New Lower Bounds on the Periodic Crosscorrelation of QAM Codes with Arbitrary Energy

Abstract

In this paper, we consider a recent technique of Levenshtein [9] which was introduced to prove improved lower bounds on aperiodic correlation of sequence families over the complex roots of unity. We first give a new proof of the Welch [12] lower bound on the periodic correlation of complex roots of unity sequences using a modification of Levenshtein’s technique. This improves the Welch lower bound in the case that the family size is “large enough” compared to sequence length. Our main result is an improved lower bound on the periodic complex valued sequences with QAM-type alphabets. This result improves an earlier result by Boztaş [2]. To achieve this result, we extend the Levenshtein technique in a new direction.

Here, we assume that sequences are drawn from energy “shells” which are no more than PSK signal sets with the scaling determining the signal energy. It turns out that, if the weights are associated with the energy of the sequence in question, the Levenshtein technique can be modified to obtain lower bounds.

The new bound is a non-convex function of a set of suitably chosen “weights.” We demonstrate that our results improve those of [2] with a concrete example.