Partial permutation decoding is a natural generalization of permutation decoding that enables error correction without fully exploiting the code’s error-correcting capabilities. The efficiency of partial permutation decoding is maximized when the size of the s-PD set reaches the lower bound \(s+1\) . In 2019, Barrolleta and Villanueva provided a sufficient condition for obtaining s-PD sets of size \(s+1\) for partial permutation decoding. In this paper, we present a necessary and sufficient condition on the cycle structure of a permutation to generate an s-PD set of size \(s+1\) for linear codes over finite fields. As applications, we apply our general results to cyclic and quasi-cyclic codes, deriving new and specific criteria for constructing s-PD sets of size \(s+1\) . Our results naturally generalize and improve the corresponding results in the aforementioned paper. Several examples are also included to illustrate our findings.
