On generalization of sonar sequences and intermodulation-free frequencies assignment sequences (Q1920028)

\(A= (a_1,a_2, \dots, a_J)\) is an \((h,p,J)\) \(K\)-sequence iff \(\sum^J_{j=1} x_j= \sum^J_{j=1} y_j= h\), \(\sum^J_{j=1} a_jx_j= \sum^J_{j=1} a_jy_j\), \(|\sum^J_{j=1} jx_j-\sum^J_{j =1} jy_j |< p\) implies that \(x_j= y_j\) for \(1\leq j\leq J\), where \(x_1,x_2, \dots, x_J\), \(y_1,y_2, \dots,y_J\), \(a_1,a_2, \dots,a_J\) are nonnegative integers, \(p,h\) are positive integers. Let \(m(A)= \max \{a_j \mid 1\leq j \leq J\}\), \(M(h,p,J) =\min \{m(A) \mid A\) is an \((h,p,J)\) \(K\)-sequence\}. In this paper a new lower bound on \(M(h,p,J)\) is proved which improves some known results.