Related papers: Functional codes arising from rank $n$ Hermitian v…
We compute an explicit rank bound on the Picard group of the compact surfaces, which can serve as the base of an elliptic Calabi-Yau variety with canonical singularities. To bound the Picard rank from above, we develop a novel strategy in…
We initiate the study of intersecting surface operators/defects in four-dimensional quantum field theories (QFTs). We characterize these defects by coupled 4d/2d/0d theories constructed by coupling the degrees of freedom localized at a…
We introduce a framework which allows to systematically and arbitrarily scale the code distance of local fermion-to-qubit encodings in one and two dimensions without growing the weights of stabilizers. This is achieved by embedding…
We determine the maximum number of $\mathbb{F}_q$-rational points that a nonsingular threefold of degree $d$ in a projective space of dimension $4$ defined over $\mathbb{F}_q$ may contain. This settles a conjecture by Homma and Kim…
An $n\times n$ matrix $X$ is called completely positive semidefinite (cpsd) if there exist $d\times d$ Hermitian positive semidefinite matrices $\{P_i\}_{i=1}^n$ (for some $d\ge 1$) such that $X_{ij}= {\rm Tr}(P_iP_j),$ for all $i,j \in \{…
Let $K=Q(\sqrt{-\ell})$ be an imaginary quadratic field with ring of integers $\O_K$, where $\ell$ is a square free integer such that $\ell\equiv 3 \mod 4$ and $C=[n, k]$ be a linear code defined over $\O_K/2\O_K$. The level $\ell$ theta…
We continue the study of classical and quantum low-density parity check (LDPC) codes from a physical perspective. We focus on constructive approaches and formulate a general framework for systematically constructing codes with various…
We study new Willmore-type variational problem for a hypersurface $M$ in $\mathbb{R}^{n+1}$ equipped with an $s$-dimensional foliation ${\cal F}$. Its general version is the Reilly-type functional $WF_{n,s}=\int_M F(\sigma^{\cal…
A real hypersurface in $\mathbb{C}^2$ is said to be Reinhardt if it is invariant under the standard $\mathbb{T}^2$-action on $\mathbb{C}^2$. Its CR geometry can be described in terms of the curvature function of its ``generating curve'',…
One of the main objectives of quantum error-correction theory is to construct quantum codes with optimal parameters and properties. In this paper, we propose a class of 2-generator quasi-cyclic codes and study their applications in the…
Recently, it was discovered by several authors that a $q$-ary optimal locally recoverable code, i.e., a locally recoverable code archiving the Singleton-type bound, can have length much bigger than $q+1$. This is quite different from the…
In this paper, we study properties of rank metric codes in general and maximum rank distance (MRD) codes in particular. For codes with the rank metric, we first establish Gilbert and sphere-packing bounds, and then obtain the asymptotic…
We construct locally recoverable codes with hierarchy from surfaces in $\mathbb{A}^3$ admitting a fibration by curves of Artin-Schreier or Kummer type. We derive the parameters of our codes by leveraging the geometry and arithmetic of the…
For any affine-variety code we show how to construct an ideal whose solutions correspond to codewords with any assigned weight. We classify completely the intersections of the Hermitian curve with lines and parabolas (in the…
We study the functional calculus properties of generators of $C_{0}$-groups under type and cotype assumptions on the underlying Banach space. In particular, we show the following. Let $-iA$ generate a $C_{0}$-group on a Banach space $X$…
We study the dual linear code of points and generators on a non-singular Hermitian variety $\mathcal{H}(2n+1,q^2)$. We improve the earlier results for $n=2$, we solve the minimum distance problem for general $n$, we classify the $n$…
An upper bound for the maximum number of rational points on an hypersurface in a projective space over a finite field has been conjectured by Tsfasman and proved by Serre in 1989. The analogue question for hypersurfaces on weighted…
We prove an upper bound for the number of rational points of bounded height on irreducible affine hypersurfaces. More precisely, given an irreducible polynomial $f \in \mathbb{Z}[X_1, \dots, X_n]$, we prove an upper bound on the number of…
In the present paper, we have found new upper bounds for chromatic numbers for integer lattices and some rational spaces and other lattices. In particular, we have proved that for any concrete critical distance $d$ the chromatic number of…
We provide a construction for quantum codes (hermitian-self-orthogonal codes over GF(4)) starting from cyclic codes over GF(4^m). We also provide examples of these codes some of which meet the known bounds for quantum codes.