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Publisher: Cambridge University Press
Languages: English
Types: Article
Subjects: QA
A terrace for Zm is a particular type of sequence formed from the m elements of Zm. For m\ud odd, many procedures are available for constructing power-sequence terraces for Zm; each terrace of this\ud sort may be partitioned into segments, of which one contains merely the zero element of Zm, whereas\ud every other segment is either a sequence of successive powers of an element of Zm or such a sequence\ud multiplied throughout by a constant. We now refine this idea to show that, for m=n−1, where n is an odd prime power, there are many ways in which power-sequences in Zn can be used to arrange the elements of Zn \ {0} in a sequence of distinct entries i, 1 ≤ i ≤ m, usually in two or more segments, which becomes a terrace for Zm when interpreted modulo m instead of modulo n. Our constructions provide terraces for Zn-1 for all prime powers n satisfying 0 < n < 300 except for n = 125, 127 and 257.
  • The results below are discovered through our pilot algorithms. Let us know how we are doing!

    • I. Anderson and D. A. Preece, Locally balanced change-over designs, Util. Math. 62 (2002), 33-59.
    • I. Anderson and D. A. Preece, Power-sequence terraces for Zn where n is an odd prime power, Discr. Math. 261 (2003), 31-58.
    • I. Anderson and D. A. Preece, Some narcissistic half-and-half power-sequence Zn terraces with segments of different lengths, Congr. Numer. 163 (2003), 5-26.
    • I. Anderson and D. A. Preece, Narcissistic half-and-half power-sequence terraces for Zn with n = pqt, Discr. Math. 279 (2004), 33-60.
    • I. Anderson and D. A. Preece, Some power-sequence terraces for Zpq with as few segments as possible, Discr. Math. 293 (2005), 29-59.
    • Soc. B 46 (1984), 323-334.
    • R. A. Bailey, M. A. Ollis and D. A. Preece, Round-dance neighbour designs from terraces, Discr. Math. 266 (2003), 69-86.
    • DS10 (available at www.combinatorics.org/surveys/).
    • M. A. Ollis and D. A. Preece, Sectionable terraces and the (generalised) Oberwolfach problem, Discr. Math. 266 (2003), 399-416.
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