A New Weight Vector for a Tighter Levenshtein Bound on Aperiodic Correlation

The Levenshtein bound on aperiodic correlation, which is a function of the weight vector, is tighter than the Welch bound for sequence sets over the complex roots of unity when M ≥ 4 and n ≥ 2, where M denotes the set size and n the sequence length. Although it is known that the tightest Levenshtein bound is equal to the Welch bound for M ∈ {1,2}, it is unknown whether the Levenshtein bound can be tightened for M=3, and Levenshtein, in his paper published in 1999, postulated that the answer may be negative. A new weight vector is proposed in this paper, which leads to a tighter Levenshtein bound for M=3, n ≥ 3 and M ≥ 4, n ≥ 2. In addition, the explicit form of the weight vector (which is derived by relating the quadratic minimization to the Chebyshev polynomials of the second kind) in Levenshtein´s paper is given. Interestingly, this weight vector also yields a tighter Levenshtein bound for M=3, n ≥ 3 and M ≥ 4, n ≥ √M, a fact not noticed by Levenshtein.