来自欧几里得和厄米特对偶包含循环码的最优纯量子 $(r,\delta)$-LRC
Optimal Pure Quantum $(r,δ)$-LRCs from Euclidean and Hermitian Dual-Containing Cyclic Codes
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中文总结 AI 辅助
本文利用欧几里得和厄米特对偶包含循环码构造了多族最优纯量子局部可恢复码，满足量子Singleton型界，并覆盖了现有构造未达到的参数范围。
中文摘要 AI 辅助
局部可恢复码（LRC）将全局纠错与仅通过访问有限数量的其他坐标来修复少量擦除坐标的能力相结合。受其量子对应物的启发，我们构造了几族最优循环 $(r,\delta)$-LRC，这些码是欧几里得或厄米特对偶包含的。在欧几里得情形下，我们在 $\mathbb{F}_q$ 上获得了四族最优对偶包含循环 $(r,\delta)$-LRC，其最小距离范围延伸至超过局部距离 $\delta$。在厄米特情形下，当 $(r+\delta-1)\mid(q^2-1)$ 时，我们在 $\mathbb{F}_{q^2}$ 上推导出类似的厄米特对偶包含族。此外，我们针对 $(r+\delta-1)\mid(q^2+1)$ 的情形，利用对称定义集和奇数 $\delta$ 开发了一种不同的构造，该构造产生最小距离为 $\ell+2$、$\delta+2$、$2\delta-2$ 和 $2\delta$ 的最优码。然后，欧几里得 Calderbank-Shor-Steane（CSS）和厄米特稳定子构造在 $\mathbb{F}_q$ 上给出相应的量子循环 $(r,\delta)$-LRC。进一步地，所得的稳定子码是纯的；它们满足相关的量子 Singleton 型界，因此是最优的。我们还与先前的工作进行了比较，强调了我们的结果所覆盖的参数范围和最小距离区间，这些是现有构造未达到的。显式示例说明了这些构造并验证了对偶包含性和纯性条件。
英文摘要
Locally recoverable codes (LRCs) combine global error correction with the ability to repair a small number of erased coordinates by accessing only a limited number of other coordinates. Motivated by their quantum counterparts, we construct several families of optimal cyclic $(r,δ)$-LRCs that are Euclidean or Hermitian dual-containing. In the Euclidean case, we obtain four families of optimal dual-containing cyclic $(r,δ)$-LRCs over $\mathbb{F}_q$ with a range of minimum distances extending beyond the local distance $δ$. In the Hermitian case, we derive analogous families over $\mathbb{F}_{q^2}$ that are Hermitian dual-containing, when $(r+δ-1)\mid(q^2-1)$. In addition, we develop a distinct construction for the case $(r+δ-1)\mid(q^2+1)$ using symmetric defining sets and odd $δ$, which yields optimal codes with minimum distances $\ell+2$, $δ+2$, $2δ-2$, and $2δ$. The Euclidean Calderbank-Shor-Steane (CSS) and Hermitian stabilizer constructions then give corresponding quantum cyclic $(r,δ)$-LRCs over~$\mathbb{F}_q$. Further, the resulting stabilizer codes are pure; they meet the relevant quantum Singleton-type bound, and are therefore \textit{optimal}. We also provide a comparison with previous works, highlighting the parameter regimes and minimum distance ranges covered by our results that are not attained by existing constructions. Explicit examples illustrate the constructions and verify the dual-containment and purity conditions.
