Mixed-level covering, locating, and detecting arrays via cyclotomy (Q6616800)

This paper discusses the construction of covering arrays using cyclotomic methods over finite fields of order \(q\) and divisors \(v\) of \(q-1\). In [\textit{C. J. Colbourn}, Des. Codes Cryptography 55, No. 2--3, 201--219 (2010; Zbl 1215.05019)] it is pointed out that additive translates of a cyclotomic vector generate a \(q \times q\) cyclotomic array with \(v\) symbols, and for any positive integer \(t\), there exist sufficiently large values of \(q\) such that these arrays are always covering arrays of strength \(t\). However, asymptotically, these cyclotomic arrays tend to have too many rows to be competitive with other covering array constructions. Despite this, for small values of \(t\), cyclotomic arrays yield the smallest known covering arrays for several practical parameter sets. The paper extends the cyclotomic strategy of [Colbourn, loc. cit.] in three different directions. The paper also explores the extension of these ideas to generate covering arrays of higher index, as well as locating and detecting arrays with large separation. Computational results show that cyclotomic arrays with the same order \(q\) but different values of \(v\) can be combined to produce mixed-level covering, locating, and detecting arrays.\N\NFor the entire collection see [Zbl 1540.05004].