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We determine a necessary and sufficient condition for the existence of semicyclic holey group divisible designs with block size three and group type (n,mt). New recursive constructions on semicyclic incomplete holey group divisible designs are introduced to settle this problem completely. [ABSTRACT FROM AUTHOR]

Frame difference families, which can be obtained via a careful use of cyclotomic conditions attached to strong difference families, play an important role in direct constructions for resolvable balanced incomplete block designs. We establish asymptotic existences for several classes of frame difference families. As corollaries new infinite families of 1-rotational (pq+1,p+1,1)-RBIBDs over Fp+×Fq+ are derived, and the existence of (125q+1,6,1)-RBIBDs is discussed. We construct (v, 8, 1)-RBIBDs for v∈{624,1576,2976,5720,5776,10200,14176,24480}, whose existence were previously in doubt. As applications, we establish asymptotic existences for an infinite family of optimal constant composition codes and an infinite family of strictly optimal frequency hopping sequences. [ABSTRACT FROM AUTHOR]

We consider the existence problem for a semi-cyclic holey group divisible design of type [ABSTRACT FROM AUTHOR]

In this paper we discuss the existence problem for a semi-cyclic holey group divisible design of type $$(n,m^t)$$ with block size $$3$$ , which is denoted by a $$3$$ -SCHGDD of type $$(n,m^t)$$ . When $$n=3$$ , a $$3$$ -SCHGDD of type $$(3,m^t)$$ is equivalent to a $$(3,mt;m)$$ -cyclic holey difference matrix, denoted by a $$(3,mt;m)$$ -CHDM. It is shown that there is a $$(3,mt;m)$$ -CHDM if and only if $$(t-1)m\equiv 0\ (\mathrm{mod}\ 2)$$ and $$t\ge 3$$ with the exception of $$m\equiv 0\ (\mathrm{mod}\ 2)$$ and $$t=3$$ . When $$n\ge 4$$ , the case of $$t$$ odd is considered. It is established that if $$t\equiv 1\ (\mathrm{mod}\ 2)$$ and $$n\ge 4$$ , then there exists a $$3$$ -SCHGDD of type $$(n,m^t)$$ if and only if $$t\ge 3$$ and $$(t-1)n(n-1)m\equiv 0\ (\mathrm{mod}\ 6)$$ with some possible exceptions of $$n=6$$ and $$8$$ . The main results in this paper have been used to construct optimal two-dimensional optical orthogonal codes with weight $$3$$ and different auto- and cross-correlation constraints by the authors recently. [ABSTRACT FROM AUTHOR]

Strong difference families of special types are introduced to produce new relative difference families from the point of view of both asymptotic existences and concrete examples. As applications, group divisible designs of type 3 0 u with block size 6 are discussed, r -rotational balanced incomplete block designs with block size 6 are derived for r ∈ { 6 , 10 } , and several classes of optimal optical orthogonal codes with weight 5, 6, 7, or 8 are obtained. [ABSTRACT FROM AUTHOR]