Related papers: Isometry-Dual Flags of Many-Point AG Codes
A flag of codes $C_0 \subsetneq C_1 \subsetneq \cdots \subsetneq C_s \subseteq {\mathbb F}_q^n$ is said to satisfy the {\it isometry-dual property} if there exists ${\bf x}\in (\mathbb{F}_q^*)^n$ such that the code $C_i$ is {\bf…
Consider a complete flag $\{0\} = C_0 < C_1 < \cdots < C_n = \mathbb{F}^n$ of one-point AG codes of length $n$ over the finite field $\mathbb{F}$. The codes are defined by evaluating functions with poles at a given point $Q$ in points…
A flag $C_0 \subsetneq C_1 \cdots \subsetneq C_s \subsetneq {\mathbb F}_q^n $ of linear codes is said to be self-orthogonal if the duals of the codes in the flag satisfy $C_{i}^\perp=C_{s-i}$, and it is said to satisfy the isometry-dual…
Consider a sequence of AG codes evaluating at a set of evaluation points $P_1,\dots,P_n$ the functions having only poles at a defining point $Q$, with the sequence of codes satisfying the isometry-dual condition (i.e. containing at the same…
Given a finite field F_q and a positive integer n, a flag is a sequence of nested F_q-subspaces of a vector space F_q^n and a flag code is a nonempty collection of flags. The projected codes of a flag code are the constant dimension codes…
It is proved in a reference (Fan, Lin, IEEE TIT, vol.67, pp.5016-5025) that the self-dual (LCD respectively) dihedral codes over a finite field~$F$ with ${|F|=q}$ are asymptotically good if $q$ is even (odd respectively). In this paper, we…
For a linear code $C$ over a finite field, if its dual code $C^{\perp}$ is equivalent to itself, then the code $C$ is said to be {\it isometry-dual}. In this paper, we first confirm a conjecture about the isometry-dual MDS elliptic codes…
The aim of this work is to study the dual and the algebraic dual of an evaluation code using standard monomials and indicator functions. We show that the dual of an evaluation code is the evaluation code of the algebraic dual. We develop an…
In this work we investigate the problem of producing iso-dual algebraic geometry (AG) codes over a finite field $\mathbb{F}_q$ with $q$ elements. Given a finite separable extension $\mathcal{M}/\mathcal{F}$ of function fields and an…
In this paper, we investigate the existence of self-dual MRD codes $C\subset L^n$, where $L/F$ is an arbitrary field extension of degree $m\geq n$. We then apply our results to the case of finite fields, and prove that if $m=n$ and…
A kind of self-dual quasi-abelian codes of index $2$ over any finite field $F$ is introduced. By counting the number of such codes and the number of the codes of this kind whose relative minimum weights are small, such codes are proved to…
Two families of complementary codes over finite fields $\mathbb{F}_q$ are studied, where $q=r^2$ is square: i) Hermitian complementary dual linear codes, and ii) trace Hermitian complementary dual subfield linear codes. Necessary and…
Given $\mathbb{F}_q$ the finite field with $q$ elements and an integer $n\geq 2$, a flag is a sequence of nested subspaces of $\mathbb{F}_q^n$ and a flag code is a nonempty set of flags. In this context, the distance between flags is the…
We show that every self-orthogonal code over $\mathbb F_q$ of length $n$ can be extended to a self-dual code, if there exists self-dual codes of length $n$. Using a family of Galois towers of algebraic function fields we show that over any…
We extend Witt's theorem to several kinds of simultaneous isometries of subspaces. We determine sufficient and necessary conditions for the extension of an isometry of subspaces $\phi:E\to E'$ to an isometry $\phi_V:V\to V'$ that also sends…
It is known that a linear two-weight code $C$ over a finite field $\F_q$ corresponds both to a multiset in a projective space over $\F_q$ that meets every hyperplane in either $a$ or $b$ points for some integers $a<b$, and to a strongly…
Let $\mathbb{F}_q$ be a finite field with $q$ elements and denote by $\theta : \mathbb{F}_q\to\mathbb{F}_q$ an automorphism of $\mathbb{F}_q$. In this paper, we deal with linear codes of $\mathbb{F}_q^n$ invariant under a semi-linear map…
We call a linear code $C$ with length $n$ over a field $F$, a linear complementary equi-dual code, when there exists a linear code $D$ over $F$ such that $D$ is permutation equivalent to $C^\perp$ and $(C,D)$ is a linear complementary pair…
Flag codes generalize constant dimension codes by considering sequences of nested subspaces with prescribed dimensions as codewords. A comprehensive construction, which unites cyclic orbit flag codes, yields two families of flag codes on…
Double polycirculant codes are introduced here as a generalization of double circulant codes. When the matrix of the polyshift is a companion matrix of a trinomial, we show that such a code is isodual, hence formally self-dual. Numerical…