Ulam Sphere Size Analysis for Permutation and Multipermutation Codes Correcting Translocation Errors

TL;DR

This paper introduces a new method using Young Tableaux to analyze permutation and multipermutation sphere sizes in the Ulam metric, proving non-existence of perfect codes and establishing bounds for code sizes.

Contribution

It presents a novel approach to calculating sphere sizes with Young Tableaux and extends the analysis to multipermutations, providing bounds on code sizes in the Ulam metric.

Findings

Young Tableaux method for sphere size calculation

Non-existence of non-trivial perfect permutation codes

Bounds on multipermutation code sizes

Abstract

Permutation and multipermutation codes in the Ulam metric have been suggested for use in non-volatile memory storage systems such as flash memory devices. In this paper we introduce a new method to calculate permutation sphere sizes in the Ulam metric using Young Tableaux and prove the non-existence of non-trivial perfect permutation codes in the Ulam metric. We then extend the study to multipermutations, providing tight upper and lower bounds on multipermutation Ulam sphere sizes and resulting upper and lower bounds on the maximal size of multipermutation codes in the Ulam metric.