# Quantum codes with linear distance

I was wondering if there exists any family of quantum codes (encoding any number of logical qubits) with linear distance.

I know of:

1 - Families of topological codes with linear distance when we restrict perturbations to be local on some lattice (here)

2 - Quantum codes with linear distance encoding a single qubit (I don't have any reference though)

It is known that random quantum codes almost surely have linear distance, and in fact almost surely attain the so-called quantum Gilbert-Varshamov bound. I list two families of random quantum codes:

1. random subspaces (the least structure)
2. random stabilizer codes (impose at least some stabilizer structure)

Concatenated quantum codes can also attain the Gilbert-Varshamov bound. For example, one can concatenate the quantum Reed-Solomon outer code with random inner codes.

For explicit codes with large linear distance, there are the algebraic geometry codes, and also the quantum Justesen codes, and also many more. When making quantum codes on qudits, provided that the dimension of the qudit is more than 7, the algebraic geometry codes outperform the random codes almost surely

Kitaev's toric code does not have linear distance. The ratio of the distance to the number of qubits approaches zero as the number of qubits approaches infinity.

• Given the age of the original question, I figure I'll ask this on your recent answer: when you speak of linear distance, and you have an error correcting code that encodes $k$ logical qubits into $n$ physical qubits with distance $d$ between codewords, what exactly is linear? Commented Aug 4, 2017 at 14:58
• Suppose that we have a quantum code $\mathcal C_i$ on $n_i$ qubits that encodes $k_i$ logical qubits that has distance $d_i$. The relative distance of the code $\mathcal C_i$ is then defined to be $d_i/n_i$. Let $\{C_i\}_{i=1}^{\infty}$ be a family of quantum codes such that $\lim_{i \to \infty} n_i = \infty$. Then the asymptotic relative distance of the family of codes is $\delta = \lim_{i \to \infty} \delta_i$. If this asymptotic relative distance is strictly positive, then we say that this family of codes has a linear distance. Commented Aug 5, 2017 at 22:43