线性子空间的公共补与MRD码的稀疏性
Common Complements of Linear Subspaces and the Sparseness of MRD Codes
摘要：
受秩度量码理论应用的启发，我们研究了利用有限域上的子空间族的基数及其交结构来估计该子空间族的公共补数的问题.我们推导出这个数的上界和下界，以及它们的渐近版本，因为字段大小趋于无穷大。然后，我们使用这些界限来描述的一般行为的共同补充相对于稀疏性和密度，显示的决定性属性是是否要补充的空间的数量是可以忽略不计的字段大小。通过将我们的结果专门化到矩阵空间，我们得到了MRD码在秩度量上的个数的上界和下界。特别是，我们回答了一个悬而未决的问题，在编码理论，证明MRD码是稀疏的所有参数集的字段大小的增长，只有很少的例外。我们还研究了MRD码的密度，因为它们的列数趋于无穷大，获得一个新的渐近界。然后，利用数论中欧拉函数的性质，我们证明了对于大多数参数集，我们的界比已知结果有所改进。最后，我们通过建立秩度量码的密度函数的一般结构性质来总结本文。
英文摘要：
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.
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