Why can't edit distance be solved as L1 distance? Given two strings $x$ and $y$ over the alphabet $\Sigma$ one defines the edit-distance $\text{ed}(x,y)$ as the minimum number of substitutions, insertions and deletions of characters required to transform $x$ into $y$. 

If $m = \max ( \text{len}(x),\text{len}(y) )$ then padding with appropriate zeroes on the smaller string one can define $x$ and $y$ as both being elements of $\Sigma^m$. Now both $x$ and $y$ are vertices of the cube $\Sigma^m$. 

Now isn't $\text{ed}(x,y)$ the shortest distance between $x$ and $y$ in this cube graph? Why isn't edit distance found by finding shortest paths in this cube?
Consider the two strings 01010101 and 10101010. They have distance 8 on the cube but their edit distance is 2.