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

Zilong Liu, Udaya Parampalli, Yong Liang Guan, Serdar Boztas
2014 IEEE Transactions on Information Theory  
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 1999 paper, postulated that the answer may be
more » ... nswer 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.
doi:10.1109/tit.2013.2293493 fatcat:reqyesbqcbekddhxy2eqpz6fru