Year: 2025
Type: article
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 method has been 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 Sn of all permutations on {1, ..., n} with the minimum distance at least d ∈ {1, ..., (n/2)} under the Kendall τ-metric. By presenting an algorithm and two theorems, we improve the known lower and upper bounds for P(n, d). In particular, we show that P(n, d) = 4 for all n ≥ 6 and 3/5 (n/2) < d ≤ 2/3 (n/2). Additionally, we prove that for any prime number n and integer r ≤ n/6, P(n, 3) ≤ (n − 1)! − n − 6r/√n2 − 8rn + 20r2 √(n − 1)!/n(n − r)!. This result greatly improves the upper bound of P(n, 3) for all primes n ≥ 37.
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Awards: G400339533, G5482153549
OpenAlex ID: W4409475285