On completing three cyclically generated transversals to a latin square

Cavenagh, Nicholas J., Hamalainen, Carlo and Nelson, Adrian M. (2009) On completing three cyclically generated transversals to a latin square. Finite Fields and Their Applications, 15 3: 294-303. doi:10.1016/j.ffa.2008.05.009


Author Cavenagh, Nicholas J.
Hamalainen, Carlo
Nelson, Adrian M.
Title On completing three cyclically generated transversals to a latin square
Journal name Finite Fields and Their Applications   Check publisher's open access policy
ISSN 1071-5797
1090-2465
Publication date 2009-06-01
Sub-type Article (original research)
DOI 10.1016/j.ffa.2008.05.009
Open Access Status
Volume 15
Issue 3
Start page 294
End page 303
Total pages 10
Place of publication Maryland Heights, MO, United States
Publisher Academic Press
Language eng
Subject 2604 Applied Mathematics
2602 Algebra and Number Theory
2614 Theoretical Computer Science
2200 Engineering
Abstract Let P be a partial latin square of prime order p > 7 consisting of three cyclically generated transversals. Specifically, let P be a partial latin square of the form:P = {(i, c + i, s + i), (i, c + i, s + i), (i, c + i, s + i) | 0 ≤ i < p} for some distinct c, c, c and some distinct s, s, s. In this paper we show that any such P completes to a latin square which is diagonally cyclic. Equivalently, we prove that for each prime p > 7, every partial transversal of size 3 in the addition table for the integers modulo p can be completed to a full transversal.
Formatted abstract
Let P be a partial latin square of prime order p > 7 consisting of three cyclically generated transversals. Specifically, let P be a partial latin square of the form:

P = {(i, c + i, s + i), (i, c′ + i, s′ + i), (i, c″ + i, s″ + i) | 0 ≤ i < p}

for some distinct c, c′, c″ and some distinct s, s′, s″. In this paper we show that any such P completes to a latin square which is diagonally cyclic. Equivalently, we prove that for each prime p > 7, every partial transversal of size 3 in the addition table for the integers modulo p can be completed to a full transversal.
Keyword Latin square
Diagonally cyclic latin square
Transversal
Complete mapping
Q-Index Code C1
Q-Index Status Provisional Code
Institutional Status UQ

Document type: Journal Article
Sub-type: Article (original research)
Collection: School of Mathematics and Physics
 
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Created: Thu, 03 Sep 2009, 18:11:49 EST by Mr Andrew Martlew on behalf of Mathematics