In this paper, we propose techniques for the construction of frequency-coding sequences that give rise to frequency-coded waveforms having ambiguity functions with a clear area - containing no sidelobes - in a connected region surrounding the main lobe. These constructed sequences are called pushing sequences. First, two important properties of pushing sequences are investigated: the group D/sub 4/ dihedral symmetry property and the frequency omission property. Using the group D/sub 4/ dihedral symmetry property, we show how to construct additional pushing sequences from a given pushing sequence. Using the frequency omission property, we show how to construct pushing sequences of any length N and design proper frequency-coded waveforms that meet specific constraints in the frequency domain. Next, we use the Lempel T/sub 4/ construction of Costas sequences to construct pushing sequences with power 1. Finally, we show how to construct pushing sequences with any desired power using Lee codewords. Because these arbitrary-power pushing sequences constructed using Lee codewords do not have the Costas property, we derive expressions for the pattern of hits in the geometric array. Based on this, the general form of the positions and levels of all the sidelobe peaks are derived.