射影空间中半径$R$、余维数$tR$和$tR+\frac{R}{2}$及饱和集的新覆盖码
New covering codes of radius $R$, codimension $tR$ and $tR+\frac{R}{2}$, and saturating sets in projective spaces
摘要：
长度函数$\ell_q(r,R)$是余维$r$和覆盖半径$R$的$ q $元线性码的最小长度。在这项工作中，我们得到了新的建设性的上界$\ell_q(r,R)$所有$R\ge4$，$r=tR$，$t\ge2$，也为所有甚至$R\ge2$，$r=tR+\frac{R}{2}$，$t\ge1$。新的界限提供了无限家庭的新的覆盖码固定$R$和增加余维。新的界限改进了已知的界限。我们提出了一个在射影空间$\mathrm{PG}(2ρ+1,q)$中对于所有$ρ\ge0$的极小$ρ$ -饱和$((ρ+1)q+1)$ -集的一般正则构造（称为"线+椭圆“）.这样的集合对应于覆盖半径$R=ρ+1$的$[Rq+1,Rq+1-2R,3]_qR$局部最优$^1$码。基于这些码关于球形胶囊$^1$的组合性质，我们给出了码余维提升的构造，得到了具有余维$r=tR$，$t\ge2$的新的表面覆盖$^1$码的无限族。此外，我们还得到了射影平面$\mathrm{PG}(2,q^2)$上新的1-饱和集，并在此基础上构造了具有固定偶半径$R\ge2$和余维数$r=tR+\frac{R}{2}$，$t\ge1$的无限码族.（$^1$参见第1节中的定义）
英文摘要：
The length function $\ell_q(r,R)$ is the smallest length of a $ q $-ary linear code of codimension $r$ and covering radius $R$. In this work we obtain new constructive upper bounds on $\ell_q(r,R)$ for all $R\ge4$, $r=tR$, $t\ge2$, and also for all even $R\ge2$, $r=tR+\frac{R}{2}$, $t\ge1$. The new bounds are provided by infinite families of new covering codes with fixed $R$ and increasing codimension. The new bounds improve upon the known ones. We propose a general regular construction (called ``Line+Ovals'') of a minimal $ρ$-saturating $((ρ+1)q+1)$-set in the projective space $\mathrm{PG}(2ρ+1,q)$ for all $ρ\ge0$. Such a set corresponds to an $[Rq+1,Rq+1-2R,3]_qR$ locally optimal$^1$ code of covering radius $R=ρ+1$. Basing on combinatorial properties of these codes regarding to spherical capsules$^1$, we give constructions for code codimension lifting and obtain infinite families of new surface-covering$^1$ codes with codimension $r=tR$, $t\ge2$. In addition, we obtain new 1-saturating sets in the projective plane $\mathrm{PG}(2,q^2)$ and, basing on them, construct infinite code families with fixed even radius $R\ge2$ and codimension $r=tR+\frac{R}{2}$, $t\ge1$. ($^1$ see the definitions in Section 1)
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