射影空间中的线性非二元覆盖码与饱和集
Linear nonbinary covering codes and saturating sets in projective spaces
摘要：
设A_{R,q}表示一类覆盖码，其中覆盖半径R和基域的大小q是固定的，而码长趋于无穷大.本文考虑了A_{R,q}族的无限集，其中R是固定的，但q在一个无限素幂集上，并研究了A_{R,q}族的渐近覆盖密度对q的依赖性.证明了对于A_{R,q}的覆盖密度的上极限，最佳可能性为O（q）.本文的主要成果是对任意覆盖半径R >= 2构造了族A_{R,q}的渐近最优无限集。我们首先证明了，对于给定的R，为了得到族的最优无限集，构造R个无限族A_{R,q} ^{（0）}，A_{R,q} ^{（1）}，.，A_{R,q} ^{（R-1）}，使得对于所有u >= u_{0}，族A_{R,q} ^{（v）}包含余维r_{u} =Ru+v和长度f_{q} ^{v}（r_{u}）的码，其中f_{q} ^{v}（r）=O（q^{（r-R）/R），并且u_{0}是常数。然后，我们能够为任何覆盖半径R >= 2构造所需的族A_{R,q} ^{（v）}，其中q的范围在R次幂的（无限）集合上。对于这些族A_{R,q} ^{（v）}中的每一个，覆盖密度的下极限由与q无关的常数从上限定。
英文摘要：
Let A_{R,q} denote a family of covering codes, in which the covering radius R and the size q of the underlying Galois field are fixed, while the code length tends to infinity. In this paper, infinite sets of families A_{R,q}, where R is fixed but q ranges over an infinite set of prime powers are considered, and the dependence on q of the asymptotic covering densities of A_{R,q} is investigated. It turns out that for the upper limit of the covering density of A_{R,q}, the best possibility is O(q). The main achievement of the present paper is the construction of asymptotic optimal infinite sets of families A_{R,q} for any covering radius R >= 2. We first showed that for a given R, to obtain optimal infinite sets of families it is enough to construct R infinite families A_{R,q}^{(0)},A_{R,q}^{(1)},...,A_{R,q}^{(R-1)} such that, for all u >= u_{0}, the family A_{R,q}^{(v)} contains codes of codimension r_{u}=Ru+v and length f_{q}^{v}(r_{u}) where f_{q}^{v}(r)=O(q^{(r-R)/R) and u_{0} is a constant. Then, we were able to construct the needed families A_{R,q}^{(v)} for any covering radius R >= 2, with q ranging over the (infinite) set of R-th powers. For each of these families A_{R,q}^{(v)}, the lower limit of the covering density is bounded from above by a constant independent of q.
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