Difficulty in understanding the definition of this sequence Let $G$ be a finite group and $J=\langle g_1,\cdots g_k\rangle$ be a sequence of group elements. For any $\delta \ge 1$, $J$ is said to be a cube generating sequence for $G$ with closeness parameter $\delta$, if the probability distribution $D_J$ on $G$ given by $g_1^{\epsilon_1}\cdots g_k^{\epsilon_k},$ where each $\epsilon_i$ is independent and uniformally distributed in {0,1}, is $\delta$-close to the uniform distribution in the $L_2$ norm.
Question: What is the difference between ordinary generating set and generating set given abobve?
 A: The main difference is that any generating set will not give you something "close" to the uniform distribution. Consider the group $\mathbb{Z}^2$ with standard generators $a,b$. Choose as a generating set $b,b^{-1},a^{-1}$ and $a,a^2,a^3,...,a^k$ for some big $k$.
This generating set will not be "cube generating" according to your terminology. More precisely, for any given $\delta$, you can choose $k$ so that the generating set is not $\delta$-cube generating.
The reason is that you will have more chance to pick an element with a huge power of $a$ than any element in the group.
If you are familiar with free groups, this is even clearer in the free group generated by $a$ and $b$ with the same generating set.
Now, the second difference is that it is not clear for me that a "cube generating" sequence is actually a generating set, since you only require some closeness to the uniform distribution. It seems that a $\delta$-cube generating sequence for $G$ is a $\delta'$-cube generating sequence for $G\times F$, where $F$ is a finite group. 
