同态扩张
Homomorphism Extension
摘要：
我们定义了同态扩张（HomExt）问题：给定一个群$G$，一个子群$M \leq G$和一个同态$φ: M \to H$，判定是否存在一个同态$\widetildeφ: G\to H$扩张$φ$，即，$\widetildeφ|_M = φ$。这个问题出现在列表解码同态码的背景下，但也是独立的利益，无论是作为一个问题，在计算群论和作为一个新的和自然的NP问题的不稳定的复杂性状态。我们考虑$H=S_m$（度为$m$的对称群）的情况，即，$φ: G \to H$是一组$m$元素上的$G$ -操作。我们假设$G\le S_n$是由生成元列表给出的置换群。我们刻画的等价类的多维Oracle子集和问题的扩展。由此我们推断，对于有界$G$，HomExt问题可以在多项式时间内解决。我们的主要结果涉及的情况下，$G=A_n$（交替组的程度$n$）为变量$n$的$M$在$G$的指数是有界的聚$(n)$。我们解决了这种情况下，在多项式时间的所有$m < 2^{n-1}/\sqrt{n}$。这是与同态码直接相关的情况（Babai，Black和Wuu，arXiv 2018）;它被用作该论文中主要算法之一的组成部分。
英文摘要：
We define the Homomorphism Extension (HomExt) problem: given a group $G$, a subgroup $M \leq G$ and a homomorphism $φ: M \to H$, decide whether or not there exists a homomorphism $\widetildeφ: G\to H$ extending $φ$, i.e., $\widetildeφ|_M = φ$. This problem arose in the context of list-decoding homomorphism codes but is also of independent interest, both as a problem in computational group theory and as a new and natural problem in NP of unsettled complexity status. We consider the case $H=S_m$ (the symmetric group of degree $m$), i.e., $φ: G \to H$ is a $G$-action on a set of $m$ elements. We assume $G\le S_n$ is given as a permutation group by a list of generators. We characterize the equivalence classes of extensions in terms of a multidimensional oracle subset-sum problem. From this we infer that for bounded $G$ the HomExt problem can be solved in polynomial time. Our main result concerns the case $G=A_n$ (the alternating group of degree $n$) for variable $n$ under the assumption that the index of $M$ in $G$ is bounded by poly$(n)$. We solve this case in polynomial time for all $m < 2^{n-1}/\sqrt{n}$. This is the case with direct relevance to homomorphism codes (Babai, Black, and Wuu, arXiv 2018); it is used as a component of one of the main algorithms in that paper.
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