# Is the Steane code the only $[\![7,1,3]\!]$ CSS code?

Is the Steane code the only $$[\![7,1,3]\!]$$ CSS code?

This paper claims there are 10 non-equivalent $$[\![7,1,3]\!]$$ stabilizer codes. How many of these are CSS codes? Is it just the Steane code?

The CSS construction takes a classical linear $$[n,k_1]$$ code $$C_1$$ and a classical linear $$[n,k_2]$$ code $$C_2\subset C_1$$ such that $$C_1$$ and $$C_2^\perp$$ both have code distance at least $$d$$, and yields a quantum stabilizer code $$[\![n, k_1-k_2, d]\!]$$. Thus, the question can be answered by counting suitable pairs of classical codes. By the classical Hamming bound $$k_1,k_2\leqslant 4$$, so there are only three cases to consider.

## Case one: $$[7,2]+[7,1]$$

A classical $$[7,1]$$ code consists of two code words $$a$$ and $$b$$. By a simple counting argument, we can find two distinct $$i,j\in\{0,1,\dots,6\}$$ such that $$a_i=a_j$$ and $$b_i=b_j$$. But then the $$7$$-bit string $$e_{i,j}$$ with $$1$$ on positions $$i$$ and $$j$$ and zeroes everywhere else belongs to $$C_2^\perp$$. However, its Hamming weight is $$2$$, so $$C_2^\perp$$ cannot have distance $$3$$. Therefore, there are no suitable classical codes with $$k_1=2$$ and $$k_2=1$$.

## Case two: $$[7,3]+[7,2]$$

The above argument applies in this case, too, albeit we now have to use the fact that the code is linear. In more detail, a classical linear $$[7,2]$$ code $$C_2$$ is a subspace of $$\mathbb{Z}_2^7$$ spanned by two linearly independent $$7$$-bit strings $$a$$ and $$b$$. Once again, we can find two distinct positions $$i,j\in\{0,1,\dots,6\}$$ such that $$a_i=a_j$$ and $$b_i=b_j$$. But then $$e_{i,j}\in C_2^\perp$$, so $$C_2^\perp$$ has distance at most $$2$$ and once again we find no suitable classical codes with $$k_1=3$$ and $$k_2=2$$.

## Case three: $$[7,4]+[7,3]$$

Let $$G\in\mathbb{Z}_2^{4\times 7}$$ be the generator matrix of a $$[7,4]$$ code $$C_1$$. By adding and rearranging rows we can transform the matrix into the standard form $$G=[I_4|P]\tag1$$ where $$I_4$$ is the $$4\times 4$$ identity matrix and $$P\in\mathbb{Z}_2^{4\times 3}$$. But $$C_1$$ has distance at least $$3$$, so every row of $$P$$ must have Hamming weight at least two. For the same reason, the rows of $$P$$ must be distinct. But there are only four possible $$3$$-bit strings that satisfy these requirements: $$111$$, $$110$$, $$101$$, and $$011$$. Therefore, up to permutations of rows and columns, the generator matrix is $$G=\begin{bmatrix} 1 & 0 & 0 & 0 & 1 & 1 & 1\\ 0 & 1 & 0 & 0 & 1 & 1 & 0\\ 0 & 0 & 1 & 0 & 1 & 0 & 1\\ 0 & 0 & 0 & 1 & 0 & 1 & 1\\ \end{bmatrix}.\tag2$$ This fixes the $$C_1$$ code. Our task is then reduced to the task of enumerating all $$3$$-dimensional subspaces of $$C_1$$ whose dual is a linear code of distance at least three. However, $$C_1$$ is a set of mere sixteen elements, so there are fewer than $${15 \choose 3}=455$$ eligible subspaces. The exhaustive search is left as a simple coding exercise for the reader.

• I see, this is very nice and not only do these sorts of arguments prove that the Steane code is the only $[\![7,1,3]\!]$ CSS code but they also show that no $[\![6,1,3]\!]$ CSS code exists (or generally no $[\![n,1,3]\!]$ CSS code exists for $n \leq 6$). Commented Mar 4 at 13:29
• Interesting that this paper arxiv.org/abs/quant-ph/9908010 seems to already reference "the 7-qubit CSS code" as if it were known to be unique Commented Jun 18 at 19:37