Some Z(n-2) terraces from Z(n) power-sequences, n being an odd prime power

Anderson, I. and Preece, D.A. (2010) Some Z(n-2) terraces from Z(n) power-sequences, n being an odd prime power. Glasgow Mathematical Journal, 52(01), pp. 65-85. (doi:10.1017/S0017089509990164)

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Publisher's URL: http://dx.doi.org/10.1017/S0017089509990164

Abstract

A terrace for Z(m) is an arrangement (a(1), a(2),..., a(m)) of the m elements of Z(m) Such that the sets of differences a(i+l) - a(i) and a(i) - a(i+1) (i = 1, 2,..., m - 1) between them contain each element of Z(m) \ {0} exactly twice. For in odd, many procedures are available for constructing power-seqUence terraces for Z(m); each such terrace may be partitioned into segments, one of which contains merely the zero element of Z(m), whereas each other segment is either (a) a sequence of successive powers of an element of Z(m) or (b) such a sequence Multiplied throughout by a constant. We now adapt this idea by using power-sequences in Z(n), where n is an odd prime power, to obtain terraces for Z(m), where m = n - 2. We write each element from Z(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 Z(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 Z(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 Z(n-2) terraces for n = 3(r) (r > 1) and for some values n = p(2) where p is prime.

Item Type:Articles
Status:Published
Refereed:Yes
Glasgow Author(s) Enlighten ID:Anderson, Dr Ian
Authors: Anderson, I., and Preece, D.A.
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Glasgow Mathematical Journal
ISSN:0017-0895

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