当没有奇偶校验函数时，我们能有多接近奇偶校验函数？
How close can we come to a parity function when there isn't one?
摘要：
考虑一个群G，使得没有同态f：G到{+1，-1}。在这种情况下，我们离这样的同态有多近？证明了如果f的期望为零，则f（xy）= f（x）f（y）的概率至多为1/2（1+1/sqrt{d}），其中x，y是独立于G的一致选择，d是G的最小非平凡不可约表示的维数.例如，对于交错群A_n，d=n-1.另一方面，A_n包含一个同构于S_{n-2}的子群，其奇偶函数可以推广得到一个f，使得该概率为1/2（1+1/{n \choose 2}）.因此，f可以比从A_n到{+1，-1}的随机函数“更同态”的程度介于O（n^{-1/2}）和Omega（n^{-2}）之间。
英文摘要：
Consider a group G such that there is no homomorphism f:G to {+1,-1}. In that case, how close can we come to such a homomorphism? We show that if f has zero expectation, then the probability that f(xy) = f(x) f(y), where x, y are chosen uniformly and independently from G, is at most 1/2(1+1/sqrt{d}), where d is the dimension of G's smallest nontrivial irreducible representation. For the alternating group A_n, for instance, d=n-1. On the other hand, A_n contains a subgroup isomorphic to S_{n-2}, whose parity function we can extend to obtain an f for which this probability is 1/2(1+1/{n \choose 2}). Thus the extent to which f can be "more homomorphic" than a random function from A_n to {+1,-1} lies between O(n^{-1/2}) and Omega(n^{-2}).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
