论文标题
线性子空间的常见补充和MRD代码的稀疏度
Common Complements of Linear Subspaces and the Sparseness of MRD Codes
论文作者
论文摘要
在对等级码理论的应用中，我们研究了根据家族的基数及其交叉结构来估计一个子空间家族的共同补充的数量。我们为此数字得出了上限和下限，以及它们的渐近版本，因为田间大小趋向于无穷大。然后，我们使用这些界限来描述公共补充在稀疏性和密度方面的一般行为，表明决定性属性是相对于场大小而言，要补充的空间数量是否可以忽略不计。通过将我们的结果专门为基质空间，我们获得了等级度量中MRD代码数量的上限和下限。特别是，我们回答了编码理论中的一个空旷的问题，证明随着场大小的增长，所有参数集的MRD代码稀疏，只有很少的例外。我们还研究了MRD代码的密度，因为它们的列数倾向于无穷大，获得了新的渐近结合。然后，使用数字理论的Euler函数的属性，我们表明我们的界限改进了大多数参数集的已知结果。我们通过建立等级码密度函数的一般结构特性来结束论文。
Motivated by applications to the theory of rank-metric codes, we study the problem of estimating the number of common complements of a family of subspaces over a finite field in terms of the cardinality of the family and its intersection structure. We derive upper and lower bounds for this number, along with their asymptotic versions as the field size tends to infinity. We then use these bounds to describe the general behaviour of common complements with respect to sparseness and density, showing that the decisive property is whether or not the number of spaces to be complemented is negligible with respect to the field size. By specializing our results to matrix spaces, we obtain upper and lower bounds for the number of MRD codes in the rank metric. In particular, we answer an open question in coding theory, proving that MRD codes are sparse for all parameter sets as the field size grows, with only very few exceptions. We also investigate the density of MRD codes as their number of columns tends to infinity, obtaining a new asymptotic bound. Using properties of the Euler function from number theory, we then show that our bound improves on known results for most parameter sets. We conclude the paper by establishing general structural properties of the density function of rank-metric codes.
