In recent years, simplicial complexes have gained considerable attention as a useful tool for constructing distance-optimal codes over finite fields. In this article, we construct four infinite families of linear codes over the ring $\mathcal{R}=\mathbb{F}_{q}+u\mathbb{F}_{q}$ with $u^2=0$ using simplicial complexes with one or two maximal elements, and completely determine their Lee weight distributions via exponential-sum techniques. By employing a Gray map on $\mathcal{R}$, we obtain infinite families of distance-optimal codes over $\mathbb{F}_{q}$, including a near-Griesmer family, and establish sufficient conditions for their minimality. Furthermore, we investigate the corresponding subfield codes and derive sufficient conditions for their distance-optimality and minimality, yielding infinite families of Griesmer and near-Griesmer codes.
Linear complementary dual (LCD) codes are linear codes that intersect their dual trivially. We study double circulant (Euclidean) LCD codes over the finite chain ring $R=\mathbb F_2+u\mathbb F_2$, where $u^2=0$. Using the Chinese Remainder Theorem and the factorization of $x^n-1$, we derive an explicit enumeration form...
Fuglede's conjecture in $\mathbb{F}_p^d$ says that a nonempty set tiles by translation if and only if it is spectral. We prove the conjecture in $\mathbb{F}_2^8$ and $\mathbb{F}_{11}^3$. Over $\mathbb{F}_2$ we show more generally that the Fourier support of a nonconstant indicator of density different from one half con...
We investigate the linear codes derived from the zero-divisor graph $\Gamma(R)$ of the finite local ring $R = \mathbb{F}_p[x]/\langle x^4 \rangle$ for an odd prime $p$. Using an equitable partition of $Z^*(R)$, we determine the parameters of the binary incidence code of $\Gamma(R)$ and construct induced bipartite subgr...
The canonical $\mathbb{Z}_{2^s}$-linear Hadamard codes are the Gray images of the $\mathbb{Z}_{2^s}$-additive codes. Their rank, the dimension of the binary linear span, was known in closed form only for $s\le 3$, and for $s \ge 4$ only through computer computations for small lengths. We give a method that computes the...
We study the Jacobi spectrum and low-index rigidity of complex Hopf fibrations. We first give a representation-theoretic description of the Jacobi operator of a homogeneous Riemannian fibration and apply it to \[ \pi:(\mathbb{S}^{2n+1},g_\epsilon)\longrightarrow(\mathbb{C}P^n,h), \] where $g_\epsilon$ is a Berger metri...
We extend the explicit octonionic approach to groups of type $\mathrm{E}_6$ to the twisted groups ${}^2\mathrm{SE}_{6,K}(F)$ associated with a quadratic Galois extension $K/F$. These groups act on the Albert space over $K$, preserving the Dickson--Freudenthal determinant and a Hermitean form. When the field norm $K^{\t...
J. Bray, Yegor Stepanov, Robert A. Wilson· 0 citations
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