Tilings with $n$-Dimensional Chairs and their Applications to Asymmetric Codes

TL;DR

This paper explores tilings of n-dimensional chairs and their applications in designing codes for asymmetric errors and write-once memories, providing constructions and non-existence proofs.

Contribution

It introduces a new connection between tilings of n-dimensional chairs and coding theory, including constructions and non-existence results for perfect codes.

Findings

Established an equivalence between lattice tilings and generalized splitting sequences.

Provided constructions for perfect codes correcting asymmetric errors with limited-magnitude.

Proved cases where such perfect codes cannot exist.

Abstract

An $n$-dimensional chair consists of an $n$-dimensional box from which a smaller $n$-dimensional box is removed. A tiling of an $n$-dimensional chair has two nice applications in coding for write-once memories. The first one is in the design of codes which correct asymmetric errors with limited-magnitude. The second one is in the design of $n$ cells $q$-ary write-once memory codes. We show an equivalence between the design of a tiling with an integer lattice and the design of a tiling from a generalization of splitting (or of Sidon sequences). A tiling of an $n$-dimensional chair can define a perfect code for correcting asymmetric errors with limited-magnitude. We present constructions for such tilings and prove cases where perfect codes for these type of errors do not exist.