{"id":"f00aafd7-bda2-44ce-ba15-da7f9aa0c21e","arxiv_id":"2411.19101","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"New error-erasure decoders for interleaved linearized Reed-Solomon codes correct full errors, row erasures, and column erasures up to a probabilistic radius of s/(s+1)(n-k) with O(s n^2) average complexity.","lead":"The paper presents new syndrome-based decoders for vertically and horizontally interleaved linearized Reed-Solomon codes, handling full errors and row/column erasures together. These codes matter for post-quantum cryptography and network coding, and the new decoders offer faster error-erasure correction than previous approaches.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The error-erasure decoder's failure-probability bound is explicitly deferred in Theorems 4 and 8; the central probabilistic claim for mixed errors rests on an unproved reduction to error-only syndromes.","rationale":"The reader's weakest_assumption matches my independent reading. The error-only decoders (Theorems 2 and 6, Lemmas 2 and 4) are derived in full: the key equation is proved, the coefficient matrix is factored into ·X̂, and the rank-distribution bound follows from the referenced result in [35]. The error-erasure theorems, by contrast, contain an explicit 'we will present the details in follow-up work' location in the proof of the failure-probability statement. This is not a matter of external consensus; it is an internally unsupported step in the main argument. The paper itself flags it, which strengthens the concern rather than weakening it. The empirical section does not cover the error-erasure setting, so the claimed tightness of the error-erasure bounds is untested. I therefore do not see a reason to move the reader's CONDITIONAL verdict: the contribution is real but the central error-erasure probabilistic guarantees are not yet proven. My concrete test targets exactly the missing factorization; if an independent derivation succeeds, the condition can be lifted to ACCEPT. If it fails, the probabilistic radius and guaranteed radius for mixed errors would need revision. The verdict is unchanged from the reader's assessment because the same gap was identified and appropriately conditionalized.","tokens_in":50606,"tokens_out":2287,"duration_ms":24267,"concrete_test":"Re-derive the coefficient matrix of the VILRS error-erasure key equation (32) for a concrete minimal instance, e.g. s=1, t_F=1, t_R=1, t_C=1, n−k=4. Write the matrix entries θ^{-(ν−1)}(s_CR,j,l−ν+1) explicitly in terms of a_F,j, x_F, x_R, x_C, and the known matrices A_R and B_C, and check whether the stacked matrix factorizes as (a matrix depending only on the full-error values) times a full-rank t_F×t_F matrix that is independent of the unknown error locators, in the same way S = ·X̂ in Lemma 2. If the factorization does not hold or requires additional assumptions on t_R,j and t_C, the failure-probability bound from Lemma 2 does not carry over to the erasure case.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's headline error-erasure claims are conditional on an assertion that the proofs of Theorems 4 and 8 explicitly defer. In Theorem 4's proof, the authors state that the coefficient vectors of the auxiliary component-syndrome polynomials s_CR,j 'can be interpreted as modified component syndromes corresponding to an error-only transmission,' but add that 'this is a nontrivial endeavor and we will present the details in follow-up work.' The same deferral appears verbatim in the proof of Theorem 8. This reduction is the sole justification for: (i) the probabilistic failure bound Pr_fail ≤ κ_q^{ℓ+1} q^{-m((s+1)(τ_max−τ*)+1)} in the error-erasure case, and (ii) the guaranteed radius t_F ≤ (1/2)(n−k−max_j t_{R,j}−t_C) for VILRS (and the analogous HILRS radius). Without this reduction, neither the probabilistic decoding radius nor the guaranteed error-erasure radius is derived from Lemma 2 or Lemma 4; the key equation (30)/(55) involves s_CR,j, not the original component syndromes, and the rank-distribution argument in Lemmas 2/4 does not obviously apply to the modified system. The simulations in Section V only test the error-only bound, not the error-erasure bound, so they provide no independent support for the deferred claim. This is a genuine gap in the argument for the paper's central contribution, not a stylistic issue.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies vertically and horizontally interleaved linearized Reed–Solomon (VILRS and HILRS) codes in the sum-rank metric. It presents syndrome-based error-only decoders for both families, with a guaranteed decoding radius of (n−k)/2 and a probabilistic radius of s(n−k)/(s+1), together with explicit failure-probability bounds. It then formulates error-erasure decoders that jointly handle full errors, row erasures, and column erasures, claiming a guaranteed radius t_F ≤ (n−k−t_R−t_C)/2 and a probabilistic radius t_F ≤ s/(s+1)(n−k−t_R−t_C), with the exact radius depending on the interleaving direction. The paper further gives average-case complexity estimates and Monte Carlo simulations for the error-only failure probability.","tokens_in":50897,"tokens_out":6380,"duration_ms":64523,"significance":"If the main claims are established, the paper provides a useful unified treatment of vertical and horizontal interleaving for LRS codes and extends syndrome-based decoding to mixed error-erasure channels. The error-only decoders are derived in detail: Theorem 1, Theorem 5, Lemma 2, and Lemma 4 contain full proofs of the key equations and of the rank-based failure-probability bounds, and the simulations in Section V give quantitative evidence for the tightness of those bounds. The advertised central novelty, however, is the error-erasure generalization, and that part currently rests on an explicitly deferred reduction. The paper would be a solid contribution once that missing argument is supplied.","major_comments":[{"comment":"","section":"Section III-D, proof of Theorem 4"},{"comment":"","section":"Section IV-D, proof of Theorem 8"},{"comment":"","section":"Section V, Table I"}],"minor_comments":[{"comment":"","section":"Algorithm 4, line 20"},{"comment":"","section":"Section V, Eq. (63)"},{"comment":"","section":"Sections III-D and IV-D, Eqs. (33) and (58)"},{"comment":"","section":"Abstract and Conclusion"}],"recommendation":"major_revision","confidential_remarks":"The main advertised contribution—the error-erasure decoders and their probabilistic decoding radii—is stated in Theorems 4 and 8 but its proof is explicitly deferred to follow-up work. This is not a minor presentation issue; it is the central technical claim of the paper. I would ask the editor to require the missing reduction to be included in the manuscript before acceptance. The error-only portions and the unificational presentation are promising, and the gap appears fillable, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Worth your time if you work in sum-rank-metric decoding. The paper gives the first syndrome-based and first error-erasure decoders for vertically and horizontally interleaved linearized Reed–Solomon codes, with a genuinely unified treatment of the two interleaving types. The error-only decoders are derived in full: key equations, rank arguments, failure-probability bounds, and complexity claims all check out. The ELP/ESP duality between vertical and horizontal interleaving is clearly explained and is a nice expository contribution. The fast subroutines section is a real asset.\n\nThe soft spot is exactly where the reader and stress-test put it. In the proofs of Theorems 4 and 8, the authors need to show that the auxiliary syndrome polynomials s_CR,j (resp. s_RC,j) behave like component syndromes of an error-only instance. They state this can be shown by suitably generalizing [12], but they defer the details to follow-up work. The guaranteed radius t_F <= 1/2(n-k-max_j t_R,j - t_C) and the probabilistic failure bound for the error-erasure case both ride on that reduction. Without it, Lemma 2/4 do not apply to the modified key equation, and the simulations in Section V test only the error-only bound. This is a real gap, not a style complaint. It is also clearly addressable: the pattern from [12] likely goes through, and the authors seem to know the route. But as written, the paper's central claims are conditional.\n\nThe self-citations are not a problem: [35] and [19] are prior published results with independent derivations, and the reliance is legitimate. The paper is honest about the deferral, which I respect, but honesty does not make the theorem proven.\n\nWho is this for: coding theorists working on sum-rank metrics, network coding, and code-based cryptography. A serious referee should engage. My recommendation: send it to review, but the referee should insist on either a full proof of the reduction or a revision that states the error-erasure radius and failure claims as conditional or conjectural and adjusts the abstract accordingly. The error-only results are solid enough to stand on their own.","headline":"Solid syndrome-based decoders for VILRS and HILRS, but the headline error-erasure guarantees rest on an explicitly deferred reduction, so treat those radius and failure claims as conditional.","tokens_in":51442,"tokens_out":1640,"would_cite":true,"duration_ms":17907,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94B35","94B27","11T71"],"pacs":[],"model":"deepseek-v4-flash","headline":"Syndrome-based decoding of vertically and horizontally interleaved linearized Reed–Solomon codes can correct full errors, row erasures, and column erasures together, with guaranteed and probabilistic unique-decoding radii and quantified…","keywords":["linearized Reed–Solomon codes","interleaved codes","sum-rank metric","error-erasure decoding","syndrome-based decoding","skew polynomials","vertical interleaving","horizontal interleaving"],"falsifier":"For parameters such as $q=3$, $m=4$, $\\ell=2$, $n=(4,4)$, $k=3$, $s=4$, run the VILRS error-erasure decoder with $t_{\\mathcal{F}}=4$ plus row and column erasures chosen so that $\\tau^*_{\\mathrm{vert}}=\\tau_{\\max}=4$; if the observed failure rate exceeds the predicted bound $\\kappa_q^{\\ell+1}q^{-m((s+1)(\\tau_{\\max}-\\tau^*)+1)}$ by more than the factor of about 3–5 seen in the paper's error-only simulations, the unproven reduction in Theorem 4 does not hold.","tokens_in":50400,"feed_emoji":"📡","tokens_out":13192,"duration_ms":101383,"temperature":0.7,"pith_summary":"Vertically and horizontally interleaved linearized Reed–Solomon (LRS) codes can each be decoded by a syndrome-based algorithm that handles full errors, row erasures, and column erasures in one joint key equation instead of treating erasures as a separate pre-processing step. The paper claims guaranteed unique decoding whenever the number of full errors satisfies $t_{\\mathcal{F}}\\le \\tfrac{1}{2}(n-k-t_{\\mathcal{R}}-t_{\\mathcal{C}})$ (up to the exact per-component terms), and probabilistic unique decoding up to $t_{\\mathcal{F}}\\le \\tfrac{s}{s+1}(n-k-t_{\\mathcal{R}}-t_{\\mathcal{C}})$ with an explicit failure-probability bound. These are claimed to be the first syndrome-based and the first error-erasure decoders for both interleaved code families, and they unify the vertical and horizontal settings through a dual error-locator / error-span polynomial framework. If the claims hold, erasure side information translates directly into decoding power, and the interleaving gain $s/(s+1)$ known from rank-metric codes now covers joint error-erasure scenarios in the sum-rank metric.","feed_headline":"Decoders correct interleaved LRS codes with row and column erasures","feed_subtitle":"Guaranteed decoding when full errors stay within half the leftover redundancy; probabilistic decoding goes further.","key_machinery":"The machinery lives in the skew-polynomial ring $\\mathbb{F}_{q^m}[x;\\theta^{-1}]$ with generalized operator evaluation, and in two dual key equations. For VILRS codes the central object is the error-locator polynomial (ELP), the minimal skew polynomial vanishing on the error locators; the decoder sets up the ELP key equation $\\lambda_{\\mathcal{F}}\\cdot s_{\\mathcal{CR},j}\\equiv \\psi_j \\pmod{x^{n-k}}$, where $s_{\\mathcal{CR},j}$ is an auxiliary component-syndrome polynomial built from the received syndrome together with the known erasure information. For HILRS codes the central object is the error-span polynomial (ESP), with the dual key equation $\\sigma_{\\mathcal{F}}\\cdot s_{\\mathcal{RC},j}\\equiv \\omega_j \\pmod{x^{n-k}}$; the decoder first recovers error values instead of error locations. Partial ELPs and ESPs encode the known column spaces of row erasures and the known row spaces of column erasures, and these partial polynomials are multiplied into the auxiliary syndromes. The key equations are solved by multisequence skew-feedback shift-register synthesis; roots of the recovered skew polynomial are found with a Skachek–Roth-like algorithm; and the remaining linear systems, which have generalized Moore matrices, are solved with a Gabidulin-like algorithm.","core_discovery":"The paper's central claim is that a single syndrome-based decoder, built from a Berlekamp–Massey-like key equation, recovers the transmitted codeword of a VILRS or HILRS code from an additive sum-rank error of weight $\\tau=t_{\\mathcal{F}}+t_{\\mathcal{R}}+t_{\\mathcal{C}}$, decomposed into full errors, row erasures, and column erasures. For vertical interleaving the key object is the error-locator polynomial (ELP), and the decoder first recovers error locations; for horizontal interleaving it is the error-span polynomial (ESP), and the decoder first recovers error values. Erasure knowledge is incorporated directly: known column spaces of row erasures produce partial component ESPs, and known row spaces of column erasures produce partial ELPs, which are multiplied into auxiliary component-syndrome polynomials in the key equation. The paper establishes guaranteed unique decoding for $t_{\\mathcal{F}}\\le \\tfrac{1}{2}(n-k-\\max_j t_{\\mathcal{R},j}-t_{\\mathcal{C}})$ in the vertical case and $t_{\\mathcal{F}}\\le \\tfrac{1}{2}(n-k-t_{\\mathcal{R}}-\\max_j t_{\\mathcal{C},j})$ in the horizontal case, and states probabilistic unique decoding for larger weights up to $t_{\\mathcal{F}}\\le \\tfrac{s}{s+1}(n-k-\\bar{t}_{\\mathcal{R}}-t_{\\mathcal{C}})$ (with the roles of $\\mathcal{R}$ and $\\mathcal{C}$ swapped for horizontal interleaving). The failure probability is bounded by $\\kappa_q^{\\ell+1} q^{-m((s+1)(\\tau_{\\max}-\\tau^*)+1)}$, where $\\tau^*$ is the effective error weight and $\\tau_{\\max}=\\tfrac{s}{s+1}(n-k)$; Monte Carlo simulations for error-only decoding confirm the bound is tight within a constant factor.","pith_inferences":["A completed proof of the deferred reduction would close the gap in Theorems 4 and 8 and would give the same rigorous failure bound for non-interleaved LRS error-erasure decoding, where the bound is currently inherited from the rank-metric setting.","The joint treatment of erasures is directly relevant to code-based cryptography: side information about an error's row or column space is exactly what these decoders convert into a larger decoding radius, so security analyses of sum-rank cryptosystems should account for this erasure-aware capability.","A natural testable extension is heterogeneous interleaving, which the authors name as future work; the same ELP/ESP key-equation architecture appears flexible enough to allow different component codes per row or column without reworking the syndrome algebra.","Lifted versions of VILRS and HILRS codes could inherit the error-erasure capability for multishot network coding, where lost packets play the role of erasures and corrupted packets play the role of full errors."],"forward_implications":["VILRS and HILRS codes become the first interleaved sum-rank code families with syndrome-based error-erasure decoding, extending the known error-only decoders.","The complexity bounds $O(sn^2)$ for error-only and $\\widetilde{O}(sn^2)$ for error-erasure decoding mean the erasure-aware key equations cost no additional asymptotic order in most parameter regimes.","Each row or column erasure reduces the guaranteed full-error radius by one unit, so the decoding radius is $n-k$ minus the erasure budget, halved for full errors.","The probabilistic radius extends the decodable full-error weight by the interleaving factor $s/(s+1)$, matching the gain known from interleaved Gabidulin codes.","The unified ELP/ESP presentation exposes a systematic duality—shared row space for vertical interleaving, shared column space for horizontal—that should make future decoding results transferable between the two settings."],"supporting_citations":[{"why":"supplies the VILRS code model and the Lemma 7 rank bound used for every failure-probability estimate.","marker":"[35]"},{"why":"gives the ELP/ESP error-erasure key equations for non-interleaved LRS codes that the interleaved decoders generalize.","marker":"[19]"},{"why":"provides multisequence skew-feedback shift-register synthesis, the subroutine that solves the key equations in O(s(n−k)^2).","marker":"[28]"},{"why":"is the Gabidulin error-erasure decoder whose modified-syndrome reduction the proofs of Theorems 4 and 8 invoke to justify the failure bounds.","marker":"[12]"},{"why":"defines linearized Reed–Solomon codes and the skew-polynomial evaluation framework underlying the whole construction.","marker":"[1]"}],"fun_headline_variants":["Unified decoder for interleaved LRS codes with erasures","Interleaved LRS codes decoded with row and column erasures","Syndrome decoder handles full errors plus row and column erasures","New decoder for interleaved LRS codes: errors and erasures jointly","Error-erasure decoding for interleaved linearized RS codes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The probabilistic radii and failure bounds for the error-erasure decoders rest on the assertion, made in the proofs of Theorems 4 and 8, that the auxiliary syndrome polynomials can be rewritten as ordinary error-only syndromes; that rewriting is stated without proof and deferred to follow-up work.","fun_headline_variants_meta":{"raw":{"variants":["Unified decoder for interleaved LRS codes with erasures","Interleaved LRS codes decoded with row and column erasures","Syndrome decoder handles full errors plus row and column erasures","New decoder for interleaved LRS codes: errors and erasures jointly","Error-erasure decoding for interleaved linearized RS codes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000184,"raw_usage":{"total_tokens":1527,"prompt_tokens":1362,"completion_tokens":165,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":978,"completion_tokens_details":{"reasoning_tokens":74}},"tokens_in":978,"tokens_out":165,"duration_ms":2711,"temperature":1.0,"reasoning_tokens":74,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T10:31:50.532159+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For parameters such as $q=3$, $m=4$, $\\ell=2$, $n=(4,4)$, $k=3$, $s=4$, run the VILRS error-erasure decoder with $t_{\\mathcal{F}}=4$ plus row and column erasures chosen so that $\\tau^*_{\\mathrm{vert}}=\\tau_{\\max}=4$; if the observed failure rate exceeds the predicted bound $\\kappa_q^{\\ell+1}q^{-m((s+1)(\\tau_{\\max}-\\tau^*)+1)}$ by more than the factor of about 3–5 seen in the paper's error-only simulations, the unproven reduction in Theorem 4 does not hold.","supporting_citations":[{"cited_title":"Error-Erasur e Decoding of Linearized Reed–Solomon Codes in the Sum-Rank Metric,","cited_arxiv_id":null,"evidence_quote":"gives the ELP/ESP error-erasure key equations for non-interleaved LRS codes that the interleaved decoders generalize."},{"cited_title":"Skew-Feedba ck Shift-Register Synthesis and Decoding Interleaved Gabi dulin Codes,","cited_arxiv_id":null,"evidence_quote":"provides multisequence skew-feedback shift-register synthesis, the subroutine that solves the key equations in O(s(n−k)^2)."},{"cited_title":"Error and Erasure C orrecting Algorithms for Rank Codes,","cited_arxiv_id":null,"evidence_quote":"is the Gabidulin error-erasure decoder whose modified-syndrome reduction the proofs of Theorems 4 and 8 invoke to justify the failure bounds."},{"cited_title":"Skew and Linearized Reed–Solomo n Codes and Maximum Sum Rank Distance Codes over any Division Ring,","cited_arxiv_id":null,"evidence_quote":"defines linearized Reed–Solomon codes and the skew-polynomial evaluation framework underlying the whole construction."}],"review_version":1}