Hamming空间中球填充的腔方法
Cavity approach to sphere packing in Hamming space
摘要：
本文利用腔方法研究了Hamming空间中硬球的Packing问题。我们发现，副本对称和复本对称性破缺近似的包装是渐近相同的吉尔伯特和Varshamov的下界的最大速率。与已知的数值结果一致，副本对称方程也提出了一个结晶的解决方案，其中偶数直径的球体更有可能被发现在一个子空间（偶数或奇数）的汉明空间。这些晶体包装可以产生的递归算法，找到最大包装在超度量空间。最后，我们设计了一个基于腔方程的消息传递算法来寻找硬球的稠密填充。已知的最大填充有效地复制在非平凡范围内的尺寸和数量的领域。
英文摘要：
In this paper we study the hard sphere packing problem in the Hamming space by the cavity method. We show that both the replica symmetric and the replica symmetry breaking approximations give maximum rates of packing that are asymptotically the same as the lower bound of Gilbert and Varshamov. Consistently with known numerical results, the replica symmetric equations also suggest a crystalline solution, where for even diameters the spheres are more likely to be found in one of the subspaces (even or odd) of the Hamming space. These crystalline packings can be generated by a recursive algorithm which finds maximum packings in an ultra-metric space. Finally, we design a message passing algorithm based on the cavity equations to find dense packings of hard spheres. Known maximum packings are reproduced efficiently in non trivial ranges of dimensions and number of spheres.
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