Good code sets by spreading orthogonal vectors via Golomb rulers and Costas arrays

Abstract

Good code sets have autocorrelation functions ACF with small sidelobes, and also have small crosscorrelations. In this work, a class of good ternary codes sets are introduced. First, mutually orthogonal vectors are selected, then they are spread via a Golomb ruler. This is shown to result in such a good set. If the mutually orthogonal vectors have entries in {-1,1} or {-1,0,1}, then a ternary code set result. While there are methods of generating ternary codes, and complementary ternary codes [1–7], there is no method in prior publications of generating mutually orthogonal ternary code sets. That is one of the contributions of this work. If complex numbers with unity magnitudes are allowed, then we obtain codes with magnitudes in {0,1}. If the vectors are obtained from matrices with mutually orthogonal rows and columns, as in Hadamard matrices, or DFT matrices, then longer codes can be obtained via spreading the obtained good set via a Golomb ruler a second time. Using existing codes, such as Barker codes, and spreading them via a Golomb ruler, then compounding them with the elements of a good set, results in a new good set with higher mainlobes. The spreading could be induced via any array of any dimension with elements of magnitudes in {0,1} that have autocorrelation with unity peak sidelobes. This includes Costas arrays, in addition to Golomb rulers.