论文标题
由一类散射线性的多项式产生的线性集和MRD代码
Linear sets and MRD-codes arising from a class of scattered linearized polynomials
论文作者
论文摘要
显示了一类散射的线性化多项式覆盖无限的许多场扩展。更准确地说，$ q $ -polynomial of $ \ mathbb f_ {q^6} $，$ q \ equiv 1 \ equiv 1 \ pmod 4 $ 4 $ arxiv：1906.05611，arxiv：1910.022278均以任何$ \ ge6 $ $ \ ge6 $ $ \ mathbb f _ $ \ mathbb f_q q q q q q q q qu q q qu q q qu q q qu q qu q q qu q q qu q q qu q q qu q q qu q q que q qu q q qu q qu q q qu qu q q que 6 $ \ mathbb f_q^n $的订单$ n $。事实证明，这种$ψ（x）$及其某些功能功能被证明是分散的。特别是，这提供了射影线$ \ mathrm {pg}（1，q^n）$的新的最大散射线性集，用于$ n = 8,10 $。本文中描述的多项式导致了一个新的无限型MRD代码家族，$ \ mathbb f_q^{n \ times n} $，对于任何奇数$ q $，如果$ n \ equiv0 \ pmod4 $和任何$ q $，则最小距离$ n-1 $，任何$ q $ qu $ q $ q $ q \ eqppmod4 $ n $ n $ n $ n \ equiv2 $ pmod4 $ pmod4 $ pmod4 $ pmod4。
A class of scattered linearized polynomials covering infinitely many field extensions is exhibited. More precisely, the $q$-polynomial over $\mathbb F_{q^6}$, $q \equiv 1\pmod 4$ described in arXiv:1906.05611, arXiv:1910.02278 is generalized for any even $n\ge6$ to an $\mathbb F_q$-linear automorphism $ψ(x)$ of $\mathbb F_q^n$ of order $n$. Such $ψ(x)$ and some functional powers of it are proved to be scattered. In particular this provides new maximum scattered linear sets of the projective line $\mathrm{PG}(1,q^n)$ for $n=8,10$. The polynomials described in this paper lead to a new infinite family of MRD-codes in $\mathbb F_q^{n\times n}$ with minimum distance $n-1$ for any odd $q$ if $n\equiv0\pmod4$ and any $q\equiv1\pmod4$ if $n\equiv2\pmod4$.
