Related papers: Three Combinatorial Perspectives on Minimal Codes
Recently, much progress has been made to construct minimal linear codes due to their preference in secret sharing schemes and secure two-party computation. In this paper, we put forward a new method to construct minimal linear codes by…
In this article, we define and study a geometry and an order on the set of partitions of an even number of objects. One of the definitions involves the partition algebra, a structure of algebra on the set of such partitions depending on an…
Alon's combinatorial Nullstellensatz, and in particular the resulting nonvanishing criterion is one of the most powerful algebraic tools in combinatorics, with many important applications. In this paper we extend the nonvanishing theorem in…
Combinatorial problems are formulated to find optimal designs within a fixed set of constraints. They are commonly found across diverse engineering and scientific domains. Understanding how to best use quantum computers for combinatorial…
Linear codes play a central role in coding theory and have applications in several branches of mathematics. For error correction purposes the minimum Hamming distance should be as large as possible. Linear codes related to applications in…
The study on minimal linear codes has received great attention due to their significant applications in secret sharing schemes and secure two-party computation. Until now, numerous minimal linear codes have been discovered. However, to the…
In this paper, we propose and study $r$-minimal codes, a natural extension of minimal codes which have been extensively studied with respect to Hamming metric, rank metric and sum-rank metric. We first propose $r$-minimal codes in a general…
We present a construction for minimal blocking sets with respect to $(k-1)$-spaces in $\mathrm{PG}(n-1,q^t)$, the $(n-1)$-dimensional projective space over the finite field $\mathbb{F}_{q^t}$ of order $q^t$. The construction relies on the…
The design of block codes for short information blocks (e.g., a thousand or less information bits) is an open research problem which is gaining relevance thanks to emerging applications in wireless communication networks. In this work, we…
In this article, we present two new approaches to construct minimal linear codes of dimension $n+1$ over $\mathbb{F}_{3}$ using characteristic and ternary functions. We also obtain the weight distributions of these constructed minimal…
In connection with the needs of solving optimization problems, the development of conditional minimization methods with convenient numerical implementation continues to attract the attention of mathematicians. In this monograph we propose…
We explore a general method based on trees of elementary submodels in order to present highly simplified proofs to numerous results in infinite combinatorics. While countable elementary submodels have been employed in such settings already,…
This manuscript describes the notions of blocker and interdiction applied to well-known optimization problems. The main interest of these two concepts is the capability to analyze the existence of a combinatorial structure after some…
Trellises are crucial graphical representations of codes. While conventional trellises are well understood, the general theory of (tail-biting) trellises is still under development. Iterative decoding concretely motivates such theory. In…
Binary codes are constructed from incidence matrices of hypergraphs. A combinatroial description is given for the minimum distances of such codes via a combinatorial tool called ``eonv". This combinatorial approach provides a faster…
Recently, minimal linear codes have been extensively studied due to their applications in secret sharing schemes, two-party computations, and so on. Constructing minimal linear codes violating the Ashikhmin-Barg condition and then…
A strong blocking set in a finite projective space is a set of points that intersects each hyperplane in a spanning set. We provide a new graph theoretic construction of such sets: combining constant-degree expanders with asymptotically…
In this paper we prove tight bounds on the combinatorial and topological complexity of sets defined in terms of $n$ definable sets belonging to some fixed definable family of sets in an o-minimal structure. This generalizes the…
We show how the theory of affine geometries over the ring ${\mathbb Z}/\langle q - 1\rangle$ can be used to understand the properties of toric and generalized toric codes over ${\mathbb F}_q$. The minimum distance of these codes is strongly…
We first find the combinatorial degree of any map $f:V\to F$ where $F$ is a finite field and $V$ is a finite-dimensional vector space over $F$. We then simplify and generalize a certain construction due to Chein and Goodaire that was used…