Related papers: The second Feng-Rao number for codes coming from t…
We develop geometry-of-numbers methods to count orbits in coregular vector spaces having bounded invariants over any global field. We apply these techniques to bound the average ranks and determine average Selmer group sizes of elliptic…
Coding Theory where the alphabet is identified with the elements of a ring or a module has become an important research topic over the last 30 years. Such codes over rings had important applications and many interesting mathematical…
For numerical semigroups with three generators, we study the asymptotic behavior of weighted factorization lengths, that is, linear functionals of the coefficients in the factorizations of semigroup elements. This work generalizes many…
We generalize the geometric sequence $\{a^p, a^{p-1}b, a^{p-2}b^2,...,b^p\}$ to allow the $p$ copies of $a$ (resp. $b$) to all be different. We call the sequence $\{a_1a_2a_3\cdots a_p, b_1a_2a_3\cdots a_p, b_1b_2a_3\cdots a_p,\ldots,…
A numerical semigroup is a sub-semigroup of the natural numbers that has a finite complement. Some of the key properties of a numerical semigroup are its Frobenius number F, genus g and type t. It is known that for any numerical semigroup…
This is an expository introduction to fusion rules for affine Kac-Moody algebras, with major focus on the algorithmic aspects of their computation and the relationship with tensor product decompositions. Many explicit examples are included…
We explicitly determine all the relative generalized Hamming weights of affine Cartesian codes using the notion of footprints and results from extremal combinatorics. This generalizes the previous works on the determination of relative…
Linear codes with few weights have applications in secret sharing, authentication codes, association schemes and strongly regular graphs. In this paper, several classes of two-weight and three-weight linear codes are presented and their…
The arithmetic regularity lemma due to Green [GAFA 2005] is an analogue of the famous Szemer{\'e}di regularity lemma in graph theory. It shows that for any abelian group $G$ and any bounded function $f:G \to [0,1]$, there exists a subgroup…
We give an improved upper bound for the Gr\"unbaum--Hadwiger--Ramos problem: Let $d,n,k \in \mathbb{N}$ such that $d \geq 2^n(1+2^{k-1})$. Given $2^{n+1}$ masses on $\mathbb{R}^d$, there exist $k$ hyperplanes in $\mathbb{R}^d$ that…
Leveraging a general framework adapted from symbolic integration, a unified reduction-based algorithm for computing telescopers of minimal order for hypergeometric and q-hypergeometric terms has been recently developed. In this paper, we…
We extend the decomposition conjecture to 2d quantum field theories with a gauged $\text{Rep}(H)$ symmetry category for $H$ a finite-dimensional semisimple Hopf algebra with $\text{Rep}(G)$ trivially-acting and $\text{Vec}(\Gamma)$ the…
Self-dual codes are important because many of the best codes known are of this type and they have a rich mathematical theory. Topics covered in this survey include codes over F_2, F_3, F_4, F_q, Z_4, Z_m, shadow codes, weight enumerators,…
The main purpose of this paper is to further study the structure, parameters and constructions of the recently introduced minimal codes in the sum-rank metric. These objects form a bridge between the classical minimal codes in the Hamming…
Few-weight codes over finite chain rings are associated with combinatorial objects such as strongly regular graphs (SRGs), strongly walk-regular graphs (SWRGs) and finite geometries, and are also widely used in data storage systems and…
Generalizing supertropical algebras, we present a "layered" structure, "sorted" by a semiring which permits varying ghost layers, and indicate how it is more amenable than the "standard" supertropical construction in factorizations of…
In the last few years there has been a growing interest towards methods for statistical inference and learning based on computational geometry and, notably, tropical geometry, that is, the study of algebraic varieties over the min-plus…
In this note, we show that {\it $\gamma$-hyperelliptic} numerical semigroups of genus $g \gg \gamma$ satisfy a refinement of a well-known characteristic weight inequality due to Torres. The refinement arises from substituting the usual…
This paper studies Young diagrams of symmetric and pseudo-symmetric numerical semigroups and describes new operations on Young diagrams as well as numerical semigroups. These provide new decompositions of symmetric and pseudo-symmetric…
It is well-known that if we gauge a $\mathbb{Z}_n$ symmetry in two dimensions, a dual $\mathbb{Z}_n$ symmetry appears, such that re-gauging this dual $\mathbb{Z}_n$ symmetry leads back to the original theory. We describe how this can be…