# How to calculate the distance of stabilizer code?

How to calculate the distance of the stabilizer code [[n,k,d]]? It's better if you can make a 3-qubit example. And what's the relationship between d and Pauli group?

Your starting point is a set of stabilizers $$\{K_n\}$$ on $$N$$ qubits, satisfying $$K_n^2=I$$ and $$[K_n,K_m]=0$$. Generically, you want to consider the full set of $$4^N$$ possible tensor products of Pauli operators $$\{I,X,Y,Z\}$$ across all $$N$$ sites. Go through each of these in turn. If it does not commute with each and every $$K_n$$, discard it. If it can be written as a product of some subset of the $$K_n$$, discard it. From the set that you have left, find the term with the smallest weight (i.e. the number of terms that are not $$I$$). That's the distance.
For example, consider the stabilizers $$K_1=Z\otimes Z\otimes Z,\qquad K_2=X\otimes X\otimes I$$ I'm not writing out all 64 possible terms here, but stare at it for a minute. You'll realise that $$I\otimes I\otimes Z$$ commutes with both and cannot be written as a product of the two, so the distance is 1. The problem is that $$N=3$$ is a bit too simple an example to be able to show you too much.
A brief comment on how I would do the maths: I'd set up a computer to do it, using binary matrices. I'd describe each stabilizer generator by a row of $$2N$$ elements. The first $$N$$ are a binary string showing where the $$Z$$s are, and the second $$N$$ are a binary string showing where the $$X$$s are. Commutation is a simple linear algebra check, and similarly, we can check for the containment of stabilizers in a term by using an inner product. All calculations are performd modulo 2.