arXiv subjects

Nicholas Cavenagh

Publications and source records attributed to Nicholas Cavenagh.

7 recordsLinked to original sources

arXiv2026

Defining sets which intersect each Latin trade at least twice

A defining set of a Latin square is a partially filled-in Latin square which completes to no other Latin square of the same order. We introduce the concept of a $k$-strong defining set, in which if less than $k$ entries are deleted, the property of being a defining set is retained. Equivalently, a $k$-strong defining set intersects every Latin trade in the Latin square at least $k$ times. In the addition table for integers modulo $n$, when $n$ is even we determine the minimum size of a $k$-strong defining set for any $k$. For odd $n$ we give a construction for a minimally $2$-strong defining set. We furthermore give computational results for Latin squares of small orders.

math.CO↗

arXiv2023

Latin bitrades derived from quasigroup autoparatopisms

In 2008, Cavenagh and Dr\'{a}pal, et al, described a method of constructing Latin trades using groups. The Latin trades that arise from this construction are entry-transitive (that is, there always exists an autoparatopism of the Latin trade mapping any ordered triple to any other ordered triple). Moreover, useful properties of the Latin trade can be established using properties of the group. However, the construction does not give a direct embedding of the Latin trade into any particular Latin square. In this paper, we generalize the above to construct Latin trades embedded in a Latin square $L$, via the autoparatopism group of the quasigroup with Cayley table $L$. We apply this theory to identify non-trivial entry-transitive trades in some group operation tables as well as in Latin squares that arise from quadratic orthomorphisms

math.CO↗

arXiv2021

Row-column factorial designs with multiple levels

An {\em $m\times n$ row-column factorial design} is an arrangement of the elements of a factorial design into a rectangular array. Such an array is used in experimental design, where the rows and columns can act as blocking factors. If for each row/column and vector position, each element has the same regularity, then all main effects can be estimated without confounding by the row and column blocking factors. Formally, for any integer $q$, let $[q]=\{0,1,\dots ,q-1\}$. The $q^k$ (full) factorial design with replication $α$ is the multi-set consisting of $α$ occurrences of each element of $[q]^k$; we denote this by $α\times [q]^k$. A {\em regular $m\times n$ row-column factorial design} is an arrangement of the the elements of $α\times [q]^k$ into an $m\times n$ array (which we say is of {\em type} $I_k(m,n;q)$) such that for each row (column) and fixed vector position $i\in [q]$, each element of $[q]$ occurs $n/q$ times (respectively, $m/q$ times). Let $m\leq n$. We show that an array of type $I_k(m,n;q)$ exists if and only if (a) $q|m$ and $q|n$; (b) $q^k|mn$; (c) $(k,q,m,n)\neq (2,6,6,6)$ and (d) if $(k,q,m)=(2,2,2)$ then $4$ divides $n$. This extends the work of Godolphin (2019), who showed the above is true for the case $q=2$ when $m$ and $n$ are powers of $2$. In the case $k=2$, the above implies necessary and sufficient conditions for the existence of a pair of mutually orthogonal frequency rectangles (or $F$-rectangles) whenever each symbol occurs the same number of times in a given row or column.

math.ST↗

arXiv2018

Balanced diagonals in frequency squares

We say that a diagonal in an array is {\em $λ$-balanced} if each entry occurs $λ$ times. Let $L$ be a frequency square of type $F(n;λ^m)$; that is, an $n\times n$ array in which each entry from $\{1,2,\dots ,m\}$ occurs $λ$ times per row and $λ$ times per column. We show that if $m\leq 3$, $L$ contains a $λ$-balanced diagonal, with only one exception up to equivalence when $m=2$. We give partial results for $m\geq 4$ and suggest a generalization of Ryser's conjecture, that every latin square of odd order has a transversal. Our method relies on first identifying a small substructure with the frequency square that facilitates the task of locating a balanced diagonal in the entire array.

math.CO↗

arXiv2017

Lower bounds on the sizes of defining sets in full $n$-Latin squares and full designs

The full $n$-Latin square is the $n\times n$ array with symbols $1,2,\dots ,n$ in each cell. In this paper we show, as part of a more general result, that any defining set for the full $n$-Latin square has size $n^3(1-o(1))$. The full design $N(v,k)$ is the unique simple design with parameters $(v,k,{v-2 \choose k-2})$; that is, the design consisting of all subsets of size $k$ from a set of size $v$. We show that any defining set for the full design $N(v,k)$ has size ${v\choose k}(1-o(1))$ (as $v-k$ becomes large). These results improve existing results and are asymptotically optimal. In particular, the latter result solves an open problem given in (Donovan, Lefevre, et al, 2009), in which it is conjectured that the proportion of blocks in the complement of a full design will asymptotically approach zero.

math.CO↗

arXiv2016

On the distances between Latin squares and the smallest defining set size

In this note we show that for each Latin square $L$ of order $n\geq 2$, there exists a Latin square $L'\neq L$ of order $n$ such that $L$ and $L'$ differ in at most $8\sqrt{n}$ cells. Equivalently, each Latin square of order $n$ contains a Latin trade of size at most $8\sqrt{n}$. We also show that the size of the smallest defining set in a Latin square is $Ω(n^{3/2})$. %That is, there are constants $c$ and $n_0$ such that for any $n>n_0$ the size of the smallest defining %set of order $n$ is at least $cn^{3/2}$.

math.CO↗

arXiv2010

Multi-latin squares

A multi-latin square of order $n$ and index $k$ is an $n\times n$ array of multisets, each of cardinality $k$, such that each symbol from a fixed set of size $n$ occurs $k$ times in each row and $k$ times in each column. A multi-latin square of index $k$ is also referred to as a $k$-latin square. A $1$-latin square is equivalent to a latin square, so a multi-latin square can be thought of as a generalization of a latin square. In this note we show that any partially filled-in $k$-latin square of order $m$ embeds in a $k$-latin square of order $n$, for each $n\geq 2m$, thus generalizing Evans' Theorem. Exploiting this result, we show that there exist non-separable $k$-latin squares of order $n$ for each $n\geq k+2$. We also show that for each $n\geq 1$, there exists some finite value $g(n)$ such that for all $k\geq g(n)$, every $k$-latin square of order $n$ is separable. We discuss the connection between $k$-latin squares and related combinatorial objects such as orthogonal arrays, latin parallelepipeds, semi-latin squares and $k$-latin trades. We also enumerate and classify $k$-latin squares of small orders.

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