Articles you may be interested inCodeword stabilized quantum codes: Algorithm and structureIn this paper we propose a construction procedure of a class of topological quantum error-correcting codes on surfaces with genus g Ն 2. This generalizes the toric codes construction. We also tabulate all possible surface codes with genus 2-5. In particular, this construction reproduces the class of codes obtained when considering the embedding of complete graphs K s , for s ϵ 1 mod 4, on surfaces with appropriate genus. We also show a table comparing the rate of different codes when fixing the distance to 3-5.
In this paper we present some classes of topological quantum codes on surfaces with genus $g \geq 2$ derived from hyperbolic tessellations with a specific property. We find classes of codes with distance $d = 3$ and encoding rates asymptotically going to 1, $\frac{1}{2}$ and $\frac{1}{3}$, depending on the considered tessellation. Furthermore, these codes are associated with embedding of complete bipartite graphs. We also analyze the parameters of these codes, mainly its distance, in addition to show a class of codes with distance 4. We also present a class of codes achieving the quantum Singleton bound, possibly the only one existing under this construction.
In this paper we present six classes of topological quantum codes (TQC) on compact surfaces with genus $g\ge 2$. These codes are derived from self-dual, quasi self-dual and denser tessellations associated with embeddings of self-dual complete graphs and complete bipartite graphs on the corresponding compact surfaces. The majority of the new classes has the self-dual tessellations as their algebraic and geometric supporting mathematical structures. Every code achieves minimum distance 3 and its encoding rate is such that $\frac{k}{n} \rightarrow 1$ as $n \rightarrow \infty$, except for the one case where $\frac{k}{n} \rightarrow \frac{1}{3}$ as $n \rightarrow \infty$.
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