Related papers: Rank and Kernel of $\mathbb{F}_p$-Additive General…
Let p be an odd prime number, E an elliptic curve over a number field k, and F/k a Galois extension of degree twice a power of p. We study the Z_p-corank rk_p(E/F) of the p-power Selmer group of E over F. We obtain lower bounds for…
For each rank metric code $\mathcal{C}\subseteq \mathbb{K}^{m\times n}$, we associate a translation structure, the kernel of which is shown to be invariant with respect to the equivalence on rank metric codes. When $\mathcal{C}$ is…
A systematic way of constructing Grassmannian codes endowed with the subspace distance as lifts of matrix codes over the prime field $GF(p)$ is introduced. The matrix codes are $GF(p)$-subspaces of the ring $M_2(GF(p))$ of $2 \times 2$…
This paper extends the study of rank-metric codes in extension fields $\mathbb{L}$ equipped with an arbitrary Galois group $G = \mathrm{Gal}(\mathbb{L}/\mathbb{K})$. We propose a framework for studying these codes as subspaces of the group…
A sharp upper bound for the maximum integer not belonging to an ideal of a numerical semigroup is given and the ideals attaining this bound are characterized. Then the result is used, through the so-called Feng-Rao numbers, to bound the…
Additive codes over GF(9) that are self-dual with respect to the Hermitian trace inner product have a natural application in quantum information theory, where they correspond to ternary quantum error-correcting codes. However, these codes…
In this paper, a recent method to construct complementary sequence sets and complete complementary codes by Hadamard matrices is deeply studied. By taking the algebraic structure of Hadamard matrices into consideration, our main result…
Let $\mathcal{X}$ be a set of $(h-1)$-dimensional subspaces of $\mathrm{PG}(kh-1,q)$ with the property that every hyperplane contains at most $t$ elements of $\mathcal{X}$. We prove the upper bound $|\mathcal{X}| \leq (t-k+2)q^h + t$, and…
Inspired by the work of Zhou "On equivalence of maximum additive symmetric rank-distance codes" (2020) based on the paper of Schmidt "Symmetric bilinear forms over finite fields with applications to coding theory" (2015), we investigate the…
The higher rank numerical range is closely connected to the construction of quantum error correction code for a noisy quantum channel. It is known that if a normal matrix $A \in M_n$ has eigenvalues $a_1, \..., a_n$, then its higher rank…
In the present paper, we show that if the dimension of an arbitrary algebraic geometry code over a finite field of even characters is slightly less than half of its length, then it is equivalent to an Euclidean self-orthogonal code.…
We study the Hermitian hull-variation problem for vector rank-metric codes. Except for one parameter pair, we show that the Hermitian hull dimension of such a code can be reduced to any smaller value within its equivalence class, and in…
A subspace of matrices over $\mathbb{F}_{q^e}^{m\times n}$ can be naturally embedded as a subspace of matrices in $\mathbb{F}_q^{em\times en}$ with the property that the rank of any of its matrix is a multiple of $e$. It is quite natural to…
We ask, following Bartholdi, whether it is true that the kernel of the restriction map from the cohomology of a group G to the cohomology of a finite index subgroup H is finitely generated as an ideal. We show that in case the group has…
An additive code is an $\mathbb{F}_q$-linear subspace of $\mathbb{F}_{q^m}^n$ over $\mathbb{F}_{q^m}$, which is not a linear subspace over $\mathbb{F}_{q^m}$. Linear complementary pairs (LCP) of codes have important roles in cryptography,…
Let $p$ be an odd prime, $F$ a field with a primitive $p^2$th root of unity, and $E=F(\sqrt[p]{b_1},\sqrt[p]{b_2})$ an elementary abelian extension of degree $p^2$. This paper studies the cohomological kernel $H^n(E/F,{\mathbb Z}/p{\mathbb…
Let $f$ be the $\mathbb{F}_q$-linear map over $\mathbb{F}_{q^{2n}}$ defined by $x\mapsto x+ax^{q^s}+bx^{q^{n+s}}$ with $\gcd(n,s)=1$. It is known that the kernel of $f$ has dimension at most $2$, as proved by Csajb\'ok et al. in "A new…
Self-orthogonal codes are an important subclass of linear codes which have nice applications in quantum codes and lattices. It is known that a binary linear code is self-orthogonal if its every codeword has weight divisible by four, and a…
A non-singular complete irreducible algebraic curve $F_{k,n}$, defined over an algebraically closed field $K$, is called a generalized Fermat curve of type $(k,n)$, where $n, k \geq 2$ are integers and $k$ is relatively prime to the…
Let $K: \boldsymbol{\Omega}\times \boldsymbol{\Omega}$ be a continuous Mercer kernel defined on a compact subset of ${\mathbb R}^n$ and $\mathcal{H}_K$ be the reproducing kernel Hilbert space (RKHS) associated with $K$. Given a finite…