Cross-Error Correcting Integer Codes over $\mathbb{Z}_{2^m}$

TL;DR

This paper studies error-correcting codes over the ring ^m for correcting single-coordinate errors with bounded magnitude, providing bounds, constructions, and decoding algorithms for small lengths.

Contribution

It introduces new bounds and explicit constructions for cross-error correcting codes over ^m, including decoding algorithms for small code lengths.

Findings

Derived upper bounds on cross-error correcting codes.

Constructed linear codes for lengths 2 and 3.

Developed decoding algorithms for these codes.

Abstract

In this work we investigate codes in $\mathbb{Z}_{2^m}^n$ that can correct errors that occur in just one coordinate of the codeword, with a magnitude of up to a given parameter $t$. We will show upper bounds on these cross codes, derive constructions for linear codes and respective decoding algorithm. The constructions (and decoding algorithms) are given for length $n = 2$ and $n = 3$, but for general $m$ and $t$.