\(s\)-PD-sets for codes from projective planes \(\mathrm{PG}(2,2^h)\), \(5\leq h\leq 9\) (Q2025378)

MacWilliams in 1964 introduced the permutation decoding, an algorithm to decode error-correcting codes. Such algorithm uses PD-sets, that is sets of the code automorphism defined with respect to a given information set of the code. Not all codes can be decoded using this method, it will depend on whether or not their automorphism codes contain PD-sets. More precisely, consider a linear code \(C\subseteq \mathbb{F}_p^n\) which can correct at most \(t\) errors and consider \(I\) to be an information set for \(C\). A subset \(S\) of the automorphism group of \(C\) is said to be a PD-set for \(C\) if every \(t\)-set of coordinate positions can be moved by at least one element of \(S\) out of the information set \(I\). For \(s\leq t\), an \(s\)-PD-set is a set \(S\) of the automorphism group of \(C\) such that every \(s\)-set of coordinate positions can be moved by at least one element of \(S\) out of the information set \(I\). In this paper, the authors construct \(s\)-PD-sets with \(s \in \{2,3\}\) for the linear codes arising from the incidence matrices of points and lines of \(\mathrm{PG}(2,q)\), for \(q=2^h\) and \(h \in \{5,6,7,8,9\}\). They use a special basis for these codes which was found by \textit{P. Vandendriessche} [Finite Fields Appl. 17, No. 6, 521--531 (2011; Zbl 1248.05026)].