Quantum locally recoverable code (QLRC)[1]

Protection

A Singleton-like QLRC bound states that an \(((n,K,d))_q\) QLRC of locality \(r\) and rate \(R = \frac{\log_q K}{n}\) must have relative distance [1; Thm. 35] \begin{align} \delta = \frac{d}{n} \leq \frac{1-R}{2} - \Omega\left(\frac{1}{r}\right)~, \tag*{(1)}\end{align} implying that locality restricts the distance of the code. Random QLRCs with qudit dimension \(q = 2^{O(r)}\) achieve a relative distance that is order \(O(1/r)\) below the bound [1; Prop. 5]. Codes constructed with the help of AEL distance amplification [2,3] admit a gap of order \(O(1/r^{1/4})\) [1; Prop. 6]. Folded quantum Tamo-Barg codes yield explicit QLRCs of arbitrary prime locality \(r\), rate at least \(R\), relative distance \(\delta \geq (1-R)/2 - O(1/\sqrt{r})\), and qudit dimension \(q = n^{O(r^2)}\) [1; Cor. 64].

QLRCs have been extended to codes with intersecting recovery sets, and a Singleton-like bound has been derived for such codes [4].