Abstract
This extends a study in [1], from SETA-2010. We consider two infinite sequences of positive integers, \(A= \{a_k\} ^\infty_{k=1}\) and \(B = \{b_k\}_{k=1}^\infty\), and the related sequences \(\bar{A} = \{\alpha_j\}_{j=1}^\infty\) and \(\bar{B} = \{\beta_j\}_{j=1}^\infty\), where α j = 1 if j ∈ A, α j = 0 if j ∉ A, and β j = 1 if j ∈ B, β j = 0 if j ∉ B. We call C AB (τ) the cross-correlation of A and B where C AB (τ) is the un-normalized infinite-domain cross-correlation of \(\bar{A}\) and \(\bar{B}\), i.e.
Our interest is confined to sequence pairs A,B for which C AB (τ) is finite for all τ ∈ Z. Our interest is greater if C AB (τ) < K for some uniform bound K, for all τ ∈ Z, and especially great if K = 1. We are specifically interested in possible lower bounds on the rates of growth of A and B for a given value of K. This paper provides extensive data for the case K = 1.
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References
Golomb, S.W.: Infinite Sequences with Finite Cross-Correlation. In: Carlet, C., Pott, A. (eds.) SETA 2010. LNCS, vol. 6338, pp. 430–441. Springer, Heidelberg (2010)
Hardy, G.H., Wright, E.M.: Introduction to the Theory of Numbers, 5th edn. Oxford University Press, Oxford (1985)
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Golomb, S.W. (2012). Infinite Sequences with Finite Cross-Correlation-II. In: Helleseth, T., Jedwab, J. (eds) Sequences and Their Applications – SETA 2012. SETA 2012. Lecture Notes in Computer Science, vol 7280. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30615-0_10
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DOI: https://doi.org/10.1007/978-3-642-30615-0_10
Publisher Name: Springer, Berlin, Heidelberg
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