Error recognition in the Cantor cube

TL;DR

This paper investigates error detection protocols in the Cantor cube using topological methods, showing that all such protocols are Baire spaces and that the cube can be decomposed into thin sets analogous to parity bits.

Contribution

It introduces a topological framework for analyzing error detection protocols in the Cantor cube and demonstrates the existence of a decomposition into thin sets related to parity.

Findings

Protocols are Baire spaces and generic ones are not Borel or meager.

The Cantor cube can be decomposed into two thin sets akin to parity bits.

Decomposition relates to xor-sets from previous research.

Abstract

Based on the notion of thin sets introduced recently by T.~Banakh, Sz.~G\l{}\k{a}b, E.~Jab\l{}o\'nska and J.~Swaczyna we deliver a study of the infinite single-message transmission protocols. Such protocols are associated with a set of admissible messages (i.e. subsets of the Cantor cube $\mathbb{Z}_2^\omega$). Using Banach-Mazur games we prove that all protocols detecting errors are Baire spaces and generic (in particular maximal) ones are not neither Borel nor meager. We also show that the Cantor cube can be decomposed to two thin sets which can be considered as the infinite counterpart of the parity bit. This result is related to so-called xor-sets defined by D.~Niwi\'nski and E.~Kopczy\'nski in 2014.