关于无穷度量下置换群码的标号问题
On the Labeling Problem of Permutation Group Codes under the Infinity Metric
摘要：
最近提出了无限范数下的置换码作为用于校正秩调制方案中的有限幅度误差的编码方案。给定这样的代码，我们表明，一个简单的重新标记操作，产生一个同构的代码，可能会大大改变代码的最小距离。因此，我们可以选择一个有效的编码/解码过程的代码结构，然后通过重新标记优化代码的最小距离。我们正式定义的重新标记问题，并表明，所有的代码可以重新标记，以获得最小的距离最多为2。一个码是否可以被重新标记为距离1的决策问题是NP完全的，并且计算重新标记后的最佳可实现的最小距离被证明是难以近似的。最后，我们考虑重新标记问题的一般界限。我们具体地展示了循环群的最佳重标记距离。一般概率参数的特定情况用于显示$\agl(p)$可以被重新标记为最小距离$p-O(\sqrt{p\ln p})$。
英文摘要：
Codes over permutations under the infinity norm have been recently suggested as a coding scheme for correcting limited-magnitude errors in the rank modulation scheme. Given such a code, we show that a simple relabeling operation, which produces an isomorphic code, may drastically change the minimal distance of the code. Thus, we may choose a code structure for efficient encoding/decoding procedures, and then optimize the code's minimal distance via relabeling. We formally define the relabeling problem, and show that all codes may be relabeled to get a minimal distance at most 2. The decision problem of whether a code may be relabeled to distance 1 is shown to be NP-complete, and calculating the best achievable minimal distance after relabeling is proved hard to approximate. Finally, we consider general bounds on the relabeling problem. We specifically show the optimal relabeling distance of cyclic groups. A specific case of a general probabilistic argument is used to show $\agl(p)$ may be relabeled to a minimal distance of $p-O(\sqrt{p\ln p})$.
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