On the Construction of New Toric Quantum Codes and Quantum Burst-Error Correcting Codes

Abstract: A toric quantum error-correcting code construction procedure is presented in this work. A new class of an infinite family of toric quantum codes is provided by constructing a classical cyclic code on the square lattice $\mathbb{Z}{q}\times \mathbb{Z}{q}$ for all odd integers $q\geq 5$ and, consequently, new toric quantum codes are constructed on such square lattices regardless of whether $q$ can be represented as a sum of two squares. Furthermore this work supplies for each $q$ the polyomino shapes that tessellate the corresponding square lattices and, consequently, tile the lattice $\mathbb{Z}^{2}$. The channel without memory to be considered for these constructed toric quantum codes is symmetric, since the $\mathbb{Z}^{2}$-lattice is autodual. Moreover, we propose a quantum interleaving technique by using the constructed toric quantum codes which shows that the code rate and the coding gain of the interleaved toric quantum codes are better than the code rate and the coding gain of Kitaev's toric quantum codes for $q=2n+1$, where $n\geq 2$, and of an infinite class of Bombin and Martin-Delgado's toric quantum codes. In addition to the proposed quantum interleaving technique improves such parameters, it can be used for burst-error correction in errors which are located, quantum data stored and quantum channels with memory.