All the stabilizer codes of distance 3

Details All the stabilizer codes of distance 3 All the stabilizer codes of distance 3

2009-01-14

arxiv.org/abs/0901.1968v3

Physics

Quantum Physics

7 pages with 8 tables

Scientific paper

We give necessary and sufficient conditions for the existence of stabilizer codes $[[n,k,3]]$ of distance 3 for qubits: $n-k\ge \lceil\log_2(3n+1)\rceil+\epsilon_n$ where $\epsilon_n=1$ if $n=8\frac{4^m-1}3+\{\pm1,2\}$ or $n=\frac{4^{m+2}-1}3-\{1,2,3\}$ for some integer $m\ge1$ and $\epsilon_n=0$ otherwise. Or equivalently, a code $[[n,n-r,3]]$ exists if and only if $n\leq (4^r-1)/3, (4^r-1)/3-n\notin\lbrace 1,2,3\rbrace$ for even $r$ and $n\leq 8(4^{r-3}-1)/3, 8(4^{r-3}-1)/3-n\not=1$ for odd $r$. Given an arbitrary length $n$ we present an explicit construction for an optimal quantum stabilizer code of distance 3 that saturates the above bound.

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