{"id":"41bbb8cf-7564-41c4-b282-0193def28833","arxiv_id":"2608.10912","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"CSS quantum locally recoverable codes with intersecting recovery sets are characterized by classical codes with common recovery sets, and explicit binary families with high rates are constructed.","lead":"This paper builds quantum error-correcting codes with local recovery properties by linking them to classical codes with multiple intersecting recovery sets. It constructs explicit binary code families with high rates and derives new bounds on their size and distance.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The exact-case Singleton-like bound (Cor. 4) rests entirely on Proposition 2, whose proof sketch invokes the unstated Lemma 14 of [9]; this is the load-bearing gap.","rationale":"The paper's central positive results are the CSS equivalence in Theorem 1 and the subset-inclusion constructions. These are largely self-contained and check out: Theorem 3's dual-containment criterion is complete because H H^T=0 is equivalent to the row space of H lying in the kernel of H, i.e., C^⊥⊆C; Corollary 2's d(C^⊥)≥2 follows from nonzero columns plus dual-containment; Example 1's exactness and purity are explicitly verified with formulas. The bounds in Corollary 3 are direct translations of cited classical bounds and are not problematic. The one part that cannot be checked from the manuscript is Proposition 2, which is only a proof sketch and depends entirely on an unstated lemma from an external paper. This matches the reader's identified weakest assumption. The concern is real but localized: it threatens the exact-case Singleton-like bound and its comparison table, not the construction or the general CSS characterization. A complete proof of Proposition 2, or a counterexample found by enumeration, would settle the issue. Thus the reader's CONDITIONAL verdict remains appropriate.","tokens_in":11430,"tokens_out":28161,"duration_ms":283744,"concrete_test":"State and prove Lemma 14 of [9] for classical exact (r,t,x)-cLRCs, deriving \\bar N from first principles; then exhaustively enumerate binary linear exact (r,t,x)-cLRCs with n≤12, q=2, r≤4, t≤3, x≤3 and test whether any violates k≤n-(d-1)-\\bar N. A counterexample, or a proof requiring new cross-coordinate intersection assumptions not present in Definition 4, would refute Proposition 2; if the transfer goes through and no counterexample appears, the bound stands.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 2 is the sole support for Corollary 4, the claimed exact-case Singleton-like dimension bound. Its proof is a sketch whose key step is an appeal to Lemma 14 of [9], which is neither stated nor proved here, and no argument is given that it transfers from the quantum setting to classical exact (r,t,x)-cLRCs. The sketch requires an ordering i_1,...,i_T such that for each ℓ≥d some recovery set of i_ℓ avoids all earlier coordinates. Definition 4 controls only intersections among the t recovery sets of a single coordinate; it imposes no bound on how often one coordinate lies in another coordinate's recovery sets. The quantity \\bar N(n,r,d,⌈n p_e⌉) is asserted without derivation, so it is unclear whether it correctly accounts for cross-coordinate overlaps. If Lemma 14 or the formula for \\bar N fails to transfer, Corollary 4 and the Section VI comparison in Table II are unsupported. This does not affect the validity of Theorem 1 or the subset-inclusion construction itself, which are the more central positive results.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies CSS quantum locally recoverable codes with multiple intersecting recovery sets, the (r,t,x)-qLRC setting of Bu, Gu, and Li. Its stated contributions are threefold: (i) an iff characterization, under a dual-minimum-distance condition, of CSS (r,t,x)-qLRCs in terms of classical (r,t,x)-cLRCs with common recovery sets; (ii) an explicit binary subset-inclusion construction of dual-containing (r,t,x)-cLRCs, with a complete binomial condition for dual-containment and an exact (3,2)-family with parameters [[C(m,3), C(m,3)-2m, 4]]; and (iii) bounds for CSS (r,t,x)-qLRCs, including a Singleton-like dimension bound in the exact case and a comparison of the exact family with bounds and with the earlier Bu-Gu-Li construction.","tokens_in":11609,"tokens_out":13949,"duration_ms":128595,"significance":"If the paper's claims are correct, the main positive contribution is substantial: Theorem 1 gives a clean and useful bridge between classical and quantum locality for intersecting recovery sets, and the subset-inclusion construction of Theorems 2-3 is explicit, with a checkable dual-containment congruence and an exact infinite family whose rate approaches 1 at fixed distance 4. I found no fitted parameters and no circular reasoning: the construction is explicit and the bounds are substitutions of cited classical bounds. However, the exact-case Singleton-like bound, which is one of the advertised contributions and supports the Section VI comparison, is currently not actually proved in the manuscript, and Table II contains numerical values that do not follow from the formulas as written. These are load-bearing issues that require a major revision.","major_comments":[{"comment":"The exact-case Singleton-like bound is not proved in this manuscript. The proof of Proposition 2 is a four-step sketch whose key step is an appeal to '[9, Lemma 14]', a lemma that is neither stated nor proved here, and no argument is given that this lemma transfers from the quantum setting of [9] to classical exact (r,t,x)-cLRCs. The quantity \\bar N(n,r,d,⌈n p_e⌉) is also asserted without derivation. Because Corollary 4 and the Section VI comparison depend entirely on Proposition 2, the authors should either state and prove the needed lemma (or give a self-contained derivation of the ordering and counting arguments) and derive \\bar N, or remove the affected claims.","section":"V-B, Proposition 2"},{"comment":"The m=10 row of Table II is inconsistent with the formulas in Corollary 4 as written. With n=120, r=35, x=7, t=3, d=4, the stated p_e is 9/72 - 9/128 + 1/170 = 0.06057, so ⌈n p_e⌉=8; the condition in \\bar N becomes 36N - (N(N-1)/56)·168 ≤ 117, whose largest solution is N=4; Corollary 4 then gives κ ≤ 120 - 2(3) - 2·4 = 106, not the listed 104. I also find discrepancies for m=18,22,26 (the formula gives 792,1512,2566 respectively, while the table lists 790,1508,2564). The table and the formulas must be reconciled before the comparison in Section VI can be used as evidence.","section":"VI, Table II"},{"comment":"Even granting the existence of the unstated Lemma 14, the proof sketch of Proposition 2 requires an ordering i_1,...,i_T such that for every ℓ≥d some recovery set of i_ℓ avoids all earlier coordinates. Definition 4 only controls intersections among the t recovery sets of a single coordinate; it imposes no visible bound on how often one coordinate belongs to another coordinate's recovery sets. The ordering property is therefore nontrivial and must be proved (or derived explicitly from the cited lemma) before the puncturing argument k ≤ n - |U| can be accepted.","section":"V-B, Proposition 2 and Definition 4"}],"minor_comments":[{"comment":"The iff characterization is correctly stated as conditional on d(C⊥_ℓ)≥2, but the paper would benefit from a sentence in the conclusion noting that this assumption excludes CSS codes whose duals have weight-one codewords; in that excluded case Lemma 1 would give non-vanishing local syndromes rather than the classical condition.","section":"III, Theorem 1"},{"comment":"The notation κ~_e^* is used in the table and caption but the tilde is never defined; the caption should explicitly say that κ~_e^* denotes the CSS exact bound from Corollary 4.","section":"VI, Table II"},{"comment":"The sentence 'Since each column of H_{m,s,α} has weight C(s,α)>0, C_{m,s,α} has no weight-one codeword' is correct but would be clearer if it noted explicitly that a weight-one codeword e_i would require the i-th column of H to be zero.","section":"IV-B, Corollary 2"}],"recommendation":"major_revision","confidential_remarks":"The subset-inclusion construction and Theorem 1 appear sound and are the paper's main positive content. The exact-case bound and the associated table are not yet in provable form; if the authors can supply a self-contained proof of Proposition 2 (or remove Corollary 4 and the exact-bound column of Table II), the paper may be acceptable. The reliance on an unstated lemma from an arXiv preprint should be resolved during revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Good news first: this paper has a real result. Theorem 1 — the iff characterization of CSS (r,t,x)-qLRCs via classical (r,t,x)-cLRCs with common recovery sets, under dual minimum distance at least two — is clean and genuinely useful. It makes precise what the CSS construction can and cannot do for intersecting recovery sets. The subset-inclusion construction (Theorems 2 and 3) is also solid: the parameters r, t, x are computed carefully, and the dual-containment condition is a sharp binomial congruence. The resulting exact (3,2)-family with parameters [[C(m,3), C(m,3)-2m, 4]] and rate approaching 1 is explicit, pure, and a clear improvement in rate over the Bu–Gu–Li construction. That part deserves a serious referee.\n\nThe soft spot is exactly where the reader and stress-test put it: the exact-case Singleton-like bound. Corollary 4 rests entirely on Proposition 2, and Proposition 2 is a proof sketch whose key step is an appeal to Lemma 14 of [9], which is neither stated nor proved here. The transfer from the quantum setting to classical exact (r,t,x)-cLRCs is not argued, and the formula for Nbar is asserted without derivation. The concern about cross-coordinate overlaps — Definition 4 controls intersections among the t recovery sets of a single coordinate but imposes no bound on how often one coordinate sits in another coordinate's recovery sets — is real. So as written, Corollary 4 and the Section VI comparison in Table II are unsupported. This does not infect Theorem 1 or the construction, but it should be fixed.\n\nOne more minor thing: in Table II, the m=10 row appears inconsistent with Corollary 4 as written — applying the stated p_e and Nbar formulas gives a different CSS dimension bound than the one listed. The authors should check their arithmetic.\n\nVerdict: worth reviewing, not worth accepting as is. The reviewer should be asked to either complete the proof of Proposition 2 — state and verify the needed lemma — or downgrade Corollary 4 to a conjecture. The rest can be published after numerical cleanup.","headline":"Clean iff characterization and a good subset-inclusion construction for CSS qLRCs; the exact-case Singleton bound is a proof sketch resting on an unstated lemma and should not be relied on as written.","tokens_in":12179,"tokens_out":4533,"would_cite":true,"duration_ms":37958,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94B60","94B05","81P70"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that CSS quantum locally recoverable codes with intersecting recovery sets are exactly classical LRCs with common recovery sets, constructs binary families from subset-inclusion matrices, and derives a Singleton-like…","keywords":["quantum locally recoverable codes","CSS construction","intersecting recovery sets","availability","subset-inclusion matrices","dual-containing codes","Singleton-type bound","binary codes"],"falsifier":"Enumerate all binary exact $(r,t,x)$-cLRCs for small $n$ and check directly whether the ordering required by Lemma 14 exists; exhibit one exact code where no such ordering exists, and the Singleton-like bound in Corollary 4 has no proof. Similarly, an explicit CSS code pair with $d(C^\\perp)\\ge 2$ that is an $(r,t,x)$-qLRC but whose classical codes are not $(r,t,x)$-cLRCs with common recovery sets would refute Theorem 1.","tokens_in":11232,"feed_emoji":"⚛️","tokens_out":6762,"duration_ms":62730,"temperature":0.7,"pith_summary":"This paper tries to establish when a CSS quantum code can correct single-qudit erasures by reading only a few other qudits, when multiple such recovery sets are allowed to overlap. It proves an exact equivalence: provided the two classical codes have dual minimum distance at least two, a CSS code is an $(r,t,x)$-qLRC precisely when the underlying classical codes are $(r,t,x)$-cLRCs sharing the same recovery sets. It then constructs infinite families of binary codes from subset-inclusion matrices whose parity-check matrices are self-orthogonal, so the CSS construction applies, and it derives dimension, rate, and distance bounds for the resulting quantum codes. A special exact subfamily reaches quantum rate tending to 1 with guaranteed distance 4. The significance is that it supplies explicit quantum LRCs with overlapping recovery sets and gives a tool for transferring classical locality results to the quantum setting.","feed_headline":"Inclusion matrices produce quantum LRCs with rate near 1","feed_subtitle":"CSS codes from subset-inclusion parity checks match a dimension bound; exact family reaches distance 4.","key_machinery":"Two mechanisms carry the argument. First, the CSS local-recovery criterion is translated through the assumption $d(C^\\perp)\\ge 2$, which makes the shortened-code condition $\\sigma_{\\{i\\}}(\\pi_{R}(C))=\\{0\\}$ reduce to the same condition for both classical codes; this is why the phrase 'common recovery sets' appears. Second, the subset-inclusion matrix $H_{m,s,\\alpha}$ over $\\mathbb{F}_2$, with rows indexed by $(s-\\alpha)$-subsets of $[m]$ and columns by $s$-subsets, entry 1 iff the row set is contained in the column set, plays the role of parity-check matrix for a binary code $C_{m,s,\\alpha}$. The support of each row is a recovery set, giving locality $r=\\binom{m-s+\\alpha}{\\alpha}-1$, availability $t=\\binom{s}{\\alpha}$, and intersection parameter $x=\\binom{m-s+\\alpha-1}{\\alpha-1}-1$. Dual-containment is characterized by $H_{m,s,\\alpha}H_{m,s,\\alpha}^T=0$, i.e. by binomial congruence conditions, which enables the CSS construction.","core_discovery":"On the paper's own terms, the central discovery is a bridge between the quantum and classical world: the local-recoverability structure of a CSS code with intersecting recovery sets is captured exactly by classical codes, except for a mild nondegeneracy condition. Theorem 1 states that for $C_1,C_2$ with $d(C_1^\\perp),d(C_2^\\perp)\\ge 2$ and $C_1^\\perp\\subseteq C_2$, the code $\\mathrm{CSS}(C_1,C_2)$ is an $(r,t,x)$-qLRC if and only if $C_1$ and $C_2$ are $(r,t,x)$-cLRCs with common recovery sets. The proof reduces the quantum recovery conditions to shortened-code conditions that become empty exactly because the dual distance is at least 2. The paper then exhibits the subset-inclusion codes $C_{m,s,\\alpha}$ as $(r,t,x)$-cLRCs with explicit locality, availability, and intersection parameter, characterizes when they are dual-containing by parity conditions on binomial coefficients, and obtains binary CSS $(r,t,x)$-qLRCs. In the exact case, it proves a Singleton-like bound $\\kappa\\le n-2(d-1)-2\\bar{N}$, and constructs an exact pure family with parameters $[[\\binom{m}{3},\\binom{m}{3}-2m,4]]$ whose rate approaches 1.","pith_inferences":["Beyond the paper's claims, the parity self-orthogonality criterion for incidence matrices likely generalizes to other regular set systems, such as subspace-inclusion matrices over $\\mathbb{F}_q$, giving non-binary CSS $(r,t,x)$-qLRCs; the paper explicitly leaves the non-binary extension as future work.","The common-recovery-set obstruction suggests that optimizing the intersection parameter $x$ rather than minimizing it may be the right quantum regime: $x=0$ is impossible, and the examples show high rate comes with large $x$. One could test whether constructions with the smallest possible $x=1$ necessarily have vanishing rate, refining the tradeoff seen in [9].","Because the distance is fixed at 4 in the exact family while the upper bound grows with $m$, the construction is likely far from optimal for large distance; pushing the inclusion-matrix parameters to get distance scaling would settle whether subset-inclusion codes can approach the distance bound.","The exact-case bound relies on a lemma imported from [9]. A small exhaustive search over exact $(r,t,x)$-cLRCs would show whether the ordering lemma actually holds in all binary cases, which would confirm or falsify whether the bound transfers cleanly."],"forward_implications":["Any classical $(r,t,x)$-cLRC with common recovery sets and dual minimum distance at least 2 immediately yields a CSS $(r,t,x)$-qLRC with the same recovery sets, so classical construction techniques become quantum construction techniques under a checkable condition.","The subset-inclusion matrix family provides infinite families of binary CSS $(r,t,x)$-qLRCs with explicit parameter formulas and, for the parameter choices in Table I, high rates between 0.45 and 0.86 with guaranteed distances up to 16.","The exact $(3,2)$-subfamily is pure with distance 4, dimension $\\binom{m}{3}-2m$, and rate $1-\\frac{12}{(m-1)(m-2)}$, which tends to 1; it offers a rate advantage over the only previously known explicit exact construction while giving up on small intersection parameter.","The CSS-specific Singleton-like bound $\\kappa\\le n-2(d-1)-2\\bar{N}$ matches the general exact bound of [9] for the exact subfamily when $m\\ge 14$.","Because the exact subfamily is pure, its quantum minimum distance equals the classical distance rather than merely being lower-bounded by it, so the distance bound from Corollary 3 applies directly to the quantum code."],"supporting_citations":[{"why":"Supplies the CSS local erasure-recovery criterion used in Lemma 1 to reduce quantum recovery to shortened-code conditions.","marker":"[4]"},{"why":"Introduced $(r,t,x)$-qLRCs and exact $(r,t,x)$-qLRCs, and provides Lemma 14 on which Proposition 2's exact-case Singleton-like bound rests.","marker":"[9]"},{"why":"Defined $(r,t,x)$-cLRCs and contributes the dimension bound $k\\le n(1-p(r,t,x))$ that is translated to CSS codes in Corollary 3.","marker":"[11]"},{"why":"Provides the alphabet-dependent distance bound for $(r,t,x)$-cLRCs used in Corollary 3.","marker":"[12]"},{"why":"Gives the Wang-Zhang-Liu construction, which is the $\\alpha=1$ special case of the subset-inclusion family and links the construction to known binary $(r,t,0)$-cLRCs.","marker":"[15]"},{"why":"Provides the CSS construction itself, the mechanism converting dual-containing classical codes into quantum codes.","marker":"[21]"},{"why":"Wilson's diagonal form result gives the dimension formula for the subset-inclusion codes used in Theorem 2.","marker":"[23]"},{"why":"Supplies distance bounds for codes from subset-inclusion matrices used in Theorem 2 and Corollary 2.","marker":"[24]"}],"fun_headline_variants":["Quantum LRCs equal classical LRCs under dual distance condition","Subset-inclusion codes build CSS qLRCs with rate near 1","Exact CSS qLRC family reaches distance 4, rate near 1","Singleton-like bound for CSS qLRCs with intersecting recovery sets"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the imported Lemma 14 of [9], which guarantees that in every exact $(r,t,x)$-cLRC the coordinates can be ordered so that each coordinate's chosen recovery set avoids all previously erased coordinates; Proposition 2 is only sketched and would collapse if that lemma does not transfer to classical exact codes.","fun_headline_variants_meta":{"raw":{"variants":["Quantum LRCs equal classical LRCs under dual distance condition","Subset-inclusion codes build CSS qLRCs with rate near 1","Exact CSS qLRC family reaches distance 4, rate near 1","Singleton-like bound for CSS qLRCs with intersecting recovery sets"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000888,"raw_usage":{"total_tokens":3870,"prompt_tokens":1021,"completion_tokens":2849,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":637,"completion_tokens_details":{"reasoning_tokens":2768}},"tokens_in":637,"tokens_out":2849,"duration_ms":20828,"temperature":1.0,"reasoning_tokens":2768,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:37:09.793012+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Enumerate all binary exact $(r,t,x)$-cLRCs for small $n$ and check directly whether the ordering required by Lemma 14 exists; exhibit one exact code where no such ordering exists, and the Singleton-like bound in Corollary 4 has no proof. Similarly, an explicit CSS code pair with $d(C^\\perp)\\ge 2$ that is an $(r,t,x)$-qLRC but whose classical codes are not $(r,t,x)$-cLRCs with common recovery sets would refute Theorem 1.","supporting_citations":[{"cited_title":"On one general- ization of lrc codes with availability,","cited_arxiv_id":null,"evidence_quote":"Defined $(r,t,x)$-cLRCs and contributes the dimension bound $k\\le n(1-p(r,t,x))$ that is translated to CSS codes in Corollary 3."},{"cited_title":"On distance properties of(r, t, x)-lrc codes,","cited_arxiv_id":null,"evidence_quote":"Provides the alphabet-dependent distance bound for $(r,t,x)$-cLRCs used in Corollary 3."},{"cited_title":"Achieving arbitrary locality and availability in binary codes,","cited_arxiv_id":null,"evidence_quote":"Gives the Wang-Zhang-Liu construction, which is the $\\alpha=1$ special case of the subset-inclusion family and links the construction to known binary $(r,t,0)$-cLRCs."},{"cited_title":"Quantum error correction via codes over gf (4),","cited_arxiv_id":null,"evidence_quote":"Provides the CSS construction itself, the mechanism converting dual-containing classical codes into quantum codes."},{"cited_title":"A diagonal form for the incidence matrices of t-subsets vs. k-subsets,","cited_arxiv_id":null,"evidence_quote":"Wilson's diagonal form result gives the dimension formula for the subset-inclusion codes used in Theorem 2."},{"cited_title":"Binary codes from subset inclusion matrices,","cited_arxiv_id":null,"evidence_quote":"Supplies distance bounds for codes from subset-inclusion matrices used in Theorem 2 and Corollary 2."}],"review_version":1}