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Define the $k$-th Radon number $r_k$ of a convexity space as the smallest number (if it exists) for which any set of $r_k$ points can be partitioned into $k$ parts whose convex hulls intersect. Combining the recent abstract fractional Helly…
The hull of linear codes have promising utilization in coding theory and quantum coding theory. In this paper, we study the hull of generalized Reed-Solomon codes and extended generalized Reed-Solomon codes over finite fields with respect…
Linear codes and $t$-designs are interactive with each other. It is well known that some $t$-designs have been constructed by using certain linear codes in recent years. However, only a small number of infinite families of the extended…
Let $k\leq n$ be two positive integers and $q$ a prime power. The basic question in minimal linear codes is to determine if there exists an $[n,k]_q$ minimal linear code. The first objective of this paper is to present a new sufficient and…
A binary linear code whose permutation automorphism group has a fixed point free permutation of order $3$ is called a binary cubic code. The scope of this paper is to investigate the structural properties of binary cubic codes. Let $C$ be a…
We construct new linear codes with high minimum distance d. In at least 12 cases these codes improve the minimum distance of the previously known best linear codes for fixed parameters n,k. Among these new codes there is an optimal ternary…
Let $\mathrm{SLAut}(\mathbb{F}_{q}^{n})$ denote the group of all semilinear isometries on $\mathbb{F}_{q}^{n}$, where $q=p^{e}$ is a prime power. Matrix-product (MP) codes are a class of long classical codes generated by combining several…
In this paper, we explore a connection between binary hierarchical models, their marginal polytopes and codeword polytopes, the convex hulls of linear codes. The class of linear codes that are realizable by hierarchical models is…
In this paper, we consider the unit graph $G(\mathbb{Z}_{n})$, where $n=p_{1}^{n_{1}} \text{ or } p_{1}^{n_{1}}p_{2}^{n_{2}} \text{ or } p_{1}^{n_{1}}p_{2}^{n_{2}}p_{3}^{n_{3}}$ and $p_{1}, p_{2}, p_{3}$ are distinct primes. For any prime…
A subspace code is a nonempty set of subspaces of a vector space $\mathbb F^n_q$. Linear codes with complementary duals, or LCD codes, are linear codes whose intersection with their duals is trivial. In this paper, we introduce a notion of…
We prove that any Hermitian self-orthogonal $[n,k,d]_{q^2}$ code gives rise to an $[n,k,d]_{q^2}$ code with $\ell$ dimensional Hermitian hull for $0\le \ell \le k$. We present a new method to construct Hermitian self-orthogonal…
In this paper, we explore some properties of Galois hulls of cyclic serial codes over a chain ring and we devise an algorithm for computing all the possible parameters of the Euclidean hulls of that codes. We also establish the average…
Linear complementary-dual (LCD for short) codes are linear codes that intersect with their duals trivially. LCD codes have been used in certain communication systems. It is recently found that LCD codes can be applied in cryptography. This…
Non-overlapping codes are a set of codewords such that the prefix of each codeword is not a suffix of any codeword in the set, including itself. If the lengths of the codewords are variable, it is additionally required that every codeword…
A $q$-ary $t$-$(n,w,\lambda)$ design is a collection $\mathcal{A}$ of vectors of weight $w$ in $\mathbb{F}_{q}^{n}$ with the property that every vector of weight $t$ in $\mathbb{F}_{q}^{n}$ is contained in exactly $\lambda$ members of…
For a prime power $q$, an integer $m$ and $0\leq e\leq m-1$ we study the $e$-Galois hull dimension of Gabidulin codes $G_k(\boldsymbol{\alpha})$ of length $m$ and dimension $k$ over $\mathbb{F}_{q^m}$. Using a self-dual basis…
For odd length $n$, the cyclic codes construction over $\Re= \Z_4[v]/ \langle v^2-v \rangle$ is provided. The hulls of cyclic codes over $\Re$ are studied. The average $2$-dimension $E(n)$ of the hulls of cyclic codes over $\Re$ is also…
In this short note, we report the classification of self-dual $\mathbb{Z}_k$-codes of length $n$ for $k \le 24$ and $n \le 9$.
In this work, we prove new results concerning the combinatorial properties of random linear codes. Firstly, we prove a lower bound on the list-size required for random linear codes over $\mathbb F_q$ $\varepsilon$-close to capacity to…
Kim et al. (2021) gave a method to embed a given binary $[n,k]$ code $\mathcal{C}$ $(k = 3, 4)$ into a self-orthogonal code of the shortest length which has the same dimension $k$ and minimum distance $d' \ge d(\mathcal{C})$. We extend this…