New Bounds for Permutation Codes in Ulam Metric

TL;DR

This paper introduces improved bounds on the size of permutation codes in the Ulam metric through integer programming and probabilistic methods, advancing the understanding of code limits and providing new code constructions.

Contribution

It presents the first integer-programming upper bounds that outperform existing bounds and develops probabilistic lower bounds that surpass previous results for large minimum distances.

Findings

Integer-programming bounds improve on Singleton bounds for some lengths.

Probabilistic bounds are tighter for large minimum distances.

Computer search results offer new permutation code examples.

Abstract

New bounds on the cardinality of permutation codes equipped with the Ulam distance are presented. First, an integer-programming upper bound is derived, which improves on the Singleton-type upper bound in the literature for some lengths. Second, several probabilistic lower bounds are developed, which improve on the known lower bounds for large minimum distances. The results of a computer search for permutation codes are also presented.