On the additive cyclic structure of quasi-cyclic codes

An index $?$, length $m?$ quasi-cyclic code can be viewed as a cyclic code of length $m$ over the field ${\mathbb{F}}_{q^{?}}$ via a basis of the extension ${\mathbb{F}}_{q^{?}}∕{\mathbb{F}}_q$. However, this cyclic code is only linear over ${\mathbb{F}}_q$, making it an additive cyclic code, or an ${\mathbb{F}}_q$-linear cyclic code, over the alphabet ${\mathbb{F}}_{q^{?}}$. This approach was recently used in Shi et al. (2017) [16] to study a class of quasi-cyclic codes, and more importantly in Shi et al. (2017) [17] to settle a long-standing question on the asymptotic performance of cyclic codes. Here, we answer one of the problems posed in these two articles, and characterize those quasi-cyclic codes which have ${\mathbb{F}}_{q^{?}}$-linear cyclic images under a basis of the extension ${\mathbb{F}}_{q^{?}}∕{\mathbb{F}}_q$. Our characterizations are based on the module structure of quasi-cyclic codes, as well as on their CRT decompositions into constituents. In the case of a polynomial basis, we characterize the constituents by using the theory of invariant subspaces of operators. We also observe that analogous results extend to the case of quasi-twisted codes.     

On two upper bounds for hypersurfaces involving a Thas’ invariant

Let $X^n$ be a hypersurface in ${\mathbb{P}}^{n+1}$ with $n\ge 1$ defined over a finite field ${\mathbb{F}}_q$ of $q$ elements. In this note, we classify, up to projective equivalence, hypersurfaces $X^n$ as above which reach two elementary upper bounds for the number of ${\mathbb{F}}_q$-points on $X^n$ which involve a Thas’ invariant.     

Uniqueness of Fq-quadratic perfect nonlinear maps from Fq3 to Fq2

Let q be a power of an odd prime. We prove that all ${\mathbb{F}}_q$-quadratic perfect nonlinear maps from ${\mathbb{F}}_{q^3}$ to ${\mathbb{F}}_q^2$ are equivalent. We also give a geometric method to find the corresponding equivalence explicitly.     

Le nombre des systèmes locaux ℓ-adiques sur une courbe

Let $X_1$ be a projective, smooth, and geometrically connected curve over ${\mathbb{F}}_q$ with $q=p^n$ elements where p is a prime number, and let X be its base change to an algebraic closure of ${\mathbb{F}}_q$. The goal of this article is to announce a formula for the number of irreducible ?-adic local systems ($?\ne p$) with a fixed rank over X fixed by the Frobenius endomorphism. This number behaves like a Grothendieck–Lefschetz fixed point formula for a variety over ${\mathbb{F}}_q$, which generalises a result of Drinfeld in rank 2 and proves a conjecture of Deligne. We also sketch our method, which consists in passing to the automorphic side by Langlands correspondence, then calculating all the terms in Arthur's non-invariant trace formula and linking the geometric part of trace formula to the number of ${\mathbb{F}}_q$-points of the moduli space of stable Higgs bundles.     

A new family of Castle and Frobenius nonclassical curves

Motivated by the study of minimal value set polynomials, we construct ${\mathbb{F}}_q$-Frobenius nonclassical curves with a large number of ${\mathbb{F}}_q$-rational points. For some of these curves, we determine the Weierstrass semigroup at the unique point at infinity. In particular, we prove that they yield new examples of Castle curves.