{"id":"93c40b92-4c81-4b45-8fab-b3d009a4bda9","arxiv_id":"1908.01650","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper constructs families of minimal linear codes from characteristic functions of subsets and subspaces of finite fields, generalizing earlier constructions of Ding et al. and Xu-Qu.","lead":"Using characteristic functions of subsets of a finite field, this paper builds new families of minimal linear codes and claims a Walsh-transform characterization of minimality for defining-set codes. Such codes are relevant to secret-sharing schemes and secure two-party computation.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.2's binary minimality condition has wrong signs; it is not the p=2 case of (11) and can certify a non-minimal code, so the Section 4 characterization is unsound as written.","rationale":"The paper's headline fourth contribution is a general characterization of minimal defining-set codes, and Theorem 4.2 is its explicit binary form. The load-bearing weakness is that this theorem is false as stated: a direct specialization of the paper's own (11) produces a different condition, and a two-dimensional defining-set code provides a concrete counterexample where the printed inequality would certify a non-minimal code. This is a correctness failure of the central claim, not a stylistic or novelty dispute. The reader already rejected the paper on closely related grounds, so the verdict should remain REJECT; I mark agreement as partial because the formal 'weakest_assumption' field points at the subspace condition (8), whereas the more decisive failure is the sign error in Section 4. Credit should be given where it is due: the Section 3 constructions include worked examples and explicit weight enumerators, and the corrected binary condition may well salvage the characterization after a sign fix and re-verification. But as submitted, the announced characterization is not sound, and the reader's REJECT is appropriate without further adjustment.","tokens_in":18966,"tokens_out":16767,"duration_ms":158479,"concrete_test":"Re-run the p=2 specialization of (11) and evaluate it on D=\\{e_1,e_1+e_2\\}\\subset F_2^2; confirm that the printed Theorem 4.2 condition says 'minimal' while direct support containment shows c_{e_1} covers c_{e_2}. Then exhaustively compare the corrected inequality \\hat f_D(\\beta_1)-\\hat f_D(\\beta_1+\\beta_2)-\\hat f_D(\\beta_2) \\neq 2|D| with actual minimality for all D\\subseteq F_2^3 (or all D\\subseteq F_2^m with m\\le 4) by checking whether any nonzero codeword support contains another; if the corrected condition agrees in every case, the sign error is confirmed and the characterization is repairable.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Specializing the paper's own Theorem 4.1 to p=2 gives, from (11), the condition \\hat f_D(\\beta_1+\\beta_2)+\\hat f_D(\\beta_2)-\\hat f_D(\\beta_1) \\neq -2|D| for every F_2-independent \\beta_1,\\beta_2, equivalently \\hat f_D(\\beta_1)-\\hat f_D(\\beta_1+\\beta_2)-\\hat f_D(\\beta_2) \\neq 2|D|. Theorem 4.2 instead states \\hat f_D(\\beta_1+\\beta_2)-\\hat f_D(\\beta_1)-\\hat f_D(\\beta_2) \\neq 2|D|, which is not equivalent: both the sign of \\hat f_D(\\beta_2) and the right-hand side differ. This is not a cosmetic typo. Take D=\\{e_1,e_1+e_2\\}\\subset F_2^2. For \\beta_1=e_1 and \\beta_2=e_2, one computes \\hat f_D(e_1)=4, \\hat f_D(e_2)=0, \\hat f_D(e_1+e_2)=0, and |D|=2, so the printed inequality reads -4 \\neq 4 and predicts minimality. But the defining-set code C_D has c_{e_1}=(1,1), c_{e_2}=(0,1), and c_{e_1+e_2}=(1,0); c_{e_1} covers c_{e_2}, so C_D is not minimal. The corrected condition flags this through 4-0-0=2|D|. Since the abstract advertises the defining-set characterization as a main contribution, Theorem 4.2 as stated invalidates that central claim. The Section 3 subspace families may remain salvageable, but the announced characterization requires a sign correction and re-verification.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies minimal linear codes built from characteristic functions of subsets of F_q, both in the construction C_f = {(u f(x) - Tr(vx))_{x in F_q^*}} and in the defining-set construction C_D = {(Tr(beta x))_{x in D}}. It reports weight distributions, constructs families of binary and odd-characteristic minimal codes from subspaces, and claims a Walsh-transform characterization of minimality for defining-set codes. The main advertised contribution is the binary characterization in Theorem 4.2 and the associated sufficient condition. The paper also gives several explicit examples and weight enumerators.","tokens_in":19424,"tokens_out":19111,"duration_ms":160737,"significance":"If correct, the paper would contribute a useful Walsh-domain criterion for minimality of defining-set codes, several new infinite families with w_min/w_max below or equal to (p-1)/p, and explicit weight distributions. The paper is notable for its concrete examples, detailed weight enumerators, and the systematic use of Walsh transforms. The subspace-based constructions in Section III.B appear to be a substantive contribution. However, the central Section 4 characterization is false as stated, and the proof of Theorem 3.5 has a gap in its application of Theorem 2.2; these issues affect claims advertised in the abstract and cannot be ignored.","major_comments":[{"comment":"Theorem 4.2 is false as stated. Take D={e_1, e_1+e_2} subset F_2^2. One computes hat f_D(e_1)=4, hat f_D(e_2)=0, hat f_D(e_1+e_2)=0, and |D|=2. For beta_1=e_1 and beta_2=e_2, the printed inequality hat f_D(beta_1+beta_2)-hat f_D(beta_1)-hat f_D(beta_2) != 2|D| reads -4 != 4, so the theorem predicts minimality. But the code C_D has codewords c_{e_1}=(1,1), c_{e_2}=(0,1), and c_{e_1+e_2}=(1,0); c_{e_1} covers c_{e_2}, so C_D is not minimal. The correct binary specialization of Theorem 4.1 is hat f_D(beta_1)-hat f_D(beta_1+beta_2)-hat f_D(beta_2) != 2|D|, which flags this example. Consequently Corollary 4.3 and the abstract's claim of a defining-set characterization are unsupported as stated.","section":"Section 4, Theorem 4.2"},{"comment":"The proof of Theorem 3.5 invokes Theorem 2.2 after checking that hat f_{overline D}(h_1,h_2) +/- hat f_D(l_1,l_2) != 2^m for distinct pairs. Theorem 2.2 requires the inequality hat f(h) +/- hat f(l) != q for the same Boolean function f. Since hat f_{overline D}(w)=2-hat f_D(w), the mixed condition checked in the proof does not imply the required condition for either f_D or f_{overline D}. The minimality assertion of Theorem 3.5 is therefore not proven, and the theorem needs either a corrected proof or a corrected statement specifying which code is claimed to be minimal.","section":"Section 3.A, Theorem 3.5 proof"},{"comment":"Although the sufficient condition |hat f_D(beta)| < (2/3)|D| in Theorem 4.2 is valid in isolation, it is attached to an incorrect if-and-only-if statement. The paper should separate the correct sufficient condition from the false equivalence, and should re-derive the p=2 specialization of Theorem 4.1 before using it in Corollary 4.3 and in the examples that rely on the characterization.","section":"Section 4, Theorem 4.2 sufficient condition"}],"minor_comments":[{"comment":"Example 4 is a p=3 example but the text describes the code as \"minimal binary\"; it should say \"minimal ternary.\"","section":"Section 3, Example 4"},{"comment":"The notation \"C_{f_D}\" in the statement of Theorem 3.5 is ambiguous because the proof uses both f_D and f_{overline D}; the intended code should be stated explicitly.","section":"Section 3, Theorem 3.5 statement"},{"comment":"The formatting of Tables IV and V in the text is garbled, with weight columns and frequency columns not cleanly separated; this should be fixed in the final version.","section":"Tables IV and V"},{"comment":"Reference [19] lists the year 2003 for the IMACC proceedings, but the volume number 8308 corresponds to 2013; the year should be corrected.","section":"References"},{"comment":"The spelling of Krawtchouk polynomials is inconsistent (\"Krawchouk\" vs. \"Krawtchouk\"); please unify.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper has a concrete counterexample to its main binary characterization, and the proof of Theorem 3.5 has a clear gap. These are serious but local issues: Theorem 4.1 appears correct and the subspace constructions seem salvageable, so I recommend major revision rather than outright rejection. The authors should re-derive the p=2 specialization, update the affected corollaries and examples, and supply a valid proof or corrected statement for Theorem 3.5."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the Section 3 constructions are real and probably salvageable, but the advertised Section 4 characterization is unsound as written. The stress-test example is correct: for m=2, D={(1,0),(0,1)}, the defining-set code is [2,2,1] with c_e1=(1,1), c_e2=(0,1), c_e1+e2=(1,0); c_e1 covers c_e2, so the code is not minimal, yet Theorem 4.2's inequality holds. The sign error is not cosmetic: specializing Theorem 4.1 to p=2 gives a different condition, and Theorem 4.2 can certify non-minimal codes. Since the abstract advertises the characterization as a main contribution, that claim fails.\n\nWhat is genuinely useful: the complement-of-D trick and the subspace-union construction. Corollaries 3.3 and 3.4 give weight distributions for complements; Theorems 3.8–3.12 add families with explicit weight enumerators, including Examples 1–7. D12 for odd p appears new, and the w_min/w_max ratio bounds are computed carefully. The paper also points back to Ding et al. and Xu–Qu rather than overselling novelty.\n\nSoft spots, in order of severity:\n\n1. Theorem 4.2's condition is wrong. It is not the p=2 case of (11). The counterexample is small and unambiguous. Corollary 4.3 inherits the error. Any revision needs to fix signs and re-run the D12 minimality proof, which currently cites Theorem 4.1/4.2.\n\n2. The proof of Theorem 3.5 checks \\hat f_{\\bar D} ± \\hat f_D instead of the same function. Theorem 2.2 requires both conditions for one function. That proof needs repair; the weight tables may still be right, but the minimality conclusion lacks support as written.\n\n3. Proposition 3.7 shows condition (8) is restrictive: only single subspace, two complementary subspaces, or several half-size subspaces with m even. So “many minimal codes” overstates the range. It does not kill the families, but the reader should know.\n\n4. Minor: the Section 4 result is largely a translation of Heng–Ding–Zhou's criterion into Walsh language; the novel part is the concrete D12 class.\n\nWho gets value: researchers working on defining-set constructions and few-weight minimal codes. The examples and weight tables are checkable and potentially useful. But the current version should not be accepted: the central characterization is false and the Section 3 proof has a gap. I would send it to a competent referee only if the authors are willing to fix Section 4 or drop it; otherwise desk reject.","headline":"Section 3 has salvageable constructions, but the advertised Walsh characterization is wrong as written: Theorem 4.2's sign error certifies a non-minimal code, so the paper needs major revision before it can be trusted.","tokens_in":19915,"tokens_out":3218,"would_cite":false,"duration_ms":32659,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94B05","11T71","94B65"],"pacs":[],"model":"deepseek-v4-flash","headline":"Characteristic functions turn minimality testing and construction of linear codes into Walsh-transform computations, and yield infinite families of minimal codes with weight ratio at most $(p-1)/p$.","keywords":["minimal linear codes","characteristic functions","Walsh transform","weight distribution","defining set method","subspace constructions","Krawchouk polynomials","finite fields"],"falsifier":"Enumerate every subset $D\\subseteq\\mathbb{F}_2^m\\setminus\\{0\\}$ for a small $m$ (say $m=5$ or $6$), compute $\\hat f_D$ for all nonzero arguments, and compare the binary characterization $\\hat f_D(\\beta_1+\\beta_2)-\\hat f_D(\\beta_1)-\\hat f_D(\\beta_2)\\neq 2|D|$ with a direct brute-force verdict on whether $C_D$ is minimal; a single mismatch would disprove Theorem 4.2.","tokens_in":18786,"feed_emoji":"📡","tokens_out":15903,"duration_ms":154246,"temperature":0.7,"pith_summary":"Minimal linear codes—codes in which no nonzero codeword's support contains another codeword's support—are sought after for secret-sharing and secure two-party computation, but they are hard to certify. The paper claims that characteristic functions of subsets of a finite field turn the search into Walsh-transform control: minimality of a defining-set code is equivalent to an explicit inequality in Walsh sums, with a particularly simple form in the binary case. Complementing a defining set changes the Walsh spectrum by a fixed affine shift, so every code comes with a companion code whose weight distribution is read off from the original spectrum. The same functions, applied to unions of carefully separated subspaces, yield infinite families of minimal codes whose lightest nonzero weight is at most $(p-1)/p$ times the heaviest, and a concrete family from all vectors of Hamming weight one or two is minimal for every prime $p$.","feed_headline":"A Walsh transform decides whether a linear code is minimal","feed_subtitle":"The same characteristic-function machinery builds new minimal-code families with controlled weight ratios.","key_machinery":"The load-bearing object is the characteristic function $f_D(x)=1$ if $x\\in D$, $0$ otherwise, and its Walsh transform $\\hat f_D(w)=\\sum_{x\\in\\mathbb{F}_q}\\zeta_p^{f_D(x)-\\operatorname{Tr}(wx)}$. Two identities carry the argument: complementing $D$ affinely shifts the Walsh spectrum (Lemma 3.1), and the Walsh transform of a disjoint union is the sum of the constituent transforms (Lemma 3.2). For subspace-defined sets $D=\\cup_i E_i\\setminus\\{0\\}$, formula (9) expresses every Walsh value in terms of the subspace dimensions and orthogonality pattern, and Proposition 3.7 classifies which subspace arrangements can satisfy the required disjointness assumptions. The minimality test itself is the criterion of Theorem 1.2: a linear code is minimal if and only if $\\sum_{c\\in\\mathbb{F}_p^*}\\operatorname{wt}(a+cb)\\neq(p-1)\\operatorname{wt}(a)-\\operatorname{wt}(b)$ for every pair of linearly independent codewords $a,b$. Krawchouk polynomials supply the Walsh values for Hamming-ball defining sets such as the weight-one-or-two family.","core_discovery":"The paper's central claim, on its own terms, is that the Walsh transform of the characteristic function $f_D$ of a defining set $D$ completely governs whether the code $C_D=\\{(\\operatorname{Tr}(\\beta x))_{x\\in D}:\\beta\\in\\mathbb{F}_q\\}$ is minimal. For general $p$, $C_D$ is minimal exactly when condition (11) holds—the printed inequality involving $\\sum_{y\\in\\mathbb{F}_p^*}(\\sum_{c\\in\\mathbb{F}_p^*}\\hat f_D(y\\beta_1+yc\\beta_2)+\\hat f_D(y\\beta_2)-(p-1)\\hat f_D(y\\beta_1))$ and $(\\zeta_p-1)(p-1)|D|$—for every pair of linearly independent $\\beta_1,\\beta_2$. In the binary case this collapses to the condition $\\hat f_D(\\beta_1+\\beta_2)-\\hat f_D(\\beta_1)-\\hat f_D(\\beta_2)\\neq 2|D|$ for all distinct nonzero $\\beta_1,\\beta_2$, with the sufficient bound $|\\hat f_D(\\beta)|<2|D|/3$. The paper also claims that characteristic functions of unions of pairwise separated subspaces produce many minimal codes, including binary and odd-characteristic families reaching $w_{\\min}/w_{\\max}\\le(p-1)/p$, and that the set of vectors of Hamming weight one or two always gives a minimal defining-set code.","pith_inferences":["Editorial inference: Theorem 4.2's iff condition suggests a direct algorithmic sieve—enumerate candidate defining sets in small fields and evaluate the two Walsh expressions—rather than checking all codeword cover relations as a direct minimality test would.","Editorial inference: Proposition 3.7's three-configuration classification implies the subspace route cannot produce arbitrary geometric variety; the reported 'many minimal codes' are many parameter families within one-subspace, two-complementary-subspace, or several-half-size-subspace layouts.","Editorial inference: The weight-one-or-two family points to a general pattern: defining sets formed from low-weight Hamming layers have Krawchouk-expressible Walsh transforms, so the same proof should test weight-three or weight-four layers for minimality.","Editorial inference: Combined with the LCD-code families cited in the paper, the two-subspace case $E_2=E_1^\\perp$ turns any linear complementary dual code into a minimal code, offering a search path the paper mentions but does not develop."],"forward_implications":["In the binary case, minimality of any defining-set code can be certified by checking one Walsh-transform inequality over all distinct nonzero pairs, giving a finite and explicit test.","Every binary defining set $D$ yields a companion code from its complement whose weight distribution is obtained by replacing $\\hat f_D$ with $2-\\hat f_D$; the odd-characteristic analogue follows from Lemma 3.1 and Corollary 3.4.","When $m$ is even, unions of $s$ half-size subspaces give minimal binary codes exactly when $s$ avoids the two forbidden values, and the forbidden values are the only obstruction in that family.","The defining set $D_{12}=\\{\\beta:\\,1\\le\\operatorname{wt}(\\beta)\\le2\\}$ gives a minimal code over every prime $p$, and for $m\\ge6$ its weight ratio is at most $(p-1)/p$.","The same Walsh machinery applies to the original defining-set construction $C_D$ and to the $C_f$ construction, so the results unify two previously separate families."],"supporting_citations":[{"why":"Supplies the binary code construction, weight-distribution theorem, and Walsh minimality criterion that the characteristic-function method generalizes.","marker":"[24]"},{"why":"Provides the necessary-and-sufficient weight-sum criterion (Theorem 1.2) used to prove minimality of the new families.","marker":"[27]"},{"why":"Gives the odd-characteristic codeword weight formula that underlies the p-ary weight distributions and ratio bounds.","marker":"[32]"},{"why":"Supplies the odd-characteristic minimal-code families that the subspace construction extends.","marker":"[36]"},{"why":"Documents the binary case of the weight-one-or-two defining set, used as a baseline in Remark 7.","marker":"[38]"}],"fun_headline_variants":["Walsh transform decides minimal codes","Walsh criterion for minimal codes","Characteristic functions yield minimal codes","Minimal codes via Walsh transforms","New minimal codes from Walsh bounds"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The subspace-built weight formulas and minimality theorems rest on condition (8): every pair of chosen subspaces must intersect only at zero, and the same must hold for their orthogonal complements; Proposition 3.7 shows this assumption allows only three structural configurations, so the paper's 'many minimal codes' should be read as many parameter choices within those three shapes.","fun_headline_variants_meta":{"raw":{"variants":["Walsh transform decides minimal codes","Walsh criterion for minimal codes","Characteristic functions yield minimal codes","Minimal codes via Walsh transforms","New minimal codes from Walsh bounds"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001721,"raw_usage":{"total_tokens":6848,"prompt_tokens":1029,"completion_tokens":5819,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":645,"completion_tokens_details":{"reasoning_tokens":5761}},"tokens_in":645,"tokens_out":5819,"duration_ms":34700,"temperature":1.0,"reasoning_tokens":5761,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:09:49.395310+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Enumerate every subset $D\\subseteq\\mathbb{F}_2^m\\setminus\\{0\\}$ for a small $m$ (say $m=5$ or $6$), compute $\\hat f_D$ for all nonzero arguments, and compare the binary characterization $\\hat f_D(\\beta_1+\\beta_2)-\\hat f_D(\\beta_1)-\\hat f_D(\\beta_2)\\neq 2|D|$ with a direct brute-force verdict on whether $C_D$ is minimal; a single mismatch would disprove Theorem 4.2.","supporting_citations":[{"cited_title":"Minimal binary linear cod es,","cited_arxiv_id":null,"evidence_quote":"Supplies the binary code construction, weight-distribution theorem, and Walsh minimality criterion that the characteristic-function method generalizes."},{"cited_title":"Minimal linear codes over finite fields,","cited_arxiv_id":null,"evidence_quote":"Provides the necessary-and-sufficient weight-sum criterion (Theorem 1.2) used to prove minimality of the new families."},{"cited_title":"Linear codes with few weights from weakly regular bent functions based on a generic construction,","cited_arxiv_id":null,"evidence_quote":"Gives the odd-characteristic codeword weight formula that underlies the p-ary weight distributions and ratio bounds."},{"cited_title":"Three classes of minimal linear codes over the finite fields of odd characteristic,","cited_arxiv_id":null,"evidence_quote":"Supplies the odd-characteristic minimal-code families that the subspace construction extends."},{"cited_title":"Four families of minimal bi nary linear codes with wmin/w max ≤ 1/ 2,","cited_arxiv_id":null,"evidence_quote":"Documents the binary case of the weight-one-or-two defining set, used as a baseline in Remark 7."}],"review_version":1}