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An $ n $ qudit code always has distance bounded above by $ d \leq \frac{n+1}{2} $. The $ 5 $ qudit generalization of the Knill-Laflamme code with parameters $ [[5,1,3]]_q $ ( $ q $ denoting the size of the qudit) has stabilizer generators \begin{align*} & XZZ^\dagger X^\dagger I \\ & IXZZ^\dagger X^\dagger \\ & X^\dagger IXZZ^\dagger \\ & Z^\dagger X^\dagger IXZ \end{align*} and for any $ q $ this code saturates the bound $ d \leq \frac{n+1}{2} $ since $$ 3 = \frac{5+1}{2} $$ The three qutrit code with parameters $ [[3,1,2]]_3 $ also saturates this bound $$ 2 = \frac{3+1}{2} $$ Are there any other qudit codes that saturate this bound? Or are the $ [[5,1,3]]_q $ and $ [[3,1,2]]_3 $ codes essentially unique in this respect?

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The bound you are referring to is a special case of the quantum Singleton bound. This bound says that given a code $((n,K,d))_q$ we have $$ K \leq q^{n-2(d-1)}. $$ In the special case that we encode $k$-qu$q$its into $n$-qu$q$its (i.e., $K = q^k$) we have $$ d \leq \frac{n-k+2}{2}. $$ Thus the bound you provided is a special case of the quantum singleton bound when $k = 1$.

A code that satisfies the quantum Singleton bound with equality is called a quantum maximum-distance-separable code (or quantum MDS code). So your question is basically asking which MDS codes exist.

Qubit $q=2$ MDS codes are completely classified and there aren't that many of them. They are $[[6,0,4]]_2$, $[[5,1,3]]_2$, and the even length codes $[[2m,2m-2,2]]_2$ (also the trivial codes $[[n,n,1]]_2$ but those are sort of fake). For higher $q$, there are many more MDS codes, for example see Quantum MDS Codes over Small Fields .

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    $\begingroup$ ah ok so it looks like the only qubit code with $ d=(n+1)/2 $ is indeed the $ [[5,1,3]] $ code. But also looks like for different $ q $ there are lots of these codes like for qutrits $ q=3 $ there is the $ [[3,1,2]]_3 $ code and the $ [[5,1,3]]_3 $ code but also this paper shows there is a $ [[9,1,3]]_3 $ code. Other codes with $ d=(n+1)/2 $ described in the paper have parameters $ [[9,1,5]]_4, [[9,1,5]]_5, [[7,1,4]]_5, [[9,1,5]]_7, [[11,1,6]]_7, [[13,1,7]]_7, [[9,1,5]]_8 $. In general it seems reasonable to believe that there are $ [[9,1,5]]_q $ codes for $ q \geq 3 $. $\endgroup$ Feb 25 at 22:33
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Thinking about your first question regarding qubit codes, another bound that we know must apply (at least for non-degenerate codes) is the Quantum Hamming Bound, $$ 2^{n-k}\geq\sum_{w=0}^{t}3^w\binom{n}{w}, $$ assuming that $d=2t+1$ is odd. We make this as loose as possible by setting $k=1$, and we take $d=(n+1)/2$ (and therefore $n=4t+1$). So now we can ask for what values of $n$ is the Hamming bound satisfied? We can explicitly check for small $n$: only $n=5,9$ work. For large enough $n$ (I didn't do this carefully, but $n\geq 21$ seemed to be about right), you can inductively prove that if the Hamming bound isn't satisfied for $n$, then it isn't for $n+4$ either.

Now, we know that $n=5$ is a valid solution, so what about $n=9$? According to codetables.de, the best code $[[9,1,d]]$ has $d=3$, not $d=5$. Thus, the 5-qubit code is the only odd distance non-degenerate qubit code satisfying that bound.

I've emphasised that I've put in a non-degenerate assumption. I don't think that's too big an assumption - it's tough to find examples of degenerate codes that beat the Hamming bound and, even then, the only ones I'm aware of do so by a very small amount. So, I suspect this conclusion still holds when you include degenerate codes.

I assume (without having tried) that you could construct something similar for $q>2$.

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  • $\begingroup$ Interesting! see Eric's answer for details but it seems like the lack of a $ [[9,1,5]] $ qubit code is a bit exceptional as it appears $ [[9,1,5]]_q $ codes exist for $ q \geq 3 $ $\endgroup$ Feb 25 at 22:36

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