2025/08/23 by Diane Donovan, M. J. Grannell, Emine Şule Yazıcı · 1 voice · 1 citation
Computer Science · Decision Sciences · Engineering · #Digital Image Processing Techniques #Optimal Experimental Design Methods #graph theory and CDMA systems
paper · pdf · doi:10.1007/s10623-025-01704-x
openalex publication_date 2025/08/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/23
Abstract This paper presents a conjecture concerning the minimum possible size of a pair of maximal orthogonal partial Latin squares of a given order n . We show that in the balanced case the optimal structure is formed from a pair of partial Latin squares, each comprising three subsquares whose orders are as close as possible to one another and sum to n . Further results are obtained in unbalanced cases. The problem can be recast in terms of finding the minimum number of blocks in a maximal partial transversal design TD(4, n ), and as finding the minimum number of codewords in an n -ary code of length 4 having minimum distance 3 and covering radius 2. The conjecture is extended to sets of k maximal mutually orthogonal partial Latin squares and hence to n -ary codes of length k+2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , minimum distance k+1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> and covering radius k .