Let $v$ be a product of at most three not necessarily distinct primes. We prove that there exists no strong external difference family with more than two subsets in abelian group $G$ of order $v$, except possibly when $G=C_p^3$ and $p$ is a prime greater than $3 times 10^{12}$.
In this paper, we present some new nonexistence results on $(m,n)$-generalized bent functions, which improved recent results. More precisely, we derive new nonexistence results for general $n$ and $m$ odd or $m equiv 2 pmod{4}$, and further explicitl
y prove nonexistence of $(m,3)$-generalized bent functions for all integers $m$ odd or $m equiv 2 pmod{4}$. The main tools we utilized are certain exponents of minimal vanishing sums from applying characters to group ring equations that characterize $(m,n)$-generalized bent functions.
Let $q$ be a prime power of the form $q=12c^2+4c+3$ with $c$ an arbitrary integer. In this paper we construct a difference family with parameters $(2q^2;q^2,q^2,q^2,q^2-1;2q^2-2)$ in ${mathbb Z}_2times ({mathbb F}_{q^2},+)$. As a consequence, by appl
ying the Wallis-Whiteman array, we obtain Hadamard matrices of order $4(2q^2+1)$ for the aforementioned $q$s.
Let $q$ be a power of a prime $p$, let $k$ be a nontrivial divisor of $q-1$ and write $e=(q-1)/k$. We study upper bounds for cyclotomic numbers $(a,b)$ of order $e$ over the finite field $mathbb{F}_q$. A general result of our study is that $(a,b)leq
3$ for all $a,b in mathbb{Z}$ if $p> (sqrt{14})^{k/ord_k(p)}$. More conclusive results will be obtained through separate investigation of the five types of cyclotomic numbers: $(0,0), (0,a), (a,0), (a,a)$ and $(a,b)$, where $a eq b$ and $a,b in {1,dots,e-1}$. The main idea we use is to transform equations over $mathbb{F}_q$ into equations over the field of complex numbers on which we have more information. A major tool for the improvements we obtain over known results is new upper bounds on the norm of cyclotomic integers.
In this paper, we obtain a number of new infinite families of Hadamard matrices. Our constructions are based on four new constructions of difference families with four or eight blocks. By applying the Wallis-Whiteman array or the Kharaghani array to
the difference families constructed, we obtain new Hadamard matrices of order $4(uv+1)$ for $u=2$ and $vin Phi_1cup Phi_2 cup Phi_3 cup Phi_4$; and for $uin {3,5}$ and $vin Phi_1cup Phi_2 cup Phi_3$. Here, $Phi_1={q^2:qequiv 1pmod{4}mbox{ is a prime power}}$, $Phi_2={n^4in mathbb{N}:nequiv 1pmod{2}} cup {9n^4in mathbb{N}:nequiv 1pmod{2}}$, $Phi_3={5}$ and $Phi_4={13,37}$. Moreover, our construction also yields new Hadamard matrices of order $8(uv+1)$ for any $uin Phi_1cup Phi_2$ and $vin Phi_1cup Phi_2 cup Phi_3$.
We prove the nonexistence of lattice tilings of $mathbb{Z}^n$ by Lee spheres of radius $2$ for all dimensions $ngeq 3$. This implies that the Golomb-Welch conjecture is true when the common radius of the Lee spheres equals $2$ and $2n^2+2n+1$ is a pr
ime. As a direct consequence, we also answer an open question in the degree-diameter problem of graph theory: the order of any abelian Cayley graph of diameter $2$ and degree larger than $5$ cannot meet the abelian Cayley Moore bound.
In this paper, we make some progress towards a well-known conjecture on the minimum weights of binary cyclic codes with two primitive nonzeros. We also determine the Walsh spectrum of $Tr(x^d)$ over $F_{2^{m}}$ in the case where $m=2t$, $d=3+2^{t+1}$ and $gcd(d, 2^{m}-1)=1$.
In a recent paper [M], Mathon gives a new construction of maximal arcs which generalizes the construction of Denniston. In relation to this construction, Mathon asks the question of determining the largest degree of a non-Denniston maximal arc arisin
g from his new construction. In this paper, we give a nearly complete answer to this problem. Specifically, we prove that when $mgeq 5$ and $m eq 9$, the largest $d$ of a non-Denniston maximal arc of degree $2^d$ in PG(2,2^m) generated by a {p,1}-map is $(floor {m/2} +1)$. This confirms our conjecture in [FLX]. For {p,q}-maps, we prove that if $mgeq 7$ and $m eq 9$, then the largest $d$ of a non-Denniston maximal arc of degree $2^d$ in PG(2,2^m) generated by a {p,q}-map is either $floor {m/2} +1$ or $floor{m/2} +2$.
