{"id":"bc37aff2-e38a-4ec1-b115-8e4c469f6e56","arxiv_id":"2507.07728","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For sufficiently large minimum distances, the minimal length of linear b-symbol codes equals the Griesmer-type bound, and exact values are found for binary pair-symbol codes of dimensions 3, 4, and 5.","lead":"This paper proves that for fixed field size, dimension, and block size b, the shortest linear code in the b-symbol metric attains the Griesmer-type lower bound on length once the minimum distance is large. It also determines exact optimal lengths for binary pair-symbol codes of dimensions up to 5, with explicit generator matrices.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.16 as stated is false because it omits the k≥b hypothesis required by its own proof; the k=1, b=2 case contradicts Theorem 4.1 and the claimed formula for all large d.","rationale":"The reader's weakest_assumption explicitly included the missing k≥b condition, and my stress test confirms this is not a mere presentational gap: Theorem 3.16 as literally stated is false for k<b, and the counterexample persists for all large d. The intended k≥b version may well be correct, but the submitted theorem statement cannot stand without repair. I also note the proof's reliance on the unverified [15, Theorem 4] as a black box, which further supports conditional rather than unconditional acceptance. Since the reader already assigned a conditional verdict based on this same defect, my read does not move the verdict; I recommend keeping the paper conditional pending the statement correction and an independent check of the imported additive-code theorem.","tokens_in":24762,"tokens_out":10626,"duration_ms":118834,"concrete_test":"Analytically verify the k=1, b=2 case for a large distance: take q=2 and d=100. The [100,1]_2 code spanned by (1,...,1) has its only nonzero codeword with all 100 pair-symbol windows equal to (1,1), so its pair-symbol distance is 100 and no shorter code can exist since distance cannot exceed length, giving n_2^2(1,100)=100. Theorem 3.16's formula instead gives ceil(g_2(1,200)/3)=ceil(200/3)=67, a direct contradiction at a large d. This settles that the theorem must be restricted to k≥b, or the small-k cases handled separately.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Theorem 3.16 is stated for arbitrary parameters k, q, b, but the proof is only valid for k≥b and the statement is false without that hypothesis. The proof uses k≥b in Lemma 2.6 (association of an [n,k,d]^b_q code to a projective b-(n,k,n-d)_q system), Lemma 3.5, Proposition 3.2, and Lemma 3.14, and the final periodicity argument depends on q^{k-b}[b]_q. For k=1, b=2, a [n,1]_q code generated by a single nonzero all-ones row has every nonzero codeword (λ,...,λ) with λ≠0, and every 2-window is (λ,λ)≠0, so wt_2=n and hence n_2^q(1,d)=d for every d. The formula in Theorem 3.16 gives ceil(g_q(1,q^{b-1}d)/[b]_q)=ceil(q^{b-1}d/[b]_q), which for q=b=2 is ceil(2d/3), strictly smaller than d for all d>2. This contradicts Theorem 4.1 and, crucially, the failure persists for arbitrarily large d, so the 'sufficiently large' quantifier does not remove the problem. The central claim therefore needs an explicit k≥b hypothesis or a separate treatment of the degenerate cases; without that correction the theorem, as written, is unsound.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the minimum length n_q^b(k,d) of linear codes over F_q of dimension k and minimum b-symbol distance d. The main result, Theorem 3.16, claims that for all sufficiently large d, n_q^b(k,d) equals the ceiling of the Griesmer-type bound g_q(k,q^{b-1}d)/[b]_q, i.e. the Griesmer bound is attained. The proof uses a geometric interpretation of b-symbol codes as projective b-(n,k,s)_q systems, a construction of 'b-chains' to realize such systems as actual codes, and an existence theorem for additive codes from a companion preprint. In Section 4 the paper determines the exact values of n_2^2(k,d) for k=3,4,5 for all d, using the Griesmer bound, explicit generator matrices, and two non-existence lemmas.","tokens_in":25062,"tokens_out":18691,"duration_ms":177241,"significance":"If Theorem 3.16 were correct for all parameters, it would reduce the determination of n_q^b(k,·) to finitely many small-distance values, a qualitative analogue of the Solomon-Stiffler result for the Hamming metric. The exact tables for n_2^2(k,·), k≤5, are concrete and likely useful for storage-channel code designers. The paper is clearly written and the geometric framework is elegant. However, the main theorem as stated is false, and the construction relies on an unreviewed preprint; these issues must be fixed before the paper can be accepted.","major_comments":[{"comment":"The statement is false without the assumption k≥b. For k=1, b=2, q=2, Theorem 4.1 gives n_2^2(1,d)=d for all d≥2, while the Griesmer value is ⌈g_2(1,2d)/3⌉ = ⌈2d/3⌉, which is strictly smaller than d for every d>2. The proof uses k≥b in Lemma 2.6, Proposition 3.2, Lemma 3.5, and the parameter q^{k-b}. Therefore the theorem should be restated with the explicit hypothesis k≥b≥2, and the abstract and conclusion should be amended accordingly.","section":"Theorem 3.16"},{"comment":"The proof imports [15, Theorem 4] as a black-box existence result, and [15] is an unreviewed arXiv preprint by the same author. This theorem supplies, for every residue class d′ mod q^{k-b}[b]_q, a faithful projective b-system attaining the Griesmer bound. Since this is the main existence input for the asymptotic claim, the paper must provide a precise statement and a proof of this result, or replace it by a peer-reviewed reference (e.g. [1] or [16] if they contain the needed theorem). Without this, the central claim cannot be checked by the reader.","section":"Proof of Theorem 3.16"},{"comment":"The gluing step is only sketched. The proof asserts that the nλ b-spaces of Sλ can be interpreted as b-chains of length 1 and linked via Lemma 3.14 into a single b-chain with the same start and end, and that the resulting projective system has the exact hyperplane count (nλ−dλ)+λ′[k−b]_q. This step is load-bearing: it must explain how ordered bases of the b-spaces are chosen, how the linking chains are arranged, how the final chain is closed, and why the computed minimum distance is exactly d′+(λ+λ′)q^{k-b}[b]_q. The current text is too compressed for a proof of the main theorem.","section":"Proof of Theorem 3.16"}],"minor_comments":[{"comment":"The phrase 'he existence of a constant' should read 'the existence of a constant'; also, the parameter h in the cited [15, Theorem 4] should be explicitly identified with b.","section":"Proof of Theorem 3.16"},{"comment":"In the inductive step, the condition 'v′i /∈ ⟨v′0,...,v′i, ui+1,...,ub−2⟩' contains v′i itself and is always false; the intended condition is presumably about v′_{i+1} or v′0. Please correct.","section":"Lemma 3.14"},{"comment":"In the table header, 'g2(8, 8t+2i)' should be 'g2(4, 8t+2i)'.","section":"Proof of Theorem 4.4"},{"comment":"The phrase 'interprete' should be 'interpret', and the list 'v4, v16, v16, v1,,' contains an extra comma and a possible duplicate entry.","section":"Example 3.15"},{"comment":"The computational claims involving LinCode ('we have enumerated all 10358 even [24,5,10]_2 codes') should be accompanied by a reference to the software or a link to the data files to support reproducibility.","section":"Section 4"},{"comment":"In the proof, the phrase 'a generator matrix U of an an [n,k,≥d]^b_q code' contains a duplicated 'an'.","section":"Lemma 3.5"}],"recommendation":"major_revision","confidential_remarks":"The central result depends essentially on the author's own unpublished preprint [15], and the main theorem as stated has a clear counterexample for k<b. I would urge the editor to require a corrected statement with the k≥b hypothesis and a complete proof of the imported existence result. With those changes, the paper is a solid contribution, particularly the exact determination of n_2^2(k,·) for k≤5."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe headline is that Theorem 3.16, as written, is not true. It claims n_q^b(k,d) equals ceil(g_q(k,q^{b-1}d)/[b]_q) for all sufficiently large d, for any parameters k,q,b. But the proof only works for k>=b, and without that hypothesis the statement fails. The brute-force example: for k=1, b=2, q=2, a [n,1] code with all-ones codewords has 2-symbol distance n, so n_2^2(1,d)=d, while the formula gives ceil(2d/3), which is smaller for d>2. The 'sufficiently large' quantifier does not save it. The proof also uses q^{k-b} and b-dimensional subspaces, so the k>=b condition is genuinely needed.\n\nWhat is actually new and good: the idea of using projective b-(n,k,s) systems and b-chains to attain the Griesmer-type bound is a real step, and the exact determination of n_2^2(k,d) for k=3,4,5 looks valuable. The generator matrices are explicit, and the exceptional non-existence proofs (Lemma 4.8 and 4.9) are theoretical, not just computer claims. The paper is honest about the limits of the geometric correspondence (Example 2.7) and about the fact that the ILP searches are not shipped.\n\nThe soft spots, in proportion: the main theorem needs an explicit k>=b hypothesis or a separate treatment of k<b. The proof leans on [15, Theorem 4], an unpublished arXiv preprint by the same author, as the existence engine. That is not disqualifying, but it means the main result is only as reliable as an unverified black box. The chain-gluing lemmas (3.13, 3.14) are plausible but not fully detailed; a referee would need to check them. The exact values for n_2^2(5,d) rely on a long list of generator matrices; some non-existence claims are backed by enumeration that is not shipped, though the theoretical arguments cover the worst cases.\n\nWho is this for? People working on symbol-pair codes and short-block storage codes. The small-k tables are genuinely useful. The asymptotic result, once corrected, would complete the b-symbol analogue of Solomon-Stiffler.\n\nRecommendation: the paper deserves a serious referee, not a desk reject, because the intended result is likely correct and the small-k results are solid. But the referee should insist on fixing the theorem statement and either proving or clearly citing the additive-code result. I would not cite it until the statement is corrected.","headline":"The main theorem is false as stated (missing the k>=b hypothesis), but the intended version is plausible and the small-dimension results are solid; fix the statement and it deserves a serious referee.","tokens_in":25658,"tokens_out":4546,"would_cite":false,"duration_ms":45228,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94B05","94B65"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that, for fixed field size q, read length b, and dimension k with k ≥ b, the shortest possible length of a linear b-symbol code equals the Griesmer-type expression $\\lceil g_q(k, q^{b-1}d)/[b]_q\\rceil$ once the minimum…","keywords":["b-symbol metric","symbol-pair codes","Griesmer bound","length-optimal linear codes","projective systems","binary linear codes","Singer cycles"],"falsifier":"Find one parameter set $k\\ge b$, $q$, and $b$ together with an infinite sequence of distances $d_i\\to\\infty$ for which no linear $[\\lceil g_q(k,q^{b-1}d_i)/[b]_q\\rceil,\\,k,\\,d_i]_q^b$ code exists; a direct computer search over the residue classes modulo $q^{k-b}[b]_q$ for, say, $q=2$, $b=2$, $k=6$ beyond the ranges covered in the paper would either reproduce the formula or produce such a counterexample. Since the proof also depends on the imported additive-code theorem, independently verifying that theorem is a second potential falsifier: if some residue class lacks the promised faithful projective system, the construction has no starting point.","tokens_in":24512,"feed_emoji":"💾","tokens_out":6598,"duration_ms":72770,"temperature":0.7,"pith_summary":"This paper tries to establish that the classical Hamming-metric picture—where the Griesmer bound on code length is attained exactly for all sufficiently large minimum distances—also holds for the b-symbol read-channel metric. The claimed result is that the minimum length $n_q^b(k,d)$ of a linear code with dimension $k$ and b-symbol minimum distance $d$ coincides with the Griesmer-type bound $\\lceil g_q(k, q^{b-1}d)/[b]_q\\rceil$ for every sufficiently large $d$. If this is right, determining optimal b-symbol code lengths becomes a finite check per dimension, field, and block size rather than an infinite optimization problem. The paper also computes the complete exact function for binary pair-symbol codes in dimensions up to five, giving explicit tables and formulas.","feed_headline":"Griesmer bound is exact for b-symbol codes at large distances","feed_subtitle":"Reading b symbols at a time, optimal codes hit the theoretical length bound once distances are large.","key_machinery":"The carrying objects are projective $h-(n,k,s)_q$ systems—multisets of at most $h$-dimensional subspaces of $\\mathrm{PG}(k-1,q)$ such that every hyperplane contains at most $s$ elements—together with their faithful variant in which every element has dimension exactly $h$. A b-chain is an ordered list of vectors of length $n+b-1$ over $\\mathbb{F}_q^k$ whose consecutive b-tuples generate the b-spaces; gluing chains whose end matches the next chain's start preserves both length and the hyperplane-count parameter $s$ (Lemma 3.12). Lemma 3.14 is the pivotal gluing lemma: using Singer cycles and $\\mathrm{GL}(k,q)$ transformations it builds a chain of length $\\lambda[k]_q$ connecting any prescribed start and end while maintaining the exact projective system parameters. The Singer-cycle construction of Proposition 3.2 supplies the base codes with $n=[k]_q$ and $d=[b]_q q^{k-b}$, and Lemma 2.11 records the periodicity of the Griesmer expression that turns finitely many residue classes into the asymptotic statement.","core_discovery":"The central claim is Theorem 3.16: given parameters $k$, $q$, and $b$, one has $n_q^b(k,d) = \\lceil g_q(k, q^{b-1} d)/[b]_q\\rceil$ for all sufficiently large $d$. The intended regime is $k\\ge b$, which the proof uses even though the theorem statement does not state it. The construction proceeds by taking a faithful projective $b-(n,k,n-d)_q$ system supplied by the author's earlier result on additive codes attaining the Griesmer bound, interpreting its b-spaces as length-one b-chains, gluing them into a long b-chain with equal start and end using Singer-cycle-based articulations, and then reading the chain back as the columns of a generator matrix of a linear b-symbol code. Periodicity of the Griesmer bound then extends the construction from one representative of each residue class of $d$ modulo $q^{k-b}[b]_q$ to every sufficiently large distance in that class. For $q=b=2$ and $k\\le 5$, the paper determines $n_2^2(k,\\cdot)$ exactly, with formulas $n_2^2(k,d)=\\lceil g_2(k,2d)/3\\rceil$ for all $d\\ge 2$ when $k=3$, for $d\\ge 5$ when $k=4$, and for $d\\ge 9$ when $k=5$, plus explicit small-distance exceptional values.","pith_inferences":["The same proof structure suggests that the finite exceptional set of distances for each $k,q,b$ is periodic modulo $q^{k-b}[b]_q$, so computing one explicit threshold per residue class would turn Theorem 3.16 into an algorithm rather than an existence statement.","The b-chain formulation recasts b-symbol code construction as a directed tour problem on projective subspaces; the integer-programming model in the paper could be sharpened with subtour-elimination constraints to compute the exceptional values for larger dimensions.","If the imported additive-code existence result fails for some residue class, Theorem 3.16 would still hold for the residue classes where faithful projective systems are attested, so a useful partial version of the theorem survives.","A testable extension would be to compute $n_q^b(k,d)$ for $q=2$, $b=2$, $k=6$ using the same ILP-plus-chain method; agreement with the Griesmer formula beyond a small threshold would support the conjecture that the asymptotic regime begins very early."],"forward_implications":["The length-optimality problem for linear b-symbol codes reduces, for each fixed $k,q,b$, to checking finitely many small minimum distances; all larger distances are settled by the formula $\\lceil g_q(k,q^{b-1}d)/[b]_q\\rceil$.","The ratio $n_q^b(k,d)/\\lceil g_q(k,q^{b-1}d)/[b]_q\\rceil$ tends to $1$ as $d\\to\\infty$, so lengths and the Griesmer-type bound are asymptotically identical.","For binary pair-symbol codes, exact optimal lengths are now known for all dimensions $k\\le 5$, including the exceptional small distances where the Griesmer formula fails.","Concatenation is compatible with the formula: combining codes from Proposition 3.2 and the chain construction via Lemma 3.6 makes lengths and minimum distances additive, which is exactly what the periodicity of the bound requires.","For storage applications, reaching a large target b-symbol distance at minimum length can be done by taking finitely many Singer-cycle-type building blocks and concatenating them, with no further search needed in the asymptotic regime."],"supporting_citations":[{"why":"Supplies the black-box existence theorem for faithful projective systems attaining the Griesmer bound, the starting objects for the chain construction in Theorem 3.16.","marker":"[15]"},{"why":"Proves the Griesmer-type inequality (6) for b-symbol linear codes, which is the lower bound whose equality the paper establishes asymptotically.","marker":"[16]"},{"why":"Provides the identity rewriting the Griesmer bound as $d + \\lceil (g_q(k-b+1,d)-d)/[b]_q\\rceil$, used in Lemma 2.8 to restructure the bound.","marker":"[12]"},{"why":"States the classical Hamming-metric Griesmer bound whose b-symbol analogue is the paper's target.","marker":"[10]"},{"why":"Shows that the Hamming Griesmer bound is attained for all sufficiently large minimum distances, the classical phenomenon the paper transfers to the b-symbol metric.","marker":"[21]"},{"why":"Introduced the symbol-pair read-channel model and the pair-symbol metric that motivates the whole paper.","marker":"[3]"},{"why":"Provides the Singleton-type bound used for the small-distance exceptional values in the binary pair-symbol computations.","marker":"[7]"},{"why":"Gives the b-symbol distance generalization and the improved lower bound for cyclic codes, framing the metric and its earlier bounds.","marker":"[22]"}],"fun_headline_variants":["b-symbol codes hit Griesmer bound exactly at large distances","Optimal b-symbol codes reach Griesmer bound for large d","Griesmer bound tight for b-symbol codes at large d","Exact b-symbol code lengths for large distances from Griesmer bound","When b-symbol distances are large, Griesmer bound is exact"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes, rather than proves here, that for every residue class of the target distance there exists a configuration of b-dimensional subspaces hitting the Griesmer bound exactly, a result on additive codes imported from the author's earlier work, and it uses the dimension $k$ being at least $b$ even though the theorem statement does not say so.","fun_headline_variants_meta":{"raw":{"variants":["b-symbol codes hit Griesmer bound exactly at large distances","Optimal b-symbol codes reach Griesmer bound for large d","Griesmer bound tight for b-symbol codes at large d","Exact b-symbol code lengths for large distances from Griesmer bound","When b-symbol distances are large, Griesmer bound is exact"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000612,"raw_usage":{"total_tokens":2844,"prompt_tokens":943,"completion_tokens":1901,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":559,"completion_tokens_details":{"reasoning_tokens":1820}},"tokens_in":559,"tokens_out":1901,"duration_ms":13438,"temperature":1.0,"reasoning_tokens":1820,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T18:34:58.626257+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find one parameter set $k\\ge b$, $q$, and $b$ together with an infinite sequence of distances $d_i\\to\\infty$ for which no linear $[\\lceil g_q(k,q^{b-1}d_i)/[b]_q\\rceil,\\,k,\\,d_i]_q^b$ code exists; a direct computer search over the residue classes modulo $q^{k-b}[b]_q$ for, say, $q=2$, $b=2$, $k=6$ beyond the ranges covered in the paper would either reproduce the formula or produce such a counterexample. Since the proof also depends on the imported additive-code theorem, independently verifying that theorem is a second potential falsifier: if some residue class lacks the promised faithful projective system, the construction has no starting point.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proves the Griesmer-type inequality (6) for b-symbol linear codes, which is the lower bound whose equality the paper establishes asymptotically."},{"cited_title":"Huang, Q","cited_arxiv_id":null,"evidence_quote":"Provides the identity rewriting the Griesmer bound as $d + \\lceil (g_q(k-b+1,d)-d)/[b]_q\\rceil$, used in Lemma 2.8 to restructure the bound."},{"cited_title":"Cassuto and M","cited_arxiv_id":null,"evidence_quote":"Introduced the symbol-pair read-channel model and the pair-symbol metric that motivates the whole paper."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Singleton-type bound used for the small-distance exceptional values in the binary pair-symbol computations."},{"cited_title":"Yaakobi, J","cited_arxiv_id":null,"evidence_quote":"Gives the b-symbol distance generalization and the improved lower bound for cyclic codes, framing the metric and its earlier bounds."}],"review_version":1}