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Power-sequence terraces for Zn where n is an odd prime power

2003
*
Discrete Mathematics
*

Many elegant families of such Zn

doi:10.1016/s0012-365x(02)00459-4
fatcat:pvhksbkqfbdz3mrisjq2wxqvle
*terraces*are constructed*for*values of*n*that are*odd**prime**powers*. ... A*power*-*sequence**terrace**for*Zn*is*deÿned to be a*terrace*which can be partitioned into segments one of which contains merely the zero element of Zn, whilst each other segment*is*either (a) a*sequence*... Ollis (Queen Mary, University of London)*for*discussions that helped to stimulate the work reported in this paper. ...##
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SOME $\mathbb{Z}_{n-1}$ TERRACES FROM $\mathbb{Z}_{n}$ POWER-SEQUENCES, $n$ BEING AN ODD PRIME POWER

2007
*
Proceedings of the Edinburgh Mathematical Society
*

We now refine this idea to show that,

doi:10.1017/s0013091504000045
fatcat:3uq7imkcfffhtpsrbsn2dt4lpa
*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 ... Our constructions provide*terraces**for*Z*n*−1*for*all*prime**powers**n*satisfying 0 <*n*< 300 except*for**n*= 125, 127 and 257. ... Ollis (Marlboro College, Marlboro, VT)*for*providing the stimulus*for*the work reported in this paper. ...##
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Some da capo directed power-sequence Zn+1 terraces with n an odd prime power

2008
*
Discrete Mathematics
*

We now extend this idea by using

doi:10.1016/j.disc.2006.11.033
fatcat:4luerrwwsvce5nocf77qdf76a4
*power*-*sequences*in Z*n*to produce some*terraces**for*Z*n*+1*where**n**is**an**odd**prime**power*satisfying*n*≡ 1 or 3 (mod 8). ... A*terrace**for*Z*n**is*a particular type of*sequence*formed from the*n*elements of Z*n*. ... Z*n*+1*terraces*with*n*= p 2*where*p*is**an**odd**prime*The following three theorems provide constructions*for*Z*n*+1*terraces*with*n*= p 2*for*certain*odd**primes*. ...##
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A general approach to constructing power-sequence terraces for Zn

2008
*
Discrete Mathematics
*

It also yields

doi:10.1016/j.disc.2007.07.051
fatcat:an3hs5q4m5btrgij6usf7v7wvi
*terraces**for*some groups Z*n*with*n*= p 2*where*p*is**an**odd**prime*, and*for*some Z*n*with*n*= pq*where*p and q are distinct*primes*greater than 3. ... We now present a new general*power*-*sequence*approach that yields Z*n**terraces**for*all*odd**primes**n*less than 1000 except*for**n*= 601. ... Acknowledgement We are grateful to Dr Wilson Stothers (University of Glasgow)*for*informing us of Theorem 1.2. ...##
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Some Zn+2 terraces from Zn power-sequences, n being an odd prime

2008
*
Discrete Mathematics
*

We now extend this idea by using

doi:10.1016/j.disc.2007.07.110
fatcat:elsrpuxiwvaalgts56hfiu7iyq
*power*-*sequences*in Z*n*,*where**n**is**an**odd**prime*, to obtain*terraces**for*Z m*where*m =*n*+ 2. ... We provide Z*n*+2*terraces**for*all*odd**primes**n*satisfying 0 <*n*< 1000 except*for**n*= 127, 601, 683. ... Acknowledgement The authors warmly thank a referee who, having read the paper with very great care, made various helpful suggestions*for*correction and enhancement. ...##
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Narcissistic half-and-half power-sequence terraces for Zn with n=pqt

2004
*
Discrete Mathematics
*

Constructions are provided

doi:10.1016/s0012-365x(03)00261-9
fatcat:t4vraf2vtbctlj7lkv35kihnaq
*for*narcissistic half-and-half*power*-*sequence**terraces**for*Zn with*n*= pq t*where*p and q are distinct*odd**primes*and t*is*a positive integer. ... If*n**is**odd*, a Zn*terrace*(a1; a2; : : : ;*an*)*is*a narcissistic half-and-half*terrace*if ai − ai−1 =*an*+2−i −*an*+1−i*for*i = 2; 3; : : : ; (*n*+ 1)=2. ... Bailey (Queen Mary, University of London)*for*discussion seminal to the derivation of results in this paper, and to P.J. Cameron (Queen Mary, University of London)*for*insights into primitive -roots. ...##
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SOME n−2 TERRACES FROM n POWER-SEQUENCES, n BEING AN ODD PRIME POWER

2009
*
Glasgow Mathematical Journal
*

We now adapt this idea by using

doi:10.1017/s0017089509990164
fatcat:4z5fnvqkljh2npxkeciedjxqzm
*power*-*sequences*in ޚ*n*,*where**n**is**an**odd**prime**power*, to obtain*terraces**for*ޚ m ,*where*m =*n*− 2. ... We also provide ޚ*n*−2*terraces**for**n*= 3 r (r > 1) and*for*some values*n*= p 2 ,*where*p*is**prime*. 2000 Mathematics Subject Classification. ... Anderson and Preece [2] [3] [4] [5] gave general constructions*for*'*power*-*sequence*'*terraces**for*ޚ m ,*where*m*is**odd*. ...##
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Terraces for Small Groups
[article]

2016
*
arXiv
*
pre-print

We show that there

arXiv:1603.01496v1
fatcat:mxahhhtmc5erxm46zgmtu2xeie
*is*a*sequenceable*group*for*each*odd*order up to 555 at which there*is*a non-abelian group. ... q_2 are*prime**powers*with 3 ≤ q_1 ≤ 11 and 3 ≤ q_2 ≤ 8. ... to one of A 7 , PSL(2, q) or PGL(2, q)*for**odd**prime**powers*q > 3. ...##
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The spectrum of group-based Latin squares
[article]

2018
*
arXiv
*
pre-print

We construct

arXiv:1812.05526v1
fatcat:d5amr4aahng35pwpmmh4wgti64
*sequencings**for*many groups that are a semi-direct product of*an**odd*-order abelian group and a cyclic group of*odd**prime*order. ... It follows from these constructions that there*is*a group-based complete Latin square of order*n*if and only if*n*∈{ 1,2,4} or there*is*a non-abelian group of order*n*. ... We construct*sequencings**for*some semi-direct products Z q ⋉ A*where*A*is**an*abelian group of*odd*order and q*is**an**odd**prime*, including all possible such groups when A*is*cyclic. ...##
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Some power-sequence terraces for Zpq with as few segments as possible

2005
*
Discrete Mathematics
*

If

doi:10.1016/j.disc.2004.08.020
fatcat:7jcqgwk7urdh5favqwkwkvlrda
*n*= pq,*where*p and q are distinct*odd**primes*, the minimum number of segments*for*such a*terrace**is*3+ (*n*),*where*(*n*)*is*the ratio (*n*)/ (*n*) of the number of units in Z*n*to the maximum order of a unit ... A*power*-*sequence**terrace**for*Z*n**is*a Z*n**terrace*that can be partitioned into segments one of which contains merely the zero element of Z*n*whilst each other segment*is*either (a) a*sequence*of successive ... London)*for*very helpful discussions about primitive -roots and the properties of the units of Z*n*. ...##
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The Spectrum of Group-Based Complete Latin Squares

2019
*
Electronic Journal of Combinatorics
*

We construct

doi:10.37236/8542
fatcat:s7bcqahoxvfzxdsmantbetti3i
*sequencings**for*many groups that are a semi-direct product of*an**odd*-order abelian group and a cyclic group of*odd**prime*order. ... It follows from these constructions that there*is*a group-based complete Latin square of order $*n*$ if and only if $*n*\in \{ 1,2,4\}$ or there*is*a non-abelian group of order $*n*$. ... We construct*sequencings**for*some semi-direct products Z q A*where*A*is**an*abelian group of*odd*order and q*is**an**odd**prime*, including all possible such groups when A*is*cyclic. ...##
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Sequenceable Groups and Related Topics

2013
*
Electronic Journal of Combinatorics
*

We also look at constructions

doi:10.37236/30
fatcat:kbspizkvlrfytperbtf3niarmu
*for*row-complete latin squares that do not use*sequencings*. ... In Section 3 we consider some concepts closely related to*sequenceable*groups: R-*sequencings*, harmonious groups, supersequenceable groups (also known as super P-groups),*terraces*and the Gordon game. ... (ii) Dihedral groups D 2n*where**n**is*twice*an**odd**prime*. ...##
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Combinatorially fruitful properties of 3⋅2−1 and 3⋅2−2 modulo p

2010
*
Discrete Mathematics
*

Searches

doi:10.1016/j.disc.2008.09.046
fatcat:wuteglrp4zh4zmj2vghufk2trm
*for*such constructions have revealed that, if we wish to produce*terraces**for*Z p−1 and Z p+1*where*p*is**an**odd**prime*, then use can be made of*sequences*of*powers*, modulo p, of a and b*where*a ... Write a ≡ 3 · 2 −1 and b ≡ 3 · 2 −2 (mod p)*where*p*is**an**odd**prime*. Let c be a value that*is*congruent (mod p) to either a or b. ... Let p be*an**odd**prime*; let a = (p + 3)/2 and q = (p − 1)/2. ...##
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On terraces for abelian groups

2005
*
Discrete Mathematics
*

We adapt a construction

doi:10.1016/j.disc.2005.07.007
fatcat:yzetgxyvpvepxjljofcre37afi
*for*R-*sequencings*to give new*terraces**for*abelian groups. This enables the construction of*terraces**for*all groups of the form Z k 2 ×Z t*where*k ≥ 4 and t > 5*is**odd*. ... We also present*an*extendable*terrace**for*Z 3 2 × Z 5 . ... Theorem 3 [3, 4] Let G be a group with a normal subgroup*N*. If*N*has*odd*order and G/*N**is**terraced*then G*is**terraced*. Alternatively, if*N*has*odd*index and*N**is**terraced*then G*is**terraced*. ...##
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Sectionable terraces and the (generalised) Oberwolfach problem

2003
*
Discrete Mathematics
*

It was known in 1892, though di erent terminology was then used, that a directed

doi:10.1016/s0012-365x(02)00822-1
fatcat:rbtdw22bavb2ri2zao4cruwoaq
*terrace*with a symmetric*sequencing**for*the cyclic group of order 2n can be used to solve OP(2n + 1). ... We show how*terraces*with special properties can be used to solve OP(2; l1; l2) and OP(l1; l1; l2)*for*a wide selection of values of l1, l2 and v. ... The other essential ingredient of that proof*is*the following: Example 12. If*n**is**odd*then the LWW*terrace**for*Z*n**is**an*st-*terrace*. Theorem 11. ...
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