Infinite Sequences with Finite Cross-Correlation-II

  • Solomon W. Golomb
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7280)

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 αj = 1 if j ∈ A, αj = 0 if j ∉ A, and βj = 1 if j ∈ B, βj = 0 if j ∉ B. We call CAB(τ) the cross-correlation of A and B where CAB(τ) is the un-normalized infinite-domain cross-correlation of \(\bar{A}\) and \(\bar{B}\), i.e.

$$ C_{AB}(\tau) = \sum \limits_{i=1}^\infty \alpha_i\beta_{i+\tau},\: for all\: \tau \in Z. $$
(1)

Our interest is confined to sequence pairs A,B for which CAB(τ) is finite for all τ ∈ Z. Our interest is greater if CAB(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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    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)CrossRefGoogle Scholar
  2. 2.
    Hardy, G.H., Wright, E.M.: Introduction to the Theory of Numbers, 5th edn. Oxford University Press, Oxford (1985)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Solomon W. Golomb
    • 1
  1. 1.Electrical EngineeringUniversity of Southern CaliforniaLos AngelesUSA

Personalised recommendations