Frequency hopping is a method of transmitting signals by rapidly switching between many frequency channels, following some sequence of frequencies known to the transmitter and the receiver. This technique is used in the CDMA (code division multiple access) systems, and has many civilian and military applications. For successful transmission minimizing signal interference, we want to use sets of digital frequency sequences with minimal Hamming cross-correlation, which measures frequency overlaps with time shifts between two different sequences. We discuss a construction of a new family of one-coincidence sequences like this coming from some basic arithmetic of finite fields, which have some nice properties. This is joint work with Adib Shaar, and this talk is dedicated to his memory.