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 ) .
The effect of parental smoking on childhood asthma was investigated in which data from 302 asthmatic and 433 healthy children aged 1 to 12 years, were studied. All asthmatic patients received prick tests for common allergens. A significantly higher number of heavy parental smokers was found in asthmatic children under 6 years of age with negative prick tests (P = 0.02). A stepwise logistic regression was performed in order to verify interactions between parental smoking and other variables. It is concluded that parental smoking is an important risk factor for "prick test negative" asthmatic children aged 6 years or less.
Let d, n ∈ Z + such that 1 ≤ d ≤ n. A d-code C ⊂ F n×n q is a subset of order n square matrices with the property that for all pairs of distinct elements in C, the rank of their difference is greater than or equal to d. A d-code with as many as possible elements is called a maximum d-code.The integer d is also called the minimum distance of the code. When d < n, a classical example of such an object is the so-called generalized Gabidulin code, [7]. In [2], [16] and [13], several classes of maximum d-codes made up respectively of symmetric, alternating and hermitian matrices were exhibited. In this article we focus on such examples. Precisely, we determine their automorphism groups and solve the equivalence issue for them. Finally, we exhibit a maximum symmetric 2-code which is not equivalent to the one with same parameters constructed in [16].
A linearized polynomial f (x) ∈ F q n [x] is called scattered if for any y, z ∈ F q n , the condition zf (y) − yf (z) = 0 implies that y and z are F q -linearly dependent. In this paper two generalizations of the notion of a scattered linearized polynomial are defined and investigated. Let t be a nontrivial positive divisor of n. By weakening the property defining a scattered linearized polynomial, L-q t -partially scattered and R-q t -partially scattered linearized polynomials are introduced in such a way that the scattered linearized polynomials are precisely those which are both L-q t -and R-q t -partially scattered. They determine linear sets and maximum rank distance codes whose properties are described in this paper.
In this paper we present results concerning the stabilizer G f of the subspace, under the action of GL(2, q n ). Each G f contains the q − 1 maps (x, y) → (ax, ay), a ∈ F * q . By virtue of the results of Beard [5, 6] and Willett [25], the matrices in G f are simultaneously diagonalizable. This has several consequences: (i) the polynomials such that |G f | > q − 1 have a standard form of type n/t−1 j=0 ajx q s+jt for some s and t such that (s, t) = 1, t > 1 a divisor of n; (ii) this standard form is essentially unique; (iii) the translation plane A f associated with f (x) admits affine homologies if and only if |G f | > q − 1, and in that case the affine homologies with axis through the origin form two groups of cardinality (q t − 1)/(q − 1) that exchange axes and coaxes; (iv) no plane of type A f , f (x) a scattered polynomial not of pseudoregulus type, is a generalized André plane.
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