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Motivated by their application in the synthesis of DNA nanomaterials, geometric orthogonal codes (briefly GOCs) were introduced recently. We establish the equivalence between (n 1 × n 2 , w , 1) -GOCs and a certain type of combinatorial configurations, called (n 1 × n 2 , w , 1) -geometrical difference packings. Based on this relationship, upper bounds for the number of codewords are presented and recursive constructions are provided. As a consequence, the number of codewords in an optimal (n 1 × n 2 , 3 , 1) -GOC is determined for any positive integers n 1 and n 2 . [ABSTRACT FROM AUTHOR]

Even though Peltesohn proved that a cyclic (v, 3, 1)-design exists if and only if v ≡ 1 , 3 (mod 6) as early as 1939, the problem of determining the spectrum of cyclic (v, k, 1)-designs with k > 3 is far from being settled, even for k = 4 . This paper shows that a cyclic (v, 4, 1)-design exists if and only if v ≡ 1 , 4 (mod 12) and v ∉ { 16 , 25 , 28 } . [ABSTRACT FROM AUTHOR]

This paper introduces almost partitionable sets (APSs) to generalize the known concept of partitionable sets. These notions provide a unified frame to construct Z‐cyclic patterned starter whist tournaments and cyclic balanced sampling plans excluding contiguous units. The existences of partitionable sets and APSs are investigated. As an application, a large number of optical orthogonal codes achieving the Johnson bound or the Johnson bound minus one are constructed. [ABSTRACT FROM AUTHOR]

This paper focuses on constructions for optimal 2-D (n × m , 3 , 2 , 1) -optical orthogonal codes with m ≡ 0 (mod 4) . An upper bound on the size of such codes is established. It relies heavily on the size of optimal equi-difference 1-D (m, 3, 2, 1)-optical orthogonal codes, which is closely related to optimal equi-difference conflict avoiding codes with weight 3. The exact number of codewords of an optimal 2-D (n × m , 3 , 2 , 1) -optical orthogonal code is determined for n = 1 , 2 , m ≡ 0 (mod 4) , and n ≡ 0 (mod 3) , m ≡ 8 (mod 16) or m ≡ 32 (mod 64) or m ≡ 4 , 20 (mod 48) . [ABSTRACT FROM AUTHOR]

The study of optical orthogonal codes has been motivated by an application in an optical code-division multiple access system. From a practical point of view, compared to one-dimensional optical orthogonal codes, two-dimensional optical orthogonal codes tend to require smaller code length. On the other hand, in some circumstances only with good cross-correlation one can deal with both synchronization and user identification. These motivate the study of two-dimensional optical orthogonal codes with better cross-correlation than auto-correlation. This paper focuses on optimal two-dimensional optical orthogonal codes with the auto-correlation [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]

Given a subgroup H of a group (G , +) , a (G , H , k , 1) difference family (DF) is a set F of k -subsets of G such that { f − f ′ : f , f ′ ∈ F , f ≠ f ′ , F ∈ F } = G ∖ H. Let g Z g h be the subgroup of order h in Z g h generated by g. A (Z g h , g Z g h , k , 1) -DF is called cyclic and written as a (g h , h , k , 1) -CDF. This paper shows that for h ∈ { 2 , 3 , 6 } , there exists a (g h , h , 4 , 1) -CDF if and only if g h ≡ h (mod 12) , g ⩾ 4 and (g , h) ∉ { (9 , 3) , (5 , 6) }. As a corollary, it is shown that a 1-rotational Steiner system S (2 , 4 , v) exists if and only if v ≡ 4 (mod 12) and v ≠ 28. This solves the long-standing open problem on the existence of a 1-rotational S (2 , 4 , v). As another corollary, we establish the existence of an optimal (v , 4 , 1) -optical orthogonal code with ⌊ (v − 1) / 12 ⌋ codewords for any positive integer v ≡ 1 , 2 , 3 , 4 , 6 (mod 12) and v ≠ 25. We also give applications of our results to cyclic group divisible designs with block size four and optimal cyclic 3-ary constant-weight codes with weight four and minimum distance six. [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]