A binary linear code of length \(n\), dimension \(k\), and minimum distance at least \(d\) is called an \([n,k,d]\)-code. Let \(d(n,k) = \max \{d : \text{there exists an } [n,k,d]\text{-code}\}\). It is currently known by [6] that \(26 \leq d(66,13) \leq 28\). The nonexistence of a linear \([66,13,28]\)-code is proven.

In this paper, we completely solve the existence problem of \(\text{LOTS}(v)\) (i.e. large set of pairwise disjoint ordered triple systems of order \(v\)).

It is shown that a resolvable BIBD with block size five and index two exists whenever \(v \equiv 5 \pmod{10}\) and \(v \geq 50722395\). This result is based on an updated result on the existence of a BIBD with block size six and index unity, which leaves \(88\) unsolved cases. A construction using difference families to obtain resolvable BIBDs is also presented.

Functions \(c(n)\) and \(h(n)\) which count certain consecutive-integer partitions of a positive integer \(n\) are evaluated, and combinatorial interpretations of partitions with “\(c(n)\) copies of \(n\)” and “\(h(n)\) copies of \(n\)” are given.

J. Leech has posed the following problem: For each integer \(n\), what is the greatest integer \(N\) such that there exists a labelled tree with \(n\) nodes in which the distance between the pairs of nodes include the consecutive values \(1,2,\ldots,N\)? With the help of a computer, we get \(B(n)\) (the number \(N\) for branched trees) for \(2 \leq n \leq 10\) and lower bounds of \(B(11)\) and \(B(12)\). We also get \(U(n)\) (the number \(N\) for unbranched trees) for \(2 \leq n \leq 11\) independently, confirming some results gotten by J. Leech.

A method is presented for constructing simple partially balanced designs from \(t-(v,k,\lambda)\) designs. When the component designs satisfy a compatibility condition the result is a simple balanced design. The component designs can even be trivial (with some exceptions) with the resulting design being nontrivial. The automorphism group of the composition is given in terms of the automorphism groups of the component designs. Some previously unknown simple designs are constructed, including an infinite family of \(3\)-designs that are extremal with respect to an inequality of Cameron and Praeger. Some analogous theorems are given for difference families.

In this paper, constructions of simple cyclic \(2\)-designs are given. As a consequence, we determined the existence of simple \(2\)-\((q,k,\lambda)\) designs for every admissible parameter set \((q,k,\lambda)\) where \(q \leq 29\) is an odd prime power, with two undecided parameter sets \((q,k,\lambda) = (29,8,6)\) and \((29,8,10)\).

A map is an embedding of a graph into a surface so that each face is simply connected. Geometric duality, whereby vertices and faces are reversed, is a classic construction for maps. A generalization of map duality is given and discussed both graph and group theoretically.