Left dihedral codes over finite chain rings

Let R be a finite commutative chain ring, $D_{2n}$ be the dihedral group of size 2n and $R[D_{2n}]$ be the dihedral group ring. In this paper, we completely characterize left ideals of $R[D_{2n}]$ (called left $D_{2n}$-codes) when $\gcd (char(R),n)=1$. In this way, we explore the structure of some skew-cyclic codes of length 2 over R and also over $R\times S$, where S is an isomorphic copy of R. As a particular result, we give the structure of cyclic codes of length 2 over R. In the case where $R={\mathbb{F}}_{p^m}$ is a Galois field, we give a classification for left $D_{2N}$-codes over ${\mathbb{F}}_{p^m}$, for any positive integer N. In both cases we determine dual codes and identify self-dual ones.     

Algebraic geometry codes over abelian surfaces containing no absolutely irreducible curves of low genus

We provide a theoretical study of Algebraic Geometry codes constructed from abelian surfaces defined over finite fields. We give a general bound on their minimum distance and we investigate how this estimation can be sharpened under the assumption that the abelian surface does not contain low genus curves. This approach naturally leads us to consider Weil restrictions of elliptic curves and abelian surfaces which do not admit a principal polarization.     

Self-reciprocal and self-conjugate-reciprocal irreducible factors of xn − λ and their applications

In this paper, we present some necessary and sufficient conditions under which an irreducible polynomial is self-reciprocal (SR) or self-conjugate-reciprocal (SCR). By these characterizations, we obtain some enumeration formulas of SR and SCR irreducible factors of $x^n-\lambda$, $\lambda \in {\mathbb{F}}_q^{⁎}$, over ${\mathbb{F}}_q$, which are just open questions posed by Boripan et al. (2019). We also count the numbers of Euclidean and Hermitian LCD constacyclic codes and show some well-known results on Euclidean and Hermitian self-dual constacyclic codes in a simple and direct way.     

Polycyclic codes as invariant subspaces

Polycyclic codes are a powerful generalization of cyclic and constacyclic codes. Their algebraic structure is studied here by the theory of invariant subspaces from linear algebra. As an application, a bound on the minimum distance of these codes is derived which outperforms, in some cases, the natural analogue of the BCH bound.