Obtained a concise formula for the zero error capacity.

Abstract

This paper studies the zero error capacity of the Nearest Neighbor Error (NNE) channels with a multilevel alphabet. In the NNE channels, a transmitted symbol is a $d$-tuple of elements in $\{0,1,2,\dots, n-1 \}$. It is assumed that only one element error to a nearest neighbor element in a transmitted symbol can occur. The NNE channels can be considered as a special type of limited magnitude error channels, and it is closely related to error models for flash memories. In this paper, we derive a lower bound of the zero error capacity of the NNE channels based on a result of the perfect Lee codes. An upper bound of the zero error capacity of the NNE channels is also derived from a feasible solution of a linear programming problem defined based on the confusion graphs of the NNE channels. As a result, a concise formula of the zero error capacity is obtained using the lower and upper bounds.