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Register a new userConstructions of difference sets in nonabelian 2-groups
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Difference sets have been studied for more than 80 years. Techniques from algebraic number theory, group theory, finite geometry, and digital communications engineering have been used to establish constructive and nonexistence results. We provide a new theoretical approach which dramatically expands the class of $2$-groups known to contain a difference set, by refining the concept of covering extended building sets introduced by Davis and Jedwab in 1997. We then describe how product constructions and other methods can be used to construct difference sets in some of the remaining $2$-groups. We announce the completion of ten years of collaborative work to determine precisely which of the 56,092 nonisomorphic groups of order 256 contain a difference set. All groups of order 256 not excluded by the two classical nonexistence criteria are found to contain a difference set, in agreement with previous findings for groups of order 4, 16, and 64. We provide suggestions for how the existence question for difference sets in $2$-groups of all orders might be resolved.
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We revisit the old idea of constructing difference sets from cyclotomic classes. Two constructions of skew Hadamard difference sets are given in the additive groups of finite fields using unions of cyclotomic classes of order $N=2p_1^m$, where $p_1$ is a prime and $m$ a positive integer. Our main tools are index 2 Gauss sums, instead of cyclotomic numbers.
We revisit the problem of constructing Menon-Hadamard difference sets. In 1997, Wilson and Xiang gave a general framework for constructing Menon-Hadamard difference sets by using a combination of a spread and four projective sets of type Q in ${mathrm{PG}}(3,q)$. They also found examples of suitable spreads and projective sets of type Q for $q=5,13,17$. Subsequently, Chen (1997) succeeded in finding a spread and four projective sets of type Q in ${mathrm{PG}}(3,q)$ satisfying the conditions in the Wilson-Xiang construction for all odd prime powers $q$. Thus, he showed that there exists a Menon-Hadamard difference set of order $4q^4$ for all odd prime powers $q$. However, the projective sets of type Q found by Chen have automorphisms different from those of the examples constructed by Wilson and Xiang. In this paper, we first generalize Chens construction of projective sets of type Q by using `semi-primitive cyclotomic classes. This demonstrates that the construction of projective sets of type Q satisfying the conditions in the Wilson-Xiang construction is much more flexible than originally thought. Secondly, we give a new construction of spreads and projective sets of type Q in ${mathrm{PG}}(3,q)$ for all odd prime powers $q$, which generalizes the examples found by Wilson and Xiang. This solves a problem left open in Section 5 of the Wilson-Xiang paper from 1997.
In this paper, we give a construction of strongly regular Cayley graphs and a construction of skew Hadamard difference sets. Both constructions are based on choosing cyclotomic classes in finite fields, and they generalize the constructions given by Feng and Xiang cite{FX111,FX113}. Three infinite families of strongly regular graphs with new parameters are obtained. The main tools that we employed are index 2 Gauss sums, instead of cyclotomic numbers.
Let $G$ be a finite group and $D_{2n}$ be the dihedral group of $2n$ elements. For a positive integer $d$, let $mathsf{s}_{dmathbb{N}}(G)$ denote the smallest integer $ellin mathbb{N}_0cup {+infty}$ such that every sequence $S$ over $G$ of length $|S|geq ell$ has a nonempty $1$-product subsequence $T$ with $|T|equiv 0$ (mod $d$). In this paper, we mainly study the problem for dihedral groups $D_{2n}$ and determine their exact values: $mathsf{s}_{dmathbb{N}}(D_{2n})=2d+lfloor log_2nrfloor$, if $d$ is odd with $n|d$; $mathsf{s}_{dmathbb{N}}(D_{2n})=nd+1$, if $gcd(n,d)=1$. Furthermore, we also analysis the problem for metacyclic groups $C_pltimes_s C_q$ and obtain a result: $mathsf{s}_{kpmathbb{N}}(C_pltimes_s C_q)=lcm(kp,q)+p-2+gcd(kp,q)$, where $pgeq 3$ and $p|q-1$.
A subset $B$ of an Abelian group $G$ is called a difference basis of $G$ if each element $gin G$ can be written as the difference $g=a-b$ of some elements $a,bin B$. The smallest cardinality $|B|$ of a difference basis $Bsubset G$ is called the difference size of $G$ and is denoted by $Delta[G]$. We prove that for every $ninmathbb N$ the cyclic group $C_n$ of order $n$ has difference size $frac{1+sqrt{4|n|-3}}2le Delta[C_n]lefrac32sqrt{n}$. If $nge 9$ (and $nge 2cdot 10^{15}$), then $Delta[C_n]lefrac{12}{sqrt{73}}sqrt{n}$ (and $Delta[C_n]<frac2{sqrt{3}}sqrt{n}$). Also we calculate the difference sizes of all cyclic groups of cardinality $le 100$.
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