# Finding all small stabilizer codes

Given some choice of parameters $$[[n,k,d]]$$ with $$n$$ small, is there any computationally easy way to find all of (or at least many of) the stabilizer codes with those parameters?

For certain parameters this is easy, for example it is known that there is a unique stabilizer code corresponding to each of the parameters $$[[4,2,2]], [[5,1,3]], [[8,3,3]]$$. And for many other choices of parameters $$[[n,k,d]]$$ no codes exist at all, by the bounds given in https://arxiv.org/abs/quant-ph/9608006

But what about choices of parameters that are possible but not unique, such as $$[[5,1,2]]$$ and $$[[7,1,3]]$$?

For example this paper https://arxiv.org/abs/0709.1780 claims to find all stabilizer codes, up to equivalence, with parameters $$[[7,1,3]]$$.

A few of these I can think of off the top of my head. For example the Steane code has the weight enumerator they call $$W_8$$ and the code $$[[5,1,3]]+[[2,0,2]]$$ has the weight enumerator they call $$W_1$$, so I can write down stabilizer generators for these codes readily enough.

But in general how do I write down stabilizer generators for the many other $$[[7,1,3]]$$ codes they claim in the paper? I'm hoping there might be a nice way to get at least some of these using GUAVA? Along the lines of past answers from @unknown

Smallest distance 9 self-dual CSS code?

Example CSS codes and the properties "doubly even" and "self dual"

and even

Generators for $[[9,1,3]]$ linear quantum code

This question is partially inspired by another more recent question from @unknown about translating between stabilizer codes and graph codes

how to go from a stabilizer state to a graph

• If we translate the stabilizer codes to graph-state representations, is there any relation between the code distance and the shape of the graph? Mar 5 at 5:38