广义球面Packing界
Generalized Sphere Packing Bound
摘要：
Kulkarni和Kiyavash最近介绍了一种新的方法来建立删除纠正码大小的上限。这种方法是基于超图理论的工具。删除通道用一个超图来表示，超图的边是删除球（或球体），因此删除校正码在超图中成为匹配。因此，可以从超图的匹配数的界限获得这样的代码的大小的界限。然后调用超图理论中的经典结果来计算匹配数的上界，作为线性规划问题的解决方案。Kulkarni和Kiyavash的方法不仅适用于删除通道，而且适用于其他错误通道。本文研究了这种方法在其最一般的设置。首先，它表明，如果误差通道是规则的和对称的，那么这个上界符合球包装界，因此被称为广义球包装界。尽管这个界限是由线性规划问题明确给出的，但找到它的确切值仍然是一项具有挑战性的任务。为了简化问题的复杂性，我们提出了一种基于图自同构的技术，在许多情况下，减少了问题中的变量和约束的数量。然后，我们将此方法应用于错误通道的具体示例。我们从$Z$通道开始，并展示如何准确找到此设置的广义球体填充边界。接下来研究的是对称和非对称误差的非二进制有限幅度通道，其中我们专注于单误差的情况。我们跟进的删除和颗粒错误的渠道，并展示如何改善现有的上限为单一的删除/错误。最后，我们将此方法应用于射影空间，并在单误差情况下找到了它的广义球填充界。
英文摘要：
Kulkarni and Kiyavash recently introduced a new method to establish upper bounds on the size of deletion-correcting codes. This method is based upon tools from hypergraph theory. The deletion channel is represented by a hypergraph whose edges are the deletion balls (or spheres), so that a deletion-correcting code becomes a matching in this hypergraph. Consequently, a bound on the size of such a code can be obtained from bounds on the matching number of a hypergraph. Classical results in hypergraph theory are then invoked to compute an upper bound on the matching number as a solution to a linear-programming problem. The method by Kulkarni and Kiyavash can be applied not only for the deletion channel but also for other error channels. This paper studies this method in its most general setup. First, it is shown that if the error channel is regular and symmetric then this upper bound coincides with the sphere packing bound and thus is called the generalized sphere packing bound. Even though this bound is explicitly given by a linear programming problem, finding its exact value may still be a challenging task. In order to simplify the complexity of the problem, we present a technique based upon graph automorphisms that in many cases reduces the number of variables and constraints in the problem. We then apply this method on specific examples of error channels. We start with the $Z$ channel and show how to exactly find the generalized sphere packing bound for this setup. Next studied is the non-binary limited magnitude channel both for symmetric and asymmetric errors, where we focus on the single-error case. We follow up on the deletion and grain-error channels and show how to improve upon the existing upper bounds for single deletion/error. Finally, we apply this method for projective spaces and find its generalized sphere packing bound for the single-error case.
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