Linear sets and MRD-codes arising from a class of scattered linearized polynomials

A class of scattered linearized polynomials covering infinitely many field extensions is exhibited. More precisely, the q-polynomial over $${{\mathbb {F}}}_{q^6}$$ F q 6 , $$q \equiv 1\pmod 4$$ q ≡ 1 ( mod 4 ) described in Bartoli et al. (ARS Math Contemp 19:125–145, 2020) and Zanella and Zullo (Discrete Math 343:111800, 2020) is generalized for any even $$n\ge 6$$ n ≥ 6 to an $${{{\mathbb {F}}}_q}$$ F q -linear automorphism $$\psi (x)$$ ψ ( x ) of $${{\mathbb {F}}}_{q^n}$$ F q n of order n. Such $$\psi (x)$$ ψ ( x ) and some functional powers of it are proved to be scattered. In particular, this provides new maximum scattered linear sets of the projective line $${{\,\mathrm{{PG}}\,}}(1,q^n)$$ PG ( 1 , q n ) for $$n=8,10$$ n = 8 , 10 . The polynomials described in this paper lead to a new infinite family of MRD-codes in $${{\mathbb {F}}}_q^{n\times n}$$ F q n × n with minimum distance $$n-1$$ n - 1 for any odd q if $$n\equiv 0\pmod 4$$ n ≡ 0 ( mod 4 ) and any $$q\equiv 1\pmod 4$$ q ≡ 1 ( mod 4 ) if $$n\equiv 2\pmod 4$$ n ≡ 2 ( mod 4 ) .

How to build a catalogue of linearly evolving cosmic voids

ABSTRACT Cosmic voids provide a powerful probe of the origin and evolution of structures in the Universe because their dynamics can remain near-linear to the present day. As a result, they have the potential to connect large-scale structure at late times to early Universe physics. Existing ‘watershed’-based algorithms, however, define voids in terms of their morphological properties at low redshift. The degree to which the resulting regions exhibit linear dynamics is consequently uncertain, and there is no direct connection to their evolution from the initial density field. A recent void definition addresses these issues by considering ‘anti-haloes’. This approach consists of inverting the initial conditions of an N-body simulation to swap overdensities and underdensities. After evolving the pair of initial conditions, anti-haloes are defined by the particles within the inverted simulation that are inside haloes in the original (uninverted) simulation. In this work, we quantify the degree of non-linearity of both anti-haloes and watershed voids using the Zel’dovich approximation. We find that non-linearities are introduced by voids with radii less than $5\, \mathrm{Mpc}\, h^{-1}$, and that both anti-haloes and watershed voids can be made into highly linear sets by removing these voids.

Lineability, continuity, and antiderivatives in the non-Archimedean setting

This work focuses on the ongoing research of lineability (the search for large linear structures within certain non-linear sets) in non-Archimedean frameworks. Among several other results, we show that there exist large linear structures inside each of the following sets: (i) functions with a fixed closed subset of continuity, (ii) all continuous functions that are not Darboux continuous (or vice versa), (iii) all functions whose Dieudonné integral does not behave as an antiderivative, and (iv) functions with finite range and having antiderivative.

Translation Hyperovals and $\mathbb{F}_2$-Linear Sets of Pseudoregulus Type

In this paper, we study translation hyperovals in PG(2,qk). The main result of this paper characterises the point sets defined by translation hyperovals in the André/Bruck-Bose representation. We show that the affine point sets of translation hyperovals in PG(2,qk) are precisely those that have a scattered F2-linear set of pseudoregulus type in PG(2k−1,q) as set of directions. This correspondence is used to generalise the results of Barwick and Jackson who provided a characterisation for translation hyperovals in PG(2,q2), see [S.G. Barwick, Wen-Ai Jackson, A characterization of translation ovals in finite even order planes. Finite fields Appl. 33 (2015), 37--52.].