A Steiner quadruple system of order v (SQS(v)) is said to be have an almost spanning block design and denoted by 1-AFSQS(v) if it contains a subdesign S(2,4,v−1). In 1992, Hartman and Phelps posed a problem: Show that there exists a 1-AFSQS(v) for each v≡2(mod12). In this paper, we prove that the necessary condition for the existence of a 1-AFSQS(v) is also sufficient with a definite exception v=14 and possible exceptions v∈{86,206,374,398}.

An Hurn:x-wiley:10638539:media:jcd21640:jcd21640-math-0001 design is a triple urn:x-wiley:10638539:media:jcd21640:jcd21640-math-0002, where urn:x-wiley:10638539:media:jcd21640:jcd21640-math-0003 is a set of urn:x-wiley:10638539:media:jcd21640:jcd21640-math-0004 points, urn:x-wiley:10638539:media:jcd21640:jcd21640-math-0005 a partition of urn:x-wiley:10638539:media:jcd21640:jcd21640-math-0006 into urn:x-wiley:10638539:media:jcd21640:jcd21640-math-0007 disjoint sets of size urn:x-wiley:10638539:media:jcd21640:jcd21640-math-0008, and urn:x-wiley:10638539:media:jcd21640:jcd21640-math-0009 a set of urn:x-wiley:10638539:media:jcd21640:jcd21640-math-0010‐element transverses of urn:x-wiley:10638539:media:jcd21640:jcd21640-math-0011, such that each urn:x-wiley:10638539:media:jcd21640:jcd21640-math-0012‐element transverse of urn:x-wiley:10638539:media:jcd21640:jcd21640-math-0013 is contained in exactly one of them. In 1990, Mills determined the existence of an Hurn:x-wiley:10638539:media:jcd21640:jcd21640-math-0014 design with urn:x-wiley:10638539:media:jcd21640:jcd21640-math-0015. In this paper, an efficient construction shows that an Hurn:x-wiley:10638539:media:jcd21640:jcd21640-math-0016 exists for any integer urn:x-wiley:10638539:media:jcd21640:jcd21640-math-0017 with urn:x-wiley:10638539:media:jcd21640:jcd21640-math-0018. Consequently, the necessary and sufficient conditions for the existence of an Hurn:x-wiley:10638539:media:jcd21640:jcd21640-math-0019 design are urn:x-wiley:10638539:media:jcd21640:jcd21640-math-0020, urn:x-wiley:10638539:media:jcd21640:jcd21640-math-0021, and urn:x-wiley:10638539:media:jcd21640:jcd21640-math-0022, with a definite exception urn:x-wiley:10638539:media:jcd21640:jcd21640-math-0023.

Optical orthogonal signature pattern codes (OOSPCs) have played an important role in a novel type of optical code-division multiple-access network for 2-D image transmission. In this paper, we give four direct constructions for OOSPCs based on polynomials and rational functions over finite fields. We also use r -simple matrices to present a recursive construction for OOSPCs. These constructions yield new families of asymptotically optimal OOSPCs.

Frequency hopping sequences (FHSs) with favorable partial Hamming correlation properties have important applications in many synchronization and multiple-access systems. In this paper, we investigate constructions of FHSs and FHS sets with optimal partial Hamming correlation. We first establish a correspondence between FHS sets with optimal partial Hamming correlation and multiple partition-type balanced nested cyclic difference packings with a special property. By virtue of this correspondence, some FHSs and FHS sets with optimal partial Hamming correlation are constructed from various combinatorial structures such as cyclic difference packings, and cyclic relative difference families. We also describe a direct construction and two recursive constructions for FHS sets with optimal partial Hamming correlation. As a consequence, our constructions yield new FHSs and FHS sets with optimal partial Hamming correlation.