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Codes Cryptogr."],"published-print":{"date-parts":[[2025,12]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    This paper presents a conjecture concerning the minimum possible size of a pair of maximal orthogonal partial Latin squares of a given order\n                    <jats:italic>n<\/jats:italic>\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                    <jats:italic>n<\/jats:italic>\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,\u00a0\n                    <jats:italic>n<\/jats:italic>\n                    ), and as finding the minimum number of codewords in an\n                    <jats:italic>n<\/jats:italic>\n                    -ary code of length 4 having minimum distance 3 and covering radius 2. The conjecture is extended to sets of\n                    <jats:italic>k<\/jats:italic>\n                    maximal mutually orthogonal partial Latin squares and hence to\n                    <jats:italic>n<\/jats:italic>\n                    -ary codes of length\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$k+2$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , minimum distance\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$k+1$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    and covering radius\n                    <jats:italic>k<\/jats:italic>\n                    .\n                  <\/jats:p>","DOI":"10.1007\/s10623-025-01704-x","type":"journal-article","created":{"date-parts":[[2025,8,23]],"date-time":"2025-08-23T16:24:14Z","timestamp":1755966254000},"page":"5099-5113","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On maximal orthogonal partial Latin squares and minimal codes with specified length, minimum distance and covering radius"],"prefix":"10.1007","volume":"93","author":[{"given":"Diane M.","family":"Donovan","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mike J.","family":"Grannell","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Emine \u015eule","family":"Yaz\u0131c\u0131","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,8,23]]},"reference":[{"issue":"1","key":"1704_CR1","doi-asserted-by":"publisher","first-page":"99","DOI":"10.1109\/18.42181","volume":"35","author":"RA Brualdi","year":"1989","unstructured":"Brualdi R.A., Pless V.S., Wilson R.M.: Short codes with a given covering radius. IEEE Trans. Inform. Theory 35(1), 99\u2013109 (1989).","journal-title":"IEEE Trans. Inform. Theory"},{"key":"1704_CR2","volume-title":"Handbook of Combinatorial Designs","author":"CJ Colbourn","year":"2007","unstructured":"Colbourn C.J., Dinitz J.H.: Handbook of Combinatorial Designs, 2nd edn CRC, Boca Raton (2007).","edition":"2"},{"key":"1704_CR3","doi-asserted-by":"publisher","first-page":"419","DOI":"10.1007\/s10623-023-01314-5","volume":"92","author":"DM Donovan","year":"2024","unstructured":"Donovan D.M., Grannell M.J., Yaz\u0131c\u0131 E.\u015e: On maximal partial Latin hypercubes. Des. Codes Cryptogr. 92, 419\u2013433 (2024). https:\/\/doi.org\/10.1007\/s10623-023-01314-5.","journal-title":"Des. Codes Cryptogr."},{"issue":"1","key":"1704_CR4","doi-asserted-by":"publisher","first-page":"219","DOI":"10.1109\/18.61146","volume":"37","author":"EM Gabidulin","year":"1991","unstructured":"Gabidulin E.M., Davydov A.A., Tombak L.M.: Linear codes with covering radius 2 and other new covering codes. IEEE Trans. Inform. Theory 37(1), 219\u2013224 (1991).","journal-title":"IEEE Trans. Inform. Theory"},{"key":"1704_CR5","first-page":"225","volume-title":"Graphs, Matrices and Designs","author":"P Horak","year":"1993","unstructured":"Horak P., Rosa A.: Maximal partial Latin squares. In: Rees R.S. (ed.) Graphs, Matrices and Designs, pp. 225\u2013235. Marcel Dekker, New York (1993)."},{"key":"1704_CR6","doi-asserted-by":"publisher","first-page":"482","DOI":"10.1002\/jcd.21777","volume":"29","author":"M Meszka","year":"2021","unstructured":"Meszka M., Rosa A.: Maximal partial Room squares. J. Combin. Des. 29, 482\u2013501 (2021). https:\/\/doi.org\/10.1002\/jcd.21777.","journal-title":"J. Combin. Des."},{"issue":"2","key":"1704_CR7","first-page":"601","volume":"42","author":"J Quistorff","year":"2001","unstructured":"Quistorff J.: On codes with given minimum distance and covering radius. Beitr. Algebra Geom. 42(2), 601\u2013611 (2001).","journal-title":"Beitr. Algebra Geom."},{"key":"1704_CR8","first-page":"9","volume":"23","author":"A Rosa","year":"2015","unstructured":"Rosa A.: Maximal designs and configurations\u2014a survey. Acta Univ. M. Belii Ser. Math. 23, 9\u201325 (2015).","journal-title":"Acta Univ. M. Belii Ser. 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