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Douglas R. Stinson

Publications and source records attributed to Douglas R. Stinson.

At least 19 recordsLinked to original sources
arXiv2026

An Explication of Optimal Equidistant Codes

We discuss the problem of characterizing equidistant binary codes of a given length $n$ having largest possible distance and the maximum number of codewords. Such characterizations have been studied by several authors over the years and they involve symmetric BIBDs with certain parameters. In this primarily expository paper, we investigate the history of this problem and give a unified presentation of the main results. Perhaps surprisingly, researchers on this problem were unaware of early relevant work by Marrero and Butson from 1973. Also, it turns out that published results on characterizations of equidistant binary codes have missed one of the possible subcases when $n \equiv 2 \bmod 4$.

math.CO↗
arXiv2026

An introduction to local differential privacy protocols using block designs

The design of protocols for local differential privacy (or LDP) has been a topic of considerable research interest in recent years. LDP protocols utilise the randomised encoding of outcomes of an experiment using a transition probability matrix (TPM). Several authors have observed that balanced incomplete block designs (BIBDs) provide nice examples of TPMs for LDP protocols. Indeed, it has been shown that such BIBD-based LDP protocols provide optimal estimators. In this primarily expository paper, we give a detailed introduction to LDP protocols and their connections with block designs. We prove that a subclass of LDP protocols known as pure LDP protocols are equivalent to $(r,λ)$-designs (which contain balanced incomplete block designs as a special case). An unbiased estimator for an LDP scheme is a left inverse of the transition probability matrix. We show that the optimal estimators for BIBD-based TPMs are precisely those obtained from the Moore-Penrose inverse of the corresponding TPM. We also review some existing work on optimal LDP protocols in the context of pure protocols.

math.CO↗
arXiv2025

$λ$-fold near-factorizations of groups

We initiate the study of $λ$-fold near-factorizations of groups with $λ> 1$. While $λ$-fold near-factorizations of groups with $λ= 1$ have been studied in numerous papers, this is the first detailed treatment for $λ> 1$. We establish fundamental properties of $λ$-fold near-factorizations and introduce the notion of equivalence. We prove various necessary conditions of $λ$-fold near-factorizations, including upper bounds on $λ$. We present three constructions of infinite families of $λ$-fold near-factorizations, highlighting the characterization of two subfamilies of $λ$-fold near-factorizations. We discuss a computational approach to $λ$-fold near-factorizations and tabulate computational results for abelian groups of small order.

math.GR↗
arXiv2025

Near-factorizations of dihedral groups

We investigate near-factorizations of nonabelian groups, concentrating on dihedral groups. We show that some known constructions of near-factorizations in dihedral groups yield equivalent near-factorizations. In fact, there are very few known examples of nonequivalent near-factorizations in dihedral or other nonabelian groups; we provide some new examples with the aid of the computer. We also analyse a construction for near-factorizations in dihedral groups from near-factorizations in cyclic groups, due to Pêcher, and we investigate when nonequivalent near-factorizations can be obtained by this method.

math.GR↗
arXiv2024

Uniqueness and explicit computation of mates in near-factorizations

We show that a "mate'' $B$ of a set $A$ in a near-factorization $(A,B)$ of a finite group $G$ is unique. Further, we describe how to compute the mate $B$ very efficiently using an explicit formula for $B$. We use this approach to give an alternate proof of a theorem of Wu, Yang and Feng, which states that a strong circular external difference family cannot have more than two sets. We prove some new structural properties of near-factorizations in certain classes of groups. Then we examine all the noncyclic abelian groups of order less than $200$ in a search for a possible nontrivial near-factorization. All of these possibilities are ruled out, either by theoretical criteria or by exhaustive computer searches. (In contrast, near-factorizations in cyclic or dihedral groups are known to exist by previous results.) We also look briefly at nontrivial near-factorizations of index $λ> 1$ in noncyclic abelian groups. Various examples are found with $λ= 2$ by computer.

math.GR↗
arXiv2024

Strong External Difference Families and Classification of $α$-valuations

One method of constructing $(a^2+1, 2,a, 1)$-SEDFs (i.e., strong external difference families) in $\mathbb{Z}_{a^2+1}$ makes use of $α$-valuations of complete bipartite graphs $K_{a,a}$. We explore this approach and we provide a classification theorem which shows that all such $α$-valuations can be constructed recursively via a sequence of ``blow-up'' operations. We also enumerate all $(a^2+1, 2,a, 1)$-SEDFs in $\mathbb{Z}_{a^2+1}$ for $a \leq 14$ and we show that all these SEDFs are equivalent to $α$-valuations via affine transformations. Whether this holds for all $a > 14$ as well is an interesting open problem. We also study SEDFs in dihedral groups, where we show that two known constructions are equivalent.

math.CO↗
arXiv2024

Weak and Strong Nestings of BIBDs

We study two types of nestings of balanced incomplete block designs (BIBDs). In both types of nesting, we wish to add a point (the nested point) to every block of a $(v,k,λ)$-BIBD in such a way that we end up with a partial $(w,k+1,λ+1)$-BIBD for some $w \geq v$. In the case where $w > v$, we are introducing $w-v$ new points. This is called a weak nesting. A strong nesting satisfies the stronger property that no pair containing a new point occurs more than once in the partial $(w,k+1,λ+1)$-BIBD. In both cases, the goal is to minimize $w$. We prove lower bounds on $w$ as a function of $v$, $k$ and $λ$ and we find infinite classes of $(v,2,1)$- and $(v,3,2)$-BIBDs that have optimal nestings.

math.CO↗
arXiv2024

Nestings of BIBDs with block size four

In a nesting of a balanced incomplete block design (or BIBD), we wish to add a point (the \emph{nested point}) to every block of a $(v,k,λ)$-BIBD in such a way that we end up with a partial $(v,k+1,λ+1)$-BIBD. In the case where the partial $(v,k+1,λ+1)$-BIBD is in fact a $(v,k+1,λ+1)$-BIBD, we have a \emph{perfect nesting}. We show that a nesting is perfect if and only if $k = 2 λ+ 1$. Perfect nestings were previously known to exist in the case of Steiner triple systems (i.e., $(v,3,1)$-BIBDs) when $v \equiv 1 \bmod 6$, as well as for some symmetric BIBDs. Here we study nestings of $(v,4,1)$-BIBDs, which are not perfect nestings. We prove that there is a nested $(v,4,1)$-BIBD if and only if $v \equiv 1 \text{ or } 4 \bmod 12$, $v \geq 13$. This is accomplished by a variety of direct and recursive constructions.

math.CO↗
arXiv2024

A method of constructing pairwise balanced designs containing parallel classes

The obvious way to construct a GDD (group-divisible design) recursively is to use Wilson's Fundamental Construction for GDDs (WFC). Then a PBD (pairwise balanced design) is often obtained by adding a new point to each group of the GDD. However, after constructing such a PBD, it might be the case that we then want to identify a parallel class of blocks. In this short note, we explore some possible ways of doing this.

math.CO↗
arXiv2024

On min-base palindromic representations of powers of 2

A positive integer $N$ is \emph{palindromic in the base $b$} when $N = \sum_{i=0}^{k} c_i b^i$, $c_k\neq 0$,and $c_i=c_{k-i},\; i=0,1,2,...,k$, Focusing on powers of 2, we investigate the smallest base $b$ when $N=2^n$ is palindromic in the base $b$.

math.NT↗
arXiv2023

Circular external difference families, graceful labellings and cyclotomy

(Strong) circular external difference families (which we denote as CEDFs and SCEDFs) can be used to construct nonmalleable threshold schemes. They are a variation of (strong) external difference families, which have been extensively studied in recent years. We provide a variety of constructions for CEDFs based on graceful labellings ($α$-valuations) of lexicographic products $C_n \boldsymbol{\cdot} K_{\ell}^c$, where $C_n$ denotes a cycle of length $n$. SCEDFs having more than two subsets do not exist. However, we can construct close approximations (more specifically, certain types of circular algebraic manipulation detection (AMD) codes) using the theory of cyclotomic numbers in finite fields.

math.CO↗
arXiv2023

Dispersed graph labellings

A $k$-dispersed labelling of a graph $G$ on $n$ vertices is a labelling of the vertices of $G$ by the integers $1, \dots , n$ such that $d(i,i+1) \geq k$ for $1 \leq i \leq n-1$. $DL(G)$ denotes the maximum value of $k$ such that $G$ has a $k$-dispersed labelling. In this paper, we study upper and lower bounds on $DL(G)$. Computing $DL(G)$ is NP-hard. However, we determine the exact values of $DL(G)$ for cycles, paths, grids, hypercubes and complete binary trees. We also give a product construction and we prove a degree-based bound.

math.CO↗
arXiv2023

Bounds on data limits for all-to-all comparison from combinatorial designs

In situations where every item in a data set must be compared with every other item in the set, it may be desirable to store the data across a number of machines in such a way that any two data items are stored together on at least one machine. One way to evaluate the efficiency of such a distribution is by the largest fraction of the data it requires to be allocated to any one machine. The all-to-all comparison (ATAC) data limit for $m$ machines is a measure of the minimum of this value across all possible such distributions. In this paper we further the study of ATAC data limits. We observe relationships between them and the previously studied combinatorial parameters of fractional matching numbers and covering numbers. We also prove a lower bound on the ATAC data limit that improves on one of Hall, Kelly and Tian, and examine the special cases where equality in this bound is possible. Finally, we investigate the data limits achievable using various classes of combinatorial designs. In particular, we examine the cases of transversal designs and projective Hjelmslev planes.

math.CO↗
arXiv2023

Constructions and bounds for codes with restricted overlaps

Non-overlapping codes have been studied for almost 60 years. In such a code, no proper, non-empty prefix of any codeword is a suffix of any codeword. In this paper, we study codes in which overlaps of certain specified sizes are forbidden. We prove some general bounds and we give several constructions in the case of binary codes. Our techniques also allow us to provide an alternative, elementary proof of a lower bound on non-overlapping codes due to Levenshtein in 1964.

cs.IT↗
arXiv2023

Unconditionally Secure Non-malleable Secret Sharing and Circular External Difference Families

Various notions of non-malleable secret sharing schemes have been considered. In this paper, we review the existing work on non-malleable secret sharing and suggest a novel game-based definition. We provide a new construction of an unconditionally secure non-malleable threshold scheme with respect to a specified relation. To do so, we introduce a new type of algebraic manipulation detection (AMD) code and construct examples of new variations of external difference families, which are of independent combinatorial interest.

cs.CR↗
arXiv2022

Some new results on skew frame starters in cyclic groups

In this paper, we study skew frame starters, which are strong frame starters that satisfy an additional "skew" property. We prove three new non-existence results for cyclic skew frame starters of certain types. We also construct several small examples of previously unknown cyclic skew frame starters by computer.

math.CO↗