# Multipermutation Ulam Sphere Analysis Toward Characterizing Maximal Code Size

## TL;DR

This paper investigates the Ulam metric for multipermutation codes, analyzing sphere sizes and bounds to enhance understanding of code capacity for applications like flash memory.

## Contribution

It extends the Ulam metric analysis from permutations to multipermutations, providing new bounds and insights for code size optimization.

## Key findings

- Sphere sizes for multipermutations under the Ulam metric are characterized.
- Bounds on maximum code size are derived for multipermutation codes.
- Differences between permutation and multipermutation Ulam metrics are identified.

## Abstract

Permutation codes, in the form of rank modulation, have shown promise for applications such as flash memory. One of the metrics recently suggested as appropriate for rank modulation is the Ulam metric, which measures the minimum translocation distance between permutations. Multipermutation codes have also been proposed as a generalization of permutation codes that would improve code size (and consequently the code rate). In this paper we analyze the Ulam metric in the context of multipermutations, noting some similarities and differences between the Ulam metric in the context of permutations. We also consider sphere sizes for multipermutations under the Ulam metric and resulting bounds on code size.