Constructions of Quantum $(r,δ)$-LRCs from cyclic codes
2026-06-08 • Information Theory
Information Theory
AI summaryⓘ
The authors study ways to build quantum error-correcting codes that can fix small errors by looking at only a few parts of the data, inspired by classical locally recoverable codes (LRCs). They focus on a special type called cyclic codes that meet certain mathematical conditions to work in quantum settings. They introduce three new families of these quantum codes, with two being the best possible according to a known theoretical limit for pure codes. Their methods also allow for creating codes of any length without restrictions from the size of the underlying finite field.
locally recoverable codesquantum error correctioncyclic codesCSS codesquantum Singleton bounddual-containing codesquantum locally recoverable codesfield sizecode length
Authors
Rajendra Prasad Rajpurohit, Maheshanand Bhaintwal
Abstract
Classical $(r,δ)$ locally recoverable codes (LRCs) play a central role in distributed data storage systems as they enable an efficient recovery from erasures by accessing a small number of surviving symbols. Motivated by their prospective use in future quantum data storage and by recent theoretical progress on quantum locally recoverable codes (qLRCs), we investigate the construction of qLRCs from classical cyclic $(r,δ)$-LRCs. Our approach identifies cyclic LRCs whose defining sets satisfy a dual-containing condition, allowing them to serve as valid CSS ingredients. We present three explicit families of $(r,δ)$-qLRCs, two of which are optimal with respect to the quantum Singleton-like bound, whenever the codes are pure, thereby providing optimal examples. Additionally, the codes presented in Constructions 2 and 3 have no bound on their lengths with respect to the field size required to obtain these codes.