New properties of Sharma-Kaushik partitions (Q2923267)

The papers deals with the Sharma-Kaushik partitions (SK-partition) defined in 1977. For a positive integer \(q\), a partition \(P=(B_0,B_1, \ldots, B_m)\) of the ring \(F_q\) of integers modulo \(q\), is called SK-partition of \(F_q\) if it satisfies the following conditions, where all \(i,j\) are taken from \(\{ 0,1, \ldots, m-1 \} \): \newline 1. \(B_0=\{0\}\) and if \(a\in B_i\), then \(q-a\in B_i.\) \newline 2. If \(a\in B_i\) and \(b\in B_j\), then \(i>j\) implies \(\min\{a, q-a\}>\min\{b, q-b\}\).\newline 3. If \(i>j\), then \(|B_i|\geq |B_j|\) and \(|B_{m-1}|\geq \frac{1}{2}|B_{m-2}|\). (In the original \(|B_{m-1}|\) is written incorrectly \(|B_m-1|\).)NEWLINENEWLINEThe \(B_i\) are called classes of the partition.NEWLINENEWLINENEWLINEIn an \(x\)-translate of a set, the elements are incremented by \(x\). If the incrementation is taken modulo \(q\), we have an \(x\)-\(q\)-translate. The paper studies the conditions when a translate of a class of an SK-partition is also a class of an SK-partition. Intersections of classes with translates of classes are studied too. Finally, a formula to compute the number of \(n\)-tuples of a given weight with respect to an SK-partition is given.