On the existence of cyclic and pseudo-cyclic MDS codes (Q1268376)

A linear code \(C\) of length \(n\) and dimension \(k\) over the Galois field \(\mathbb{F}_q\), denoted as an \([n,k]_q\) code, is called MDS if its minimum distance \(d\) equals \(n-k+1\). \(C\) is called pseudo-cyclic (the names semi-cyclic and constacyclic are also in use) if there exists an \(\alpha\in \mathbb{F}_q \setminus\{0\}\) such that \((c_1,\dots, c_n)\in C\) implies that \((\alpha c_n,c_2, c_2, \dots, c_{n-1})\in C\). The paper studies conditions for the existence of pseudo-cyclic codes of dimension \(k=3,4\), or 5. Existence tables are provided for \(q\leq 64\).