Given a set of 100 objects $A$. Each with 10 properties. The property values come from a finite set, $\mathbb{Z}_5$. So the ith property of the nth object $P(n,i)\in \mathbb{Z}_5$
Now we randomly change 30 properties of the objects in this set.
This means that usually most of the objects will be the same except some of their properties have changed. So it may be easy to find the minimum number of changes.
It may become more difficult when so many properties have changed that it is now quite difficult to pair the objects back to their originals. e.g. the properties of one object have changed to be almost identical to that of another.
Is there a well known algorithm to do this? (It reminds me somewhat of error correction.)
In otherwords we want the least possible set of instructions to turn the set back into the original.
I guess one can think of this as the minimum differences between two matrices which are isomorphic with respect to row permutations. (But not collumn permutations). Whoses entries are from a finite set.
Or the minimum set of differences in coordinates between two sets of vectors if the coordinates are from a finite set.
You can solve this as a linear assignment problem on a complete bipartite graph with 100 nodes on each side. The cost of assigning old object $i$ to new object $j$ is the number of differences among the properties: $$c_{i,j}=|\{k\in \{1,\dots,10\}: P(i,k)\not= P(j,k)\}|$$