Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-05T21:42:07.995Z Has data issue: false hasContentIssue false
  • English
  • Français

SOME n−2 TERRACES FROM n POWER-SEQUENCES, n BEING AN ODD PRIME POWER

Published online by Cambridge University Press:  30 July 2009

IAN ANDERSON  and
D. A. PREECE
IAN ANDERSON
Affiliation:
Department of Mathematics, University of Glasgow, University Gardens, Glasgow G12 8QW, UK e-mail: [email protected]
D. A. PREECE
Affiliation:
Queen Mary University of London, School of Mathematical Sciences, Mile End Road, London E1 4NS, UK and Institute of Mathematics, Statistics and Actuarial Science, Cornwallis Building, University of Kent, Canterbury, Kent CT2 7NF, UK e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A terrace for m is an arrangement (a1, a2, . . . , am) of the m elements of m such that the sets of differences ai+1 − ai and ai − ai+1 (i = 1, 2, . . . , m − 1) between them contain each element of m \ {0} exactly twice. For m odd, many procedures are available for constructing power-sequence terraces for m; each such terrace may be partitioned into segments, one of which contains merely the zero element of m, whereas each other segment is either (a) a sequence of successive powers of an element of m or (b) such a sequence multiplied throughout by a constant. We now adapt this idea by using power-sequences in n, where n is an odd prime power, to obtain terraces for m, where m = n − 2. We write each element from n so that they lie in the interval [0, n − 1] and then delete 0 and n − 1 so that they leave n − 2 elements that may be interpreted as the elements of n−2. A segment of one of the new terraces may be of type (a) or (b), incorporating successive powers of 2, with each entry evaluated modulo n. Our constructions provide n−2 terraces for all odd primes n satisfying 0 < n < 1,000 except for n = 127, 241, 257, 337, 431, 601, 631, 673, 683, 911, 937 and 953. We also provide n−2 terraces for n = 3r (r > 1) and for some values n = p2, where p is prime.

Keywords

Primary 11A07secondary 05B30
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 52 , Issue 1 , January 2010 , pp. 65 - 85
Copyright
Copyright © Glasgow Mathematical Journal Trust 2009

References

REFERENCES

1.Anderson, I. and Preece, D. A., Locally balanced change-over designs, Utilitas Math. 62 (2002), 33–59.Google Scholar
2.Anderson, I. and Preece, D. A., Power-sequence terraces for n where n is an odd prime power, Discrete Math. 261 (2003), 31–58.CrossRefGoogle Scholar
3.Anderson, I. and Preece, D. A., Some narcissistic half-and-half power-sequence n terraces with segments of different lengths, Cong. Numer. 163 (2003), 5–26.Google Scholar
4.Anderson, I. and Preece, D. A., Narcissistic half-and-half power-sequence terraces for n with n = pqt, Discrete Math. 279 (2004), 33–60.CrossRefGoogle Scholar
5.Anderson, I. and Preece, D. A., Some power-sequence terraces for pq with as few segments as possible, Discrete Math. 293 (2005), 29–59.CrossRefGoogle Scholar
6.Anderson, I. and Preece, D. A., Some n−1 terraces from n power-sequences, n being an odd prime power, Proc. Edinbr. Math. Soc. 50 (2007), 527–549.CrossRefGoogle Scholar
7.Anderson, I. and Preece, D. A., Some n+2 terraces from n power-sequences, n being an odd prime, Discrete Math. 308 (2008), 4086–4107.CrossRefGoogle Scholar
8.Bailey, R. A., Quasi-complete Latin squares: Construction and randomisation, J. R. Statist. Soc. B 46 (1984), 323–334.Google Scholar
9.Hilton, A. J. W., Mays, M., Rodger, C. A. and Nash-Williams, C. StJ. A., Hamiltonian double Latin squares, J. Combin. Theory B 87 (2003), 81–129.Google Scholar
10.Ringel, G., Map color theorem (Springer, Berlin, 1974).CrossRefGoogle Scholar