Related papers: Embedding Dimension Phenomena in Intersection Comp…
Complex orthogonal designs (CODs) are used to construct space-time block codes. COD $\mathcal{O}_z$ with parameter $[p, n, k]$ is a $p \times n$ matrix, where nonzero entries are filled by $\pm z_i$ or $\pm z^*_i$, $i = 1, 2,..., k$, such…
In this paper, we study the famous Erd\H{o}s--S\'os forbidden intersection problem for words over an alphabet of size $m$: what is the maximal size of a subfamily $\mathcal{F}$ of $[m]^n$ that does not contain two vectors $x, y$ coinciding…
Binary cyclic codes are worth studying due to their applications and theoretical importance. It is an important problem to construct an infinite family of cyclic codes with large minimum distance $d$ and dual distance $d^{\perp}$. In recent…
The present paper mainly studies limits and constructions of insertion and deletion (insdel for short) codes. The paper can be divided into two parts. The first part focuses on various bounds, while the second part concentrates on…
Toric codes are a class of $m$-dimensional cyclic codes introduced recently by J. Hansen. They may be defined as evaluation codes obtained from monomials corresponding to integer lattice points in an integral convex polytope $P \subseteq…
Subspace codes have important applications in random network coding. It is interesting to construct subspace codes with both sizes, and the minimum distances are as large as possible. In particular, cyclic constant dimension subspaces codes…
Let $n=2^k-1$ and $m=2^{k-2}$ for a certain $k\ge 3$. Consider the point-line geometry of $2m$-element subsets of an $n$-element set. Maximal singular subspaces of this geometry correspond to binary simplex codes of dimension $k$. For $k\ge…
The colourful simplicial depth problem in dimension d is to find a configuration of (d+1) sets of (d+1) points such that the origin is contained in the convex hull of each set (colour) but contained in a minimal number of colourful…
Constant dimension codes (CDCs), as special subspace codes, have received extensive attention due to their applications in random network coding. The basic problem of CDCs is to determine the maximal possible size $A_q(n,d,\{k\})$ for given…
Complex orthogonal designs (CODs) play a crucial role in the construction of space-time block codes. Their real analog, real orthogonal designs (or equivalently, sum of squares composition formula) have a long history. Adams et al. (2011)…
Linear Complementary Dual codes (LCD) are binary linear codes that meet their dual trivially. We construct LCD codes using orthogonal matrices, self-dual codes, combinatorial designs and Gray map from codes over the family of rings $R_k$.…
The mixed metric dimension ${\rm mdim}(G)$ of a graph $G$ is the cardinality of a smallest set of vertices that (metrically) resolves each pair of elements from $V(G)\cup E(G)$. We say that $G$ is a max-mdim graph if ${\rm mdim}(G) = n(G)$.…
Affine-invariant codes have attracted considerable attention due to their rich algebraic structure and strong theoretical properties. In this paper, we study a family of affine-invariant codes whose defining set consists of all descendants…
In this paper, we are concerned about optimal two-dimensional optical orthogonal codes with $\lambda$ = 2. Some combinatorial constructions are presented and many infinite families of optimal two-dimensional optical orthogonal codes with…
Let $R$ be a local ring and $M$ a finitely generated $R$-module. The complete intersection dimension of $M$--defined by Avramov, Gasharov and Peeva, and denoted $\cidim_R(M)$--is a homological invariant whose finiteness implies that $M$ is…
In this work, we study the performance of Reed--Solomon codes against adversarial insertion-deletion (insdel) errors. We prove that over fields of size $n^{O(k)}$ there are $[n,k]$ Reed-Solomon codes that can decode from $n-2k+1$ insdel…
A covering code is a set of codewords with the property that the union of balls, suitably defined, around these codewords covers an entire space. Generally, the goal is to find the covering code with the minimum size codebook. While most…
Klee's measure problem (computing the volume of the union of $n$ axis-parallel boxes in $\mathbb{R}^d$) is well known to have $n^{\frac{d}{2}\pm o(1)}$-time algorithms (Overmars, Yap, SICOMP'91; Chan FOCS'13). Only recently, a conditional…
In the paper we investigate the structure of $i$-components of two classes of codes: Kerdock codes and the duals of the primitive cyclic BCH code with designed distance 5 of length $n=2^m-1$, for odd $m$. We prove that for any admissible…
We study the set of intersection sizes of a k-dimensional affine subspace and a point set of size m \in [0, 2^n] of the n-dimensional binary affine space AG(n,2). Following the theme of Erd\H{o}s, F\"uredi, Rothschild and T. S\'os, we…