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 ≡ 0 (mod 2) and t ≥ 3 with the exception of m ≡ 0 (mod 2) and t = 3. When n ≥ 4, the case of t odd is considered. It is established that if t ≡ 1 (mod 2) and n ≥ 4, then there exists a 3-SCHGDD of type (n, m t ) if and only if t ≥ 3 and (t − 1)n(n − 1)m ≡ 0 (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:We consider 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 t is odd and n = 8 or t is doubly even and t = 8, the existence problem is completely solved; when t is singly even, many infinite families are obtained. Applications of our results to twodimensional balanced sampling plans and optimal two-dimensional optical orthogonal codes are also discussed.

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Strong difference families are an interesting class of discrete structures which can be used to derive relative difference families. Relative difference families are closely related to 2-designs, and have applications in constructions for many significant codes, such as optical orthogonal codes and optical orthogonal signature pattern codes. In this paper, with a careful use of cyclotomic conditions attached to strong difference families, we improve the lower bound on the asymptotic existence results of (F p ×F q , F p ×{0}, k, λ)-DFs for k ∈ {p, p + 1}. We improve Buratti's existence results for 2-(13q, 13, λ) designs and 2-(17q, 17, λ) designs, and establish the existence of seven new 2-(v, k, λ) designs for (v, k, λ)

Bovine teeth have been considered as an excellent substitute for human teeth for dental research, however, the enamel microstructures of bovine incisors that include arrangements of prisms and interprisms, and their spatial relationships have not been well described. The aim of this study was to investigate the detail enamel microstructures of bovine incisors. Eight bovine mandibular incisors were cut into 77 pieces at eight equal intervals either in the longitudinal direction or in the horizontal direction before each piece had been tangentially cut (parallel to enameldentin junction) through the middle of the enamel thickness. All the sectioned surfaces were treated 1 M HCl for 10 sec to expose the prisms and interprisms before observation by scanning electron microscopy. The parallel enamel prisms were located in all the outer enamel, the cervical region and the incisal ridge of the bovine incisors. Most labial inner enamel and the cingulum of lingual inner enamel were composed of the Hunter-Schreger bands with the characteristics of decussating groups of prisms and decussating planes between interprisms and prisms. The interprisms were thicker in the inner enamel than in the outer enamel.