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Codes Cryptogr."],"published-print":{"date-parts":[[2025,9]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>In this paper, respectively 8, 10 and 9 mutually orthogonal Latin squares (MOLS) of sizes <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$n=54$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>n<\/mml:mi>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mn>54<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>, 96 and 108 are obtained (previously, only 5, 9 and 8 MOLS were respectively known for these values). The cases <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$n=54$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>n<\/mml:mi>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mn>54<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> and 96 are obtained by constructing separable permutation codes consisting of <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$8 \\times 54$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mn>8<\/mml:mn>\n                    <mml:mo>\u00d7<\/mml:mo>\n                    <mml:mn>54<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> and <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$10 \\times 96$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mn>10<\/mml:mn>\n                    <mml:mo>\u00d7<\/mml:mo>\n                    <mml:mn>96<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> codewords, respectively; in addition, these codes respectively have lengths 54, 96 and minimum distances 53, 95. This construction method was used in an earlier paper to obtain 6, 10 and 8 MOLS of sizes <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$n=35$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>n<\/mml:mi>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mn>35<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>, 54 and 63, respectively. The case <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$n=108$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>n<\/mml:mi>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mn>108<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> is obtained by constructing a (108,\u00a010,\u00a01) difference matrix. Also, a previous error when constructing 6 MOLS of size <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$n=45$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>n<\/mml:mi>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mn>45<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> is corrected.<\/jats:p>","DOI":"10.1007\/s10623-025-01629-5","type":"journal-article","created":{"date-parts":[[2025,5,28]],"date-time":"2025-05-28T11:18:00Z","timestamp":1748431080000},"page":"3565-3573","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Improvements for lower bounds of mutually orthogonal Latin squares of sizes 54, 96 and 108"],"prefix":"10.1007","volume":"93","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3632-9612","authenticated-orcid":false,"given":"R. 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