Bounds for General $(r,δ)$-Quantum Locally Recoverable Codes

Abstract

We study general $(r,δ)$-quantum locally recoverable codes (qLRCs) without assuming purity or a stabilizer structure, and with arbitrarily overlapping recovery blocks. Using a selection lemma for multiple-erasure recovery, we establish a Singleton-like upper bound for general quantum subspaces. When $δ=2$, this bound reduces to the general single-erasure bound of Golowich and Guruswami. An integer optimization yields an explicit remainder correction that tightens the bound by up to $δ-2$. For pure codes, local decoupling and entropy arguments extend the parameter bound of Galindo et al., originally obtained for pure stabilizer codes constructed from Euclidean or Hermitian dual-containing classical codes, to arbitrary pure quantum subspaces. We also remove the partition assumption from several existing pure-code bounds for disjoint recovery blocks and obtain a further remainder correction by optimizing the budget. Finally, we analyze three families attaining these bounds: pure codes with overlapping recovery blocks and $d=δ$, which exhibit both a strict improvement of the pure-code bound and simultaneous equality in the pure-code and general bounds; impure codes with disjoint recovery blocks attaining the general bound; and an existing family of pure codes with overlapping recovery blocks and $d>δ$, which simultaneously attains all these bounds.

The paper studies dimension–distance bounds for $(r,\delta)$ quantum locally recoverable codes when recovery blocks may overlap, without assuming purity or stabilizer structure. Each coordinate belongs to a block of size at most $r+\delta-1$ that locally corrects any $\delta-1$ erasures. Writing $t=\delta-1$, $b=r+t$, $D=d-1$, $L=n-2D$, and $A=t\lfloor(n-D)/b\rfloor$, the general bound is $k\le L-A-t\lfloor(L-A)/b\rfloor$, where $k=\log_q K$. Integer optimization improves this by at most $\delta-2$. At $\delta=2$, it recovers the Golowich–Guruswami single-erasure bound. The proof uses a two-stage selection lemma accommodating overlapping blocks, followed by construction of two disjoint correctable erasure sets and an entropy bound. For pure subspace codes, local decoupling gives $k\le n-2D-2tM-2\max\{0,v-r\}$, where $n-D=bM+v$ and $0\le v<b$. This extends existing construction-specific and disjoint-block parameter bounds to arbitrary pure subspaces. Equality conditions constrain block intersections and force local MDS structure. For $\mathbb F_q$-linear stabilizer codes at minimal locality $r=\delta$, removing fixed pure-state coordinate factors forces disjoint $\llbracket2\delta-1,1,\delta\rrbracket_q$ inner blocks concatenated with an outer stabilizer code. Consequently, $\delta\mid d$ and $k\le\lfloor n/(2\delta-1)\rfloor-2d/\delta+2$; suitable Reed–Solomon constructions attain this bound. Attaining examples include pure CSS codes with unavoidable overlaps, impure concatenated codes, and an existing overlapping family with $d>\delta$. The decomposition requires field-linear stabilizers. The full attainable parameter range, alphabet-dependent bounds without partitions, and structural extensions beyond that class remain open.