{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,4,19]],"date-time":"2025-04-19T04:08:24Z","timestamp":1745035704081,"version":"3.40.4"},"reference-count":10,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2024,7,10]],"date-time":"2024-07-10T00:00:00Z","timestamp":1720569600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,7,10]],"date-time":"2024-07-10T00:00:00Z","timestamp":1720569600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Des. Codes Cryptogr."],"published-print":{"date-parts":[[2025,4]]},"abstract":"Abstract<\/jats:title>\n Spence\u00a0[9] constructed \n \n $$\\left( \\frac{3^{d+1}(3^{d+1}-1)}{2}, \\frac{3^d(3^{d+1}+1)}{2}, \\frac{3^d(3^d+1)}{2}\\right) $$<\/jats:tex-math>\n \n \n \n \n \n 3<\/mml:mn>\n \n d<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n \n (<\/mml:mo>\n \n 3<\/mml:mn>\n \n d<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n -<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:mfrac>\n ,<\/mml:mo>\n \n \n \n 3<\/mml:mn>\n d<\/mml:mi>\n <\/mml:msup>\n \n (<\/mml:mo>\n \n 3<\/mml:mn>\n \n d<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n +<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:mfrac>\n ,<\/mml:mo>\n \n \n \n 3<\/mml:mn>\n d<\/mml:mi>\n <\/mml:msup>\n \n (<\/mml:mo>\n \n 3<\/mml:mn>\n d<\/mml:mi>\n <\/mml:msup>\n +<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:mfrac>\n <\/mml:mfenced>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-difference sets in groups \n \n $$K \\times C_3^{d+1}$$<\/jats:tex-math>\n \n \n K<\/mml:mi>\n \u00d7<\/mml:mo>\n \n C<\/mml:mi>\n 3<\/mml:mn>\n \n d<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msubsup>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> for d<\/jats:italic> any positive integer and K<\/jats:italic> any group of order \n \n $$\\frac{3^{d+1}-1}{2}$$<\/jats:tex-math>\n \n \n \n \n 3<\/mml:mn>\n \n d<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n -<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:mfrac>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>. Smith and Webster\u00a0[8] have exhaustively studied the \n \n $$d=1$$<\/jats:tex-math>\n \n \n d<\/mml:mi>\n =<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> case without requiring that the group have the form listed above and found many constructions. Among these, one intriguing example constructs Spence difference sets in \n \n $$A_4 \\times C_3$$<\/jats:tex-math>\n \n \n \n A<\/mml:mi>\n 4<\/mml:mn>\n <\/mml:msub>\n \u00d7<\/mml:mo>\n \n C<\/mml:mi>\n 3<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> by using (3,\u00a03,\u00a03,\u00a01)-relative difference sets in a non-normal subgroup isomorphic to \n \n $$C_3^2$$<\/jats:tex-math>\n \n \n C<\/mml:mi>\n 3<\/mml:mn>\n 2<\/mml:mn>\n <\/mml:msubsup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>. Drisko\u00a0[3] has a note implying that his techniques allow constructions of Spence difference sets in groups with a noncentral normal subgroup isomorphic to \n \n $$C_3^{d+1}$$<\/jats:tex-math>\n \n \n C<\/mml:mi>\n 3<\/mml:mn>\n \n d<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msubsup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> as long as \n \n $$\\frac{3^{d+1}-1}{2}$$<\/jats:tex-math>\n \n \n \n \n 3<\/mml:mn>\n \n d<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n -<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:mfrac>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> is a prime power. We generalize this result by constructing Spence difference sets in similar families of groups, but we drop the requirement that \n \n $$\\frac{3^{d+1}-1}{2}$$<\/jats:tex-math>\n \n \n \n \n 3<\/mml:mn>\n \n d<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n -<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:mfrac>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> is a prime power. We conjecture that any group of order \n \n $$\\frac{3^{d+1}(3^{d+1}-1)}{2}$$<\/jats:tex-math>\n \n \n \n \n 3<\/mml:mn>\n \n d<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n \n (<\/mml:mo>\n \n 3<\/mml:mn>\n \n d<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n -<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:mfrac>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> with a normal subgroup isomorphic to \n \n $$C_3^{d+1}$$<\/jats:tex-math>\n \n \n C<\/mml:mi>\n 3<\/mml:mn>\n \n d<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msubsup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> will have a Spence difference set (this is analogous to Dillon\u2019s conjecture in 2-groups, and that result was proved in Drisko\u2019s work). Finally, we present the first known example of a Spence difference set in a group where the Sylow 3-subgroup is nonabelian and has exponent bigger than 3. This new construction, found by computing the full automorphism group \n \n $$\\textrm{Aut}(\\mathcal {D})$$<\/jats:tex-math>\n \n \n Aut<\/mml:mtext>\n (<\/mml:mo>\n D<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> of a symmetric design associated to a known Spence difference set and identifying a regular subgroup of \n \n $$\\textrm{Aut}(\\mathcal {D})$$<\/jats:tex-math>\n \n \n Aut<\/mml:mtext>\n (<\/mml:mo>\n D<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, uses (3,\u00a03,\u00a03,\u00a01)-relative difference sets to describe the difference set.<\/jats:p>","DOI":"10.1007\/s10623-024-01446-2","type":"journal-article","created":{"date-parts":[[2024,7,10]],"date-time":"2024-07-10T10:01:42Z","timestamp":1720605702000},"page":"879-888","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["New spence difference sets"],"prefix":"10.1007","volume":"93","author":[{"given":"James A.","family":"Davis","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"John","family":"Polhill","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ken","family":"Smith","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Eric","family":"Swartz","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jordan","family":"Webster","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,7,10]]},"reference":[{"key":"1446_CR1","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511549533","volume-title":"Design Theory. 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