常秩码及其与常维码的联系
Constant-Rank Codes and Their Connection to Constant-Dimension Codes
摘要：
常维码由于其在非相干随机线性网络编码中的差错控制方面的重要性而受到关注。有限维最小距离的等维码的最大基数是多少，如何构造最大基数的最优等维码，都是有待解决的问题。在本文中，我们介绍了一种新的方法来解决这两个问题。我们首先建立了常秩码和常维码之间的联系。通过这种连接，我们表明，最佳的常数维码对应于最佳的常数秩码的矩阵有足够多的行。因此，上述两个问题分别等价于确定定秩码的最大基数和构造最优定秩码。为此，然后，我们推导出一个恒定的秩码的最大基数与给定的最小秩距离的界限，提出明确的最佳或渐近最佳的恒定秩码的建设，并建立渐近界的最大速率的恒定秩码。
英文摘要：
Constant-dimension codes have recently received attention due to their significance to error control in noncoherent random linear network coding. What the maximal cardinality of any constant-dimension code with finite dimension and minimum distance is and how to construct the optimal constant-dimension code (or codes) that achieves the maximal cardinality both remain open research problems. In this paper, we introduce a new approach to solving these two problems. We first establish a connection between constant-rank codes and constant-dimension codes. Via this connection, we show that optimal constant-dimension codes correspond to optimal constant-rank codes over matrices with sufficiently many rows. As such, the two aforementioned problems are equivalent to determining the maximum cardinality of constant-rank codes and to constructing optimal constant-rank codes, respectively. To this end, we then derive bounds on the maximum cardinality of a constant-rank code with a given minimum rank distance, propose explicit constructions of optimal or asymptotically optimal constant-rank codes, and establish asymptotic bounds on the maximum rate of a constant-rank code.
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