Improved bounds on the size of permutation codes under Kendall $τ$-metric
Farzad Parvaresh, Reza Sobhani, Alireza Abdollahi, Javad Bagherian, Fatemeh Jafari, Maryam Khatami
TL;DR
This work advances the theory of permutation codes under the Kendall $\tau$-metric within rank modulation, addressing fundamental questions about the maximal size $P(n,d)$ of codes with given minimum distance. It combines constructive coset-based methods and algorithmic search to derive new lower bounds, proves exact values such as $P(n,d)=4$ for broad distance ranges, and establishes $P(7,12)=7$ via equidistant code arguments. On the upper-bound side, it derives a prime-$n$ specific bound for $P(n,3)$ that improves prior results and yields sharper asymptotics for large prime indices. The paper also provides extensive computational appendices and demonstrates the practicality of coset-based constructions for small $n$, enhancing understanding of permutation codes in rank-modulation systems.
Abstract
In order to overcome the challenges caused by flash memories and also to protect against errors related to reading information stored in DNA molecules in the shotgun sequencing method, the rank modulation is proposed. In the rank modulation framework, codewords are permutations. In this paper, we study the largest size $P(n, d)$ of permutation codes of length $n$, i.e., subsets of the set $S_n$ of all permutations on $\{1,\ldots, n\}$ with the minimum distance at least $d\in\{1,\ldots ,\binom{n}{2}\}$ under the Kendall $τ$-metric. By presenting an algorithm and some theorems, we managed to improve the known lower and upper bounds for $P(n,d)$. In particular, we show that $P(n,d)=4$ for all $n\geq 6$ and $\frac{3}{5}\binom{n}{2}< d \leq \frac{2}{3} \binom{n}{2}$. Additionally, we prove that for any prime number $n$ and integer $r\leq \frac{n}{6}$, $ P(n,3)\leq (n-1)!-\dfrac{n-6r}{\sqrt{n^2-8rn+20r^2}}\sqrt{\dfrac{(n-1)!}{n(n-r)!}}. $ This result greatly improves the upper bound of $P(n,3)$ for all primes $n\geq 37$.