The invariant factors of some cyclic difference sets (Q1865414)

Difference sets with parameters \(\left(\frac{q^m-1}{q-1}, \frac{q^{m-1}-1}{q-1}, \frac{q^{m-2}-1}{q-1}\right)\), \(q\) prime power, are called difference sets with classical parameters. Recently, many new constructions of such difference sets with \(q=2\) have been found, and two (infinite) series for \(q=3\). One series are the HKM difference sets [see \textit{T. Helleseth} et al., Des. Codes Cryptography 23, 157-166 (2001; Zbl 1007.94013)]. The other series are the Lin difference sets (conjectured in the Ph.D. thesis of \textit{H. A. Lin} [From cyclic Hadamard difference sets to perfectly balanced sequences. University of Southern California, Los Angeles (1998)]). Both difference sets have the same \(3\)-rank. The authors compute the Smith normal form (which is a refinement of computing the \(p\)-rank), therefore showing that the designs corresponding to the difference sets are not isomorphic. The proof uses algebraic number theory, for instance Stickelberger's theorem, Gauss sums and \(p\)-adic valuations.