A Note on Low-Correlation Zone Signal Sets

In this correspondence, we present a connection between designing low-correlation zone (LCZ) sequences and the results of correlation of sequences with subfield decompositions presented in a recent book by the first two authors. This results in LCZ signal sets with huge sizes over three different alphabetic sets: finite field of size $q$, integer residue ring modulo $q$ , and the subset in the complex field which consists of powers of a primitive $q$ th root of unity. We show a connection between these sequence designs and “completely noncyclic” Hadamard matrices and a construction for those sequences. We also provide some open problems along this direction.