相关论文: The nonexistence of an additive quaternary [15,5,9…
Complete sets of mutually unbiased bases are only known to exist in prime-power dimensions. We will describe a few approaches to the problem proving the (non)-existence of four mutually unbiased bases in dimension 6. These will include the…
It's well known that the quadratic residue code over finite fields is an interesting class of cyclic codes for its higher minimum distance. Let $g$ be a positive integer and $p,p_{1},\ldots, p_{g}$ be distinct odd primes, the present paper…
By means of principal isotopes lH(a,b) of the algebra lH [Ra 99] we give an exhaustive and not repetitive description of all 4-dimensional absolute-valued algebras satisfying (x^p, x^q, x^r) = 0 for fixed integers p, q, r \in\{1,2\}. For…
One of the most fundamental topics in subspace coding is to explore the maximal possible value ${\bf A}_q(n,d,k)$ of a set of $k$-dimensional subspaces in $\mathbb{F}_q^n$ such that the subspace distance satisfies $\operatorname{d_S}(U,V) =…
We show that Kervaire invariant one elements in the homotopy groups of spheres exist only in dimensions at most 126. By Browder's Theorem, this means that smooth framed manifolds of Kervaire invariant one exist only in dimensions 2, 6, 14,…
We determine the minimal number of variables $\Gamma^*(d, K)$ which guarantees a nontrivial solution for every additive form of degree $d=4$ over the four ramified quadratic extensions $\mathbb{Q}_2(\sqrt{2}), \mathbb{Q}_2(\sqrt{10}),…
Additive codes have attracted considerable attention for their potential to outperform linear codes. However, distinguishing strictly additive codes from those that are equivalent to linear codes remains a fundamental challenge. To resolve…
We show that the non-vanishing conjecture implies the abundance conjecture when $\nu\leq 1$. We also prove the abundance conjecture in dimension $\leq 5$ when $\kappa\geq 0$ and $\nu\leq 1$ unconditionally.
Let $p$ be a prime number, $a_1, a_2, \ldots a_{4p-4}$ a sequence of elements in $(\mathbb{Z}/p\\mathbb{Z})^2$, which does not contain a subsequence of length $p$ which adds up to 0. We show that if $p$ is sufficiently large, then the…
The {\em Fibonacci cube} of dimension $n$, denoted as $\Gamma\_n$, is the subgraph of the $n$-cube $Q\_n$ induced by vertices with no consecutive 1's. In an article of 2016 Ashrafi and his co-authors proved the non-existence of perfect…
We examine the following fourth order H\'enon equation \label{pipe} \Delta^2 u = |x|^\alpha u^p \qquad \text{in}\ \IR^N, where $ 0 < \alpha$. Define the Hardy-Sobolev exponent $ p_4(\alpha):= \frac{N+4 + 2 \alpha}{N-4}$. We show that in…
A cap set in $\mathbb{F}_3^n$ is a subset that contains no three elements adding to 0. Building on a construction of Edel, a recent paper of Tyrrell gave the first improvement to the lower bound for a size of a cap set in two decades…
We show that a quartic $p$-adic form with at least $3192$ variables possesses a non-trivial zero. We also prove new results on systems of cubic, quadratic and linear forms. As an example, we show that for a system comprising two cubic forms…
New results on pentagonal geometries PENT(k,r) with block sizes k = 3 or k = 4 are given. In particular we completely determine the existence spectra for PENT(3,r) systems with the maximum number of opposite line pairs as well as those…
In this paper we provide a $4$-GDD of type $2^2 5^5$, thereby solving the existence question for the last remaining feasible type for a $4$-GDD with no more than $30$ points. We then show that $4$-GDDs of type $2^t 5^s$ exist for all but a…
Recently, simplicial complexes are used in constructions of several infinite families of minimal and optimal linear codes by Hyun {\em et al.} Building upon their research, in this paper more linear codes over the ring $\mathbb{Z}_4$ are…
This correspondence introduces two new constructive techniques to complete the determination of the sizes of optimal q-ary codes of constant weight three and distance four.
We show that the complex-hyperbolic Einstein Dehn filling compactification cannot possibly be performed in dimension four.
The Fibonacci cube of dimension n, denoted as $\Gamma$ n , is the subgraph of the n-cube 5 Q n induced by vertices with no consecutive 1's. Ashrafi and his co-authors proved the non-existence of perfect codes in $\Gamma$ n for n $\ge$ 4. As…
We prove that a putative $[72,36,16]$ code is not the image of linear code over $\ZZ_4$, $\FF_2 + u \FF_2$ or $\FF_2+v\FF_2$, thus proving that the extremal doubly even $[72,36,16]$-binary code cannot have an automorphism group containing a…