Meeting the Levenshtein bound with equality by weighted-correlation complementary set

Liu, Zi Long; Guan, Yong Liang

doi:10.1109/isit.2012.6282286

Cited by 6 publications

(5 citation statements)

References 11 publications

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“…Recently, Liu, Guan and Mow have derived the generalized Levenshtein bound for quasi-complementary sequence sets [6] and shown that it is tighter than the corresponding Welch bound [2]. In addition, Liu and Guan have shown that the Levenshtein bound is met with equality by the weighted-correlation complementary sequences [7]. The readers are referred to [8]- [12] for more information on perfect-/quasi-complementary sequences.…”

Section: Introductionmentioning

confidence: 99%

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

Liu

Parampalli

Guan

et al. 2014

IEEE Trans. Inform. Theory

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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 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.

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Section: Introductionmentioning

confidence: 99%

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

Liu

Parampalli

Guan

et al. 2014

IEEE Trans. Inform. Theory

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“…The idea of the Levenshtein bound is that the mean of the weighted aperiodic correlation squares for any sequence subset over the complex roots-of-unity is equal to or greater than that of the whole set which includes all the complex roots-of-unity sequences. In [5], Liu and Guan showed that the Levenshtein bound is met with equality by the weighted-correlation complementary sequences. In [6], Liu, Guan and Mow derived the generalized Levenshtein bound for quasi-complementary sequence set which is tighter than that of Welch [2].…”

Section: Introductionmentioning

confidence: 99%

Quadratic weight vector for tighter aperiodic Levenshtein bound

Liu

Guan

Parampalli

et al. 2013

2013 IEEE International Symposium on Information Theory

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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.

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“…However, note that the previous cited interval does not include all the values of interest (see Figure 1 for instance). As explained in [11], an open question remains on the search for set of sequences that achieve the Levenshtein bound. At this point, a first partially negative answer can be given, thanks to Corollary 1.…”

Section: Improvement Over the Existing Boundsmentioning

confidence: 99%