Related papers: On the hull of linearized polynomial codes
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…
The hull $H(C)$ of a linear code $C$ is defined by $H(C)=C \cap C^\perp$. A linear code with a complementary dual (LCD) is a linear code with $H(C)=\{0\}$. The dimension of the hull of a code is an invariant under permutation equivalence.…
The hull of a linear code is defined as the intersection of the code and its dual. This concept was initially introduced to classify finite projective planes. The hull plays a crucial role in determining the complexity of algorithms used to…
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. In this paper, we investigate general properties of linear codes associated with $\sigma$…
Given a commutative ring $R$ with identity, a matrix $A\in M_{s\times l}(R)$, and $R$-linear codes $\mathcal{C}_1, \dots, \mathcal{C}_s$ of the same length, this article considers the hull of the matrix-product codes $[\mathcal{C}_1 \dots…
The $\ell$-Galois hull $h_{\ell}(C)$ of an $[n,k]$ linear code $C$ over a finite field $\mathbb{F}_q$ is the intersection of $C$ and $C^{{\bot}_{\ell}}$, where $C^{\bot_{\ell}}$ denotes the $\ell$-Galois dual of $C$ which introduced by Fan…
We study the Hermitian hull of a particular family of generalized Reed-Solomon codes. The problem of computing the dimension of the hull is translated to a counting problem in a lattice. By solving this problem, we provide explicit formulas…
The intersection of a linear code with its dual is called the hull of the code. It is known that, for classical linear codes under the Hamming-metric, the dimension of the hull can be reduced up to equivalence. This phenomenon leads to the…
By solving a problem regarding polynomials in a quotient ring, we obtain the relative hull and the Hermitian hull of projective Reed-Muller codes over the projective plane. The dimension of the hull determines the minimum number of…
Hulls of linear codes have been of interest and extensively studied due to their rich algebraic structures and wide applications. In this paper, alternative characterizations of hulls of linear codes are given as well as their applications.…
We introduce the notion of quadratic hull of a linear code, and give some of its properties. We then show that any symmetric bilinear multiplication algorithm for a finite-dimensional algebra over a field can be obtained by…
This article explores additive codes with one-rank hull, offering key insights and constructions. The article introduces a novel approach to finding one-rank hull codes over finite fields by establishing a connection between self-orthogonal…
Constacyclic and quasi-twisted Hermitian self-dual codes over finite fields are studied. An algorithm for factorizing $x^n-\lambda$ over $\mathbb{F}_{q^2}$ is given, where $\lambda$ is a unit in $\mathbb{F}_{q^2}$. Based on this…
Recently, codes in the sum-rank metric attracted attention due to several applications in e.g. multishot network coding, distributed storage and quantum-resistant cryptography. The sum-rank analogs of Reed-Solomon and Gabidulin codes are…
The Euclidean hull of a linear code $C$ is the intersection of $C$ with its Euclidean dual $C^\perp$. The hull with low dimensions gets much interest due to its crucial role in determining the complexity of algorithms for computing the…
The Euclidean hull of a linear code $C$ is defined as $C\cap C^{\perp}$, where $C^\perp$ denotes the dual of $C$ under the Euclidean inner product. A linear code with zero hull dimension is called a linear complementary dual (LCD) code. A…
Chara et al. introduced conorm codes defined over algebraic geometry codes, but the hulls of conorm codes were not determined yet. In this paper, we study the dimension of the hull of conorm codes using the method introduced by Camps et al.…
It is well known that linear rank-metric codes give rise to q-polymatroids. Analogously to matroid theory one may ask whether a given q-polymatroid is representable by a rank-metric code. We provide an answer by presenting an example of a…
In this study, we consider the Euclidean and Galois hulls of multi-twisted (MT) codes over a finite field $\mathbb{F}_{p^e}$ of characteristic $p$. Let $\mathbf{G}$ be a generator polynomial matrix (GPM) of a MT code $\mathcal{C}$. For any…
Let $\mathcal A$ the affine algebra given by the ring $\mathbb{F}_q[X_1,X_2,\ldots,X_\ell]/ I$, where $I$ is the ideal $\langle t_1(X_1), t_2(X_2), \ldots, t_\ell(X_\ell) \rangle$ with each $t_i(X_i)$, $1\leq i\leq \ell$, being a…