Narrow Results

Given a hypergraph ${\mathscr{H}}$, we introduce a new class of evaluation toric codes called edge codes derived from ${\mathscr{H}}$. We analyze these codes, focusing on determining their basic parameters. We provide estimations for the minimum distance, particularly in scenarios involving $d$-uniform clutters. Additionally, we demonstrate that these codes exhibit self-orthogonality. Furthermore, we compute the minimum distances of edge codes for all graphs with five vertices.

Toric codes are a type of evaluation codes introduced by J. P. Hansen in 2000. They are produced by evaluating (a vector space composed by) polynomials at the points of ${({𝔽}_q^{\ast })}^s$, the monomials of these polynomials being related to a certain polytope. Toric codes related to hypersimplices are the result of the evaluation of a vector space of square-free homogeneous polynomials of degree $d$. The dimension and minimum distance of toric codes related to hypersimplices have been determined by Jaramillo et al. in 2021. The next-to-minimal weight in the case $d=1$ has been determined by Jaramillo–Velez et al. in 2023. In this work, we use tools from Gröbner basis theory to determine the next-to-minimal weight of these codes for $d$ such that $3\le d\le \frac{s-2}{2}$ or $\frac{s+2}{2}\le d<s$.

Weighted projective spaces are natural generalizations of projective spaces with a rich structure. Projective Reed–Muller codes are error-correcting codes that played an important role in reliably transmitting information on digital communication channels. In this case study, we explore the power of commutative and homological algebraic techniques to study weighted projective Reed–Muller (WPRM) codes on weighted projective spaces of the form $\mathbb{P}(1,1,b)$. We compute minimal free resolutions and thereby obtain Hilbert series for the vanishing ideal of the ${𝔽}_q$-rational points, and compute main parameters for these codes.

In this work, we will show how the topological order of the Toric Code appears when the lattice on which it is defined discretizes a three-dimensional torus. In order to do this, we will present a pedagogical review of the traditional two-dimensional Toric Code, with an emphasis on how its quasiparticles are conceived and transported. With that, we want to make clear not only how all these same quasiparticle conception and transportation fit into this three-dimensional model, but to make it clear how topology controls the degeneracy of ground state in this new situation.