Critical sets in nets and Latin squares
type: Item
instance of: scholarly article
title: Critical sets in nets and Latin squares
zbMATH Open document ID: 0810.51001
DOI: 10.1016/0378-3758(94)90167-8
author: J. A. Cooper
author: T. P. McDonough
author: V. C. Mavron
published in: Journal of Statistical Planning and Inference
publication date: 1995-04-09T00:00:00Z
review text: A partial plane \(P\) is an incidence structure consisting of a set of objects called points, together with a list of certain non-empty point subsets called lines such that any two distinct points are contained in at most one line. A subset may appear more than once in the list of lines, but such a subset must have exactly one point. \(P\) is finite if it has finitely many points and lines. It is obvious that two distinct lines of \(P\) meet in at most one point. A net \(N\) of order \(n\) and degree \(r\), where \(r\) and \(n\) are positive integers, is a finite partial plane satisfying the following conditions: i) \(N\) has exactly \(n^ 2\) points. ii) Each line consists of exactly \(n\) points. iii) The set of lines may be partitioned into \(r\) subsets, called parallel classes, each consisting of mutually disjoint lines. Lines in the same parallel class are said to be parallel. iv) Any two non-parallel lines meet in a point. v) The \(r\) parallel classes are ordered and the lines in each parallel class are ordered. A partial net of degree \(r\) is a partial plane satisfying conditions (iii) and (v) above. A Latin square \(L\) of order \(n\) over a set \(S\), where \((S) = n\), is an \(n \times n\) array of elements (the entries of \(L\)) such that each element of \(S\) appears exactly once in each row and each column. The idea of critical set in a Latin square appears to have been introduced by J. A. Nelder (1977) and has been studied by various authors. Critical sets have applications to cryptography and agriculture [see e.g. Colbourn et al., 1984; Curran and Van Rees, 1978; Nelder, 1977; Seberry, 1990; Semtaniuk, 1979; Stinson and Van Rees, 1982]. Broadly speaking, a critical set in a Latin square \(L\) is a set \(C\) of entries which uniquely determines \(L\) and is minimal with respect to this property. Putting it in another way, \(L\) is the only Latin square of that order which contains \(C\), but any proper subset of \(C\) is contained in at least one oth…
Mathematics Subject Classification ID: 51E14
Mathematics Subject Classification ID: 62K99
Mathematics Subject Classification ID: 05B15
zbMATH DE Number: 638277
zbMATH Keywords: incidence structure
zbMATH Keywords: finite partial plane
zbMATH Keywords: parallel classes
zbMATH Keywords: partial net
zbMATH Keywords: Latin square
zbMATH Keywords: critical set
zbMATH Keywords: embedding
reviewed by: Ratnakaram Nava Mohan
MaRDI profile type: Publication
full work available at URL: 0378-3758(94)90167-8
OpenAlex ID: W2021818259
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description: scientific article; zbMATH DE number 638277