GDD with equal number of even and odd blocks

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• 作者：Issa Ndungo^a ; Dinesh G. Sarvate^b ; ^{sarvated@cofc.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Group divisible designs ; Latin squares ; Even and odd GDDs ; IMOLS ; SOLS
• 刊名：Discrete Mathematics
• 出版年：2016
• 出版时间：6 April 2016
• 年：2016
• 卷：339
• 期：4
• 页码：1344-1354
• 全文大小：421 K

文摘

We prove that the necessary condition, n≡0$n\equiv 0$(mod 3), is sufficient for the existence of GDD(n,2,4;3,4$n,2,4;3,4$) except possibly for n=18$n=18$. We prove that necessary conditions for the existence of group divisible designs GDD(n,2,4;λ₁,λ₂$n,2,4;{\lambda }_1,{\lambda }_2$) with equal number of even and odd blocks are sufficient for GDD(n,2,4;5n,7(n−1))$(n,2,4;5n,7(n-1))$ for all n≥2$n\ge 2$, GDD(7s,2,4;5s,7s−1)$(7s,2,4;5s,7s-1)$ for all s$s$, GDD(5t+1,2,4;5t+1,7t)$(5t+1,2,4;5t+1,7t)$ for t≡$t\equiv$ 0(mod 2) and GDD(5t+1,2,4;2(5t+1),14t)$(5t+1,2,4;2(5t+1),14t)$ for all t$t$. To complete the existence of such GDDs, one needs to construct two more families: GDD(5t+1,2,4;5t+1,7t)$(5t+1,2,4;5t+1,7t)$ for all odd t$t$, and GDD(35s+21,2,4;5s+3,7s+4)$(35s+21,2,4;5s+3,7s+4)$ for all positive integers s$s$.