Quantum cyclic locally recoverable codes achieve optimal error correction
Optimal Pure Quantum $(r,δ)$-LRCs from Euclidean and Hermitian Dual-Containing Cyclic Codes
Information Theory
Summary
Error-correcting codes help fix lost or corrupted data by using extra information. This paper presents special cyclic codes that allow quick repair of small data losses while also providing strong overall error protection. The authors design these codes to work well with quantum information systems, ensuring they meet known limits for performance. Their constructions expand the range of parameters achievable beyond earlier methods.
What this means in practice
- •For quantum communication engineers: Build quantum error correction systems that locally repair small losses efficiently while maintaining maximal overall protection using the new optimal code constructions.
- •For classical coding theory developers: Develop classical cyclic codes with improved error and local repair capabilities by adapting the dual-containing constructions described for broader parameter ranges.
Authors
Astha Agrawal, Shayan Srinivasa Garani
Abstract
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.