Quadratic weight vector for tighter aperiodic Levenshtein bound

Abstract

The Levenshtein bound, as a function of the weight vector, is only known to be tighter than the Welch bound on aperiodic correlation for K ≥ 4, N ≥ 2, where K and N denoting the set size and the sequence length, respectively. A quadratic weight vector is proposed in this paper which leads to a tighter Levenshtein bound for K ≥ 4, N ≥ 2 and K = 3, N ≥ 4. The latter case was left open by Levensthein.

A Fourier transform approach for improving the Levenshtein's lower bound on aperiodic correlation of binary sequences

“…In [3], Levenshtein confirmed that the Welch bound cannot be improved for M = 1 and 2 by the method of Theorem 2.1. He was also pessimistic about the improvement of the Welch bound for M = 3, which was however disproved by [5] and [7]. This paper will support the conclusion of [5] and [7] by presenting a new bound for M = 3.…”

“…In this paper, we adopt the framework of Levenshtein to improve the lower bound on aperiodic correlation of binary sequences. While other literatures [3] and [5]− [8] used the weight vectors in time domain for the framework, we introduce a Fourier transform approach by taking into account them in spectral domain. The Fourier transform approach provides an alternative theoretical analysis for improving the lower bound.…”

A New Lower Bound on the Maximum Correlation of a Set With Mismatched Filters

“…This bound has been proved to be tighter than the Welch bound in some cases 2 : for N ≥ 2 with M = 2, N ≥ 3 with M = 3 and for N ≥ 2 with M ≥ 4. Some current literature [10], [12], [25] has been interested in the optimisation of this weight vector, in order to enhance the Levenshtein bound. Notably the authors of [25] have proposed an asymptotically locally optimal weight vector -in the particular case where each correlation delay is considered.…”

“…About 25 years later, a new lower bound was found by Levenshtein [8], [9] but only in the aperiodic case. At first, this bound was restricted to binary sequences, but it was later proved to hold for any sequence sets over the complex roots of unity [10] and even later for unimodular sequences [11]. This bound is tighter than the Welch one in most of the cases [9], [10], [12].…”

Generalization and Improvement of the Levenshtein Lower Bound for Aperiodic Correlation

“…THE LEVENSHTEIN BOUNDS The bound provided by Theorem 2 depends on several parameters, such as the number of sequences M and their length N, but also on the choice of the weight vector w and the number of considered delays D. Several results have been given in the literature according to the number of sequences M [5], [7], [8]. This section studies the behavior of this new bound, according to this criterion.…”