{"id":"9f376ebb-2de0-4dfd-adec-d3069271926a","arxiv_id":"2411.13375","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The complete weight hierarchy of every decreasing norm-trace code is determined by a footprint maximization, recovering the Hermitian hierarchy as a special case.","lead":"This paper gives exact formulas for the generalized Hamming weights, the full 'weight hierarchy', of error-correcting codes defined by decreasing monomial sets on norm-trace curves over finite fields. It also uses those formulas to build asymmetric quantum error-correcting codes, including impure codes whose parameters match or beat existing tables.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.2's case (2.2) is only sketched; the exact weight-hierarchy formula depends on an unstated construction for sets N with a1 ≥ u(q-2)+2.","rationale":"The paper's central theorem is credible and the main idea — attaining the footprint bound via explicit polynomial constructions — is sound. I independently checked the Gröbner-basis claim in [10, Prop. 2.2] via S-polynomial computations for several parameter sets; the initial ideal (y^{q^{s-1}}, x^{u(q−1)+1}) is correct, so the reader's weakest assumption does not appear to be the real risk. The genuine soft spot is the omitted proof of case (2.2) in Theorem 3.2: the exact equality for all N depends on an unstated construction that is asserted but not demonstrated. My small example (q=2,s=2,u=3, N={x^2y,x^2}) shows the construction can be made to work, so this is a fillable gap rather than a false theorem. Because the reader already conditioned acceptance on expanding the omitted subcase, my read does not change the verdict; I partially agree with the reader's framing, as their rationale mentioned case (2.2) but their weakest_assumption focused on the prior initial-ideal result.","tokens_in":22174,"tokens_out":58434,"duration_ms":503393,"concrete_test":"For q=2, s=2, u=3 (Hermitian curve, n=8), take M = {1, x, y, xy, x^2, x^2y} and compute all generalized Hamming weights d_r by brute-force enumeration of all r-dimensional subcodes of ev(M), comparing with 8 − max_{N⊂M, |N|=r} |Δ*(N)| for r = 1..|M|. If any r gives d_r > 8 − max|Δ*(N)|, the construction in case (2.2) fails. As an analytic check, write out the full construction for the case-(2.2) instance N = {x^2y, x^2}: exhibit F with Fin=N and |V(F)| = |Δ*(N)| = 4, verifying V(F)=V(F') for F' corresponding to N'={x^2}.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central formula d_r(ev(M)) = |X_u| − max|Δ*(N)| is proved by constructing, for every N, a polynomial set F with Fin=N and |V(F)| = |Δ*(N)|. The proof splits into four cases, and case (2.2) — a1 ≥ u(q-2)+2 with N not satisfying the strict ordering of Lemma 3.1 — is dismissed in one sentence: 'we can use the argument we used above with a subset N′ ⊂ N that satisfies the conditions in Lemma 3.1, and then increase the multiplicity of the zeroes of some of the polynomials.' This is load-bearing because if the reduction to N′ or the 'multiplicity increase' fails in any subcase, Theorem 3.2 yields only the lower bound |supp(D)| ≥ n − |Δ*(N)|, not equality. Unlike case (1.2), in case (2) the generator x^{a1+u} lies outside the footprint (a1+u > u(q−1)+1), so only divisibility-by-N′ applies; the paper does not explicitly verify that the multiplied polynomials f_j(x−α1)^{a−a_j}(y−β1)^{b−b_j} have the correct initial monomial and remain in L(M) for the modified base polynomials of case (2.1). A minor edge issue also appears in case (1.2.2): when a1=0 the formula references α1, which is undefined; this is repairable (replace the (x−α1) factor by x^{a−a1−u}) but is not stated. These are exposition gaps, not proven false; a concrete completion or small-parameter brute-force check would settle them.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies decreasing norm-trace codes, i.e., evaluation codes obtained from divisibility-closed sets of monomials on the F_{q^s}-rational points of the extended norm-trace curve X_u: x^u = Tr(y). The central result is Theorem 3.2, which asserts that d_r(ev(M)) = |X_u| - max{|Delta*(N)| : N subset M, |N|=r}, where Delta*(N) is the footprint of N together with the two ideal generators y^{q^{s-1}} and x^{min{a_1+u,u(q-1)+1}}. The proof constructs polynomial sets with prescribed initial monomials and counts their common zeroes on X_u. The paper then derives consequences for Reed-Muller-type norm-trace codes (Theorem 4.3), relative generalized Hamming weights (Theorem 5.2), and asymmetrical quantum codes (Theorem 5.5, examples, and Table 1), and it states that the Hermitian weight hierarchy of Barbero and Munuera is recovered as a special case.","tokens_in":22547,"tokens_out":9051,"duration_ms":90537,"significance":"If Theorem 3.2 is fully established, the paper gives a uniform and computationally explicit description of all generalized Hamming weights for a broad family of evaluation codes, including one-point norm-trace and Hermitian codes. The footprint maximization over monomial subsets is a clean and genuinely new approach, and the paper demonstrates its usefulness by comparing with affine Cartesian codes, by obtaining relative weight hierarchies, and by constructing impure quantum codes with parameters competitive with or better than those derived from best-known classical codes. The explicit zero-counting in cases (1.1), (1.2.1), (1.2.2), and (2.1) of Theorem 3.2 is convincing, and the examples in Sections 4 and 5 are valuable. The main weakness is that case (2.2) of Theorem 3.2 is only sketched; since that case is needed for the equality in the main formula, it must be completed before the central claim can be regarded as fully proved.","major_comments":[{"comment":"The proof of case (2.2) is not written out. The text says: 'we can use the argument we used above with a subset N' subset N that satisfies the conditions in Lemma 3.1, and then increase the multiplicity of the zeroes of some of the polynomials.' This is load-bearing: if the reduction to N' or the multiplicity-increase construction fails in any subcase, the argument gives only the lower bound |supp(D)| >= n - |Delta*(N)| from Inequality (4), not the equality claimed in Theorem 3.2. In case (2) the generator x^{a_1+u} is outside the footprint, so the construction cannot simply copy cases (1.2.1) and (1.2.2); one needs to specify the base polynomials, the factors used for each omitted monomial, and to verify that the resulting polynomials have the prescribed initial monomials and lie in L(M). Please complete this case explicitly, or prove the equality by another argument. A brute-force check for small parameters would at least settle the edge cases.","section":"Section 3, Theorem 3.2, case (2.2), page 10"},{"comment":"The displayed polynomial f_{a,b} = (x^u - gamma) * (prod_{i=1}^{a_1}(x-alpha_i)) * (x-alpha_1)^{a-a_1-u} (y-beta_1)^b is not well-defined when a_1 = 0, because alpha_1 is undefined. This case can occur, and it is easily repaired by replacing the factor (x-alpha_1)^{a-a_1-u} with x^{a-u} when a_1 = 0, but the repair is not stated. Please add this subcase and verify that the initial monomial remains x^a y^b.","section":"Section 3, Theorem 3.2, case (1.2.2), page 9"},{"comment":"The proof of Lemma 4.2 relies on [3, Prop. 3.8] without stating the proposition or its hypotheses, and the inequality r'' <= r' is asserted without proof. Since Lemma 4.2 is the key input for the exact formula in Theorem 4.3, please state the needed result from [3] explicitly (or provide a short proof) and justify the comparison of the footprints when r'' < r'. This is a verification gap rather than an observed counterexample, but it is needed for the section's main claim.","section":"Section 4, Lemma 4.2 and Theorem 4.3"}],"minor_comments":[{"comment":"There is a typo in the statement: 'M subset Delta(y^{q^{s-1}}, uu(q-1)+1)' should read 'x^{u(q-1)+1}' in the second coordinate of the footprint.","section":"Theorem 5.3, page 17"},{"comment":"The caption says 'with lambda_2 < lambda_2'; this should clearly be 'lambda_2 < lambda_1'.","section":"Table 1, caption"},{"comment":"The quantum code is printed as '[[15, 1, 7, 7]]9'; the comma between the two 7s should be a slash, consistently with the notation '[[n,k,delta_z/delta_x]]' used elsewhere.","section":"Table 1, row (8,7) for q=3, s=2, u=2"},{"comment":"The range of d is stated as '1 <= d' in Proposition 4.5 and Corollary 4.7, while Section 4 begins with '0 <= d'; please unify the hypotheses so that the boundary case d=0 is unambiguous.","section":"Section 4, Proposition 4.5 and Corollary 4.7"},{"comment":"The claim 'it is straightforward to check that Delta*(x^{11}y^3) = 61' would be easier to verify if one line of computation were included, for instance using the formula in Lemma 3.1 or its case (2.1) analogue.","section":"Example 5.6, page 18"}],"recommendation":"major_revision","confidential_remarks":"I am reasonably convinced that the main formula is correct for the cases that are explicitly checked, and the missing case (2.2) appears repairable rather than false. The paper is a good fit for an information-theory or finite-fields journal, and I do not see a circularity problem: the initial ideal from [10, Prop. 2.2] is an external, published input, not a restatement of the target result. The main task for the authors is to close the proof gap in Theorem 3.2 and to make the dependence on [3, Prop. 3.8] in Lemma 4.2 fully explicit."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my read. The paper really does something new: it determines the full weight hierarchy of decreasing norm-trace codes via a footprint maximization, where previously only minimum distances and duals were known. The recovery of the Barbero–Munuera Hermitian hierarchy is a check, not a restatement, and the relative generalized Hamming weight section plus the impure quantum code constructions are genuine additions. The main formula d_r(ev(M)) = |X_u| − max|Δ*(N)| is credible, and the proof is mostly written out: cases (1.1), (1.2.1), (1.2.2), and (2.1) contain explicit zero-counting that matches Lemma 3.1. I agree with the reader that case (2.2) is the soft spot. It is dismissed in one sentence: take a subset N' satisfying Lemma 3.1, then 'increase the multiplicity of the zeroes.' That step is load-bearing because in case (2) the generator x^{a1+u} is outside the footprint, so the argument used in (1.2) does not automatically carry over. The paper does not verify that the multiplied polynomials stay in L(M) or have the right initial monomials for the modified base polynomials. This is likely fixable—the structure strongly suggests the same construction works—but 'likely' is not a proof. A referee should ask for the subcase to be written out or for a small-parameter brute-force check. Also minor: in (1.2.2) when a1=0, α1 is undefined; replace (x−α1) by x^{a−a1−u}, and there is a typo in Theorem 5.3 ('uu(q−1)+1'). Lemma 4.2 imports [3, Prop. 3.8] without proof, which is fine for a published result. The quantum code table leans on MinT; I would not treat those entries as independently verified, but the construction logic is standard and the impure examples are interesting. The derivation is not circular: the footprint bound from [10] is an independent published result, and the target formula is new. The paper deserves a serious referee. I would send it to review, requesting the case (2.2) expansion and the small fixes. It contributes a clean result to algebraic coding theory and will be cited.","headline":"Genuinely new weight-hierarchy result for norm-trace codes, mostly well proved; the one sketched case in the main theorem is a real gap that should be filled before publication, not a fatal flaw.","tokens_in":23076,"tokens_out":2268,"would_cite":true,"duration_ms":22309,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94B05","11T71","13P25","81P70"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that the r-th generalized Hamming weight of a decreasing norm-trace code equals the number of rational points on the extended norm-trace curve minus the maximum size of a monomial footprint among all r-element subsets of…","keywords":["generalized Hamming weights","weight hierarchy","norm-trace codes","decreasing monomial codes","evaluation codes","footprint bound","quantum codes","Hermitian codes"],"falsifier":"Take the smallest non-trivial case, for example $q=3$, $s=2$, $u=2$, and exhaustively compute the generalized Hamming weights of a small decreasing code $\\operatorname{ev}(M)$ by enumerating all subcodes and comparing each $d_r$ with $|X_u| - \\max|\\Delta^*(N)|$; any mismatch would refute Theorem 3.2. A more direct check is to compute a Gröbner basis of $I(X_u)$ for such parameters: if the initial ideal is not $(y^{q^{s-1}}, x^{u(q-1)+1})$, then Inequality (4) and everything built on it collapses.","tokens_in":21988,"feed_emoji":"🧮","tokens_out":10631,"duration_ms":92792,"temperature":0.7,"pith_summary":"This paper determines the complete weight hierarchy of decreasing norm-trace codes: linear codes formed by evaluating a set of monomials that is closed under divisibility on the rational points of the extended norm-trace curve $x^u = y^{q^{s-1}} + \\cdots + y$ over $\\mathbb{F}_{q^s}$, where $u$ divides $(q^s-1)/(q-1)$. The main theorem computes the $r$-th generalized Hamming weight as the number of rational points of the curve minus the largest possible size of a monomial footprint over all $r$-element subsets of the monomial set. Because footprints can be counted directly, the weight hierarchy ceases to be a search over the exponentially many $r$-dimensional subcodes and becomes a finite combinatorial maximization. The same footprint machinery gives the relative generalized Hamming weights of nested pairs of these codes, and the first relative distances of such pairs and their duals are used, through the Calderbank-Shor-Steane construction, to build asymmetric quantum codes, several of which are impure and compare favorably with codes assembled from the best known classical codes.","feed_headline":"One formula gives every Hamming weight of norm-trace codes","feed_subtitle":"A monomial-counting rule replaces case-by-case subcode searches and yields impure quantum codes.","key_machinery":"The load-bearing object is the modified footprint $\\Delta^*(N) = \\Delta(\\{y^{q^{s-1}}, x^{\\min\\{a_1+u, u(q-1)+1\\}}\\} \\cup N)$, the set of monomials not divisible by any of those generators; here $a_1$ is the smallest $x$-exponent appearing in $N$. This object combines the two generators of the initial ideal of the vanishing ideal $I(X_u) = (\\operatorname{Tr}(y) - x^u, x^{u(q-1)+1} - x)$ with the chosen monomials themselves, and Lemma 3.1 gives a closed-form count for $|\\Delta^*(N)|$ in terms of the exponents when the monomials are ordered with increasing $x$-degree. The extra generator $x^{a_1+u}$ is what makes the footprint tight: it accounts for the S-polynomials forced by the curve equation, and without it the bound would reduce to the Cartesian-code footprint and be strictly weaker in some cases. The engine of the paper is the construction, for every allowable monomial set $N$, of polynomials with exactly those initial monomials whose zero set realizes the footprint count, which upgrades the general inequality $|\\operatorname{supp}(D)| \\ge |X_u| - |\\Delta^*(F)|$ to an equality.","core_discovery":"On the paper's own terms, the central claim is Theorem 3.2: if $M$ is a decreasing set of monomials inside the footprint $\\Delta(y^{q^{s-1}}, x^{u(q-1)+1})$ and $1 \\le r \\le |M|$, then the $r$-th generalized Hamming weight of the evaluation code $\\operatorname{ev}(M)$ is exactly $$d_r(\\operatorname{ev}(M)) = |X_u| - \\max\\{|\\$\\Delta$^*(N)| : N \\subseteq M,\\ |N| = r\\},$$ where $X_u$ is the set of $\\mathbb{F}_{q^s}$-rational points of the extended norm-trace curve and $\\Delta^*(N)$ is the footprint of $N$ together with the two initial-ideal monomials $y^{q^{s-1}}$ and $x^{\\min\\{a_1+u,\\,u(q-1)+1\\}}$, with $a_1$ the smallest $x$-exponent in $N$. The proof shows the general footprint lower bound is tight: for every $N$ one can construct $r$ polynomials whose initial monomials are precisely the elements of $N$ and whose common zero set on $X_u$ has cardinality $|\\Delta^*(N)|$. This formula specializes to the weight hierarchy of one-point norm-trace codes and, in the Hermitian case $s=2$, $u=q+1$, recovers the classical weight hierarchy of one-point Hermitian codes. The same constructive method, applied to two nested decreasing monomial sets, yields the relative generalized Hamming weights when the difference set lies above the smaller set in the monomial order, and the resulting parameters of the CSS quantum codes are tabulated for several small fields.","pith_inferences":["Beyond the paper, the same reduction should work for any family of decreasing evaluation codes on curves whose vanishing ideal has an explicit Gröbner basis; the proof of Theorem 3.2 only uses the two-generator form of $I(X_u)$ and the tightness construction, not any special norm-trace arithmetic.","The closed-form count in Lemma 3.1 suggests that the maximization over $r$-element subsets can be performed by a greedy or dynamic-programming algorithm, so the weight hierarchy of these codes is plausibly polynomial-time computable in $|M|$ for fixed $r$, something the paper does not claim.","The appearance of several impure quantum codes in the tables hints that one-point codes on norm-trace curves with $s=2$ and $u \\neq q+1$ form a systematic source of impure CSS codes, since the relative distance of the dual pair can be strictly larger than the ordinary minimum distance while the classical codes themselves remain near-optimal.","A natural testable extension is to run the maximization of Proposition 4.5 for $u=(q^s-1)/(q-1)$ with $s>2$ and small $q$ to see whether the maximizing sets always belong to the small candidate family $N_{a_1}$ described there, or whether new patterns emerge for larger $s$."],"forward_implications":["The full weight hierarchy of any decreasing norm-trace code is obtained by maximizing a single footprint function over $r$-element monomial subsets, replacing the intractable search over subcodes.","The weight hierarchy of one-point algebraic geometry codes over the extended norm-trace curve is covered, and the known Hermitian weight hierarchy of Barbero and Munuera is recovered as the special case $s=2$, $u=q+1$.","For the Reed-Muller-type codes $\\operatorname{ev}(M_{\\le d})$ with $u \\neq (q^s-1)/(q-1)$, the hierarchy agrees with that of an affine Cartesian code when $u \\le (q^{s-1}-1)/(q-1)$ and can be strictly larger otherwise, with explicit examples showing the improvement.","The relative generalized Hamming weights of nested pairs of decreasing norm-trace codes are exactly determined whenever $M_1 \\setminus M_2$ lies above $M_2$ in the monomial order, which includes all one-point AG code pairs on the curve.","The CSS construction applied to these nested pairs produces asymmetric quantum codes, several of them impure, whose parameters match or exceed those obtained from the best-known classical codes tabulated in the MinT database."],"supporting_citations":[{"why":"Supplies the vanishing ideal and initial ideal of the extended norm-trace curve used to define $\\Delta^*$, the critical prior result the footprint computations rest on.","marker":"[10]"},{"why":"Provides the generalized Hamming weight formula for affine Cartesian codes that is compared to, and matched by, the Reed-Muller-type codes in Theorem 4.3, and the maximization principle for the sets $M'(r)$.","marker":"[3]"},{"why":"Gives the weight hierarchy of one-point Hermitian codes, which the authors recover as the special case $s=2$, $u=q+1$.","marker":"[2]"},{"why":"Introduces generalized Hamming weights and their monotonicity and Singleton-type bounds, the definitions and properties the paper computes.","marker":"[45]"},{"why":"Provides the footprint bound $|V(f)| \\le |\\Delta(I,f)|$ that is the starting inequality the paper makes tight.","marker":"[17]"},{"why":"Earlier order-bound method for generalized Hamming weights of one-point AG codes; the paper's footprint approach supersedes it for norm-trace codes.","marker":"[24]"},{"why":"Prior computation of relative generalized Hamming weights of one-point AG codes via the Feng-Rao bound, which the paper tightens for norm-trace code pairs.","marker":"[19]"},{"why":"Supplies the relative footprint formula for relative generalized Hamming weights of evaluation codes used in Theorem 5.2.","marker":"[30]"}],"fun_headline_variants":["Formula reveals all Hamming weights of norm-trace codes","Norm-trace code weight hierarchy solved in one theorem","From monomial sets to quantum codes: weight hierarchy solved","Exact weight hierarchy for decreasing norm-trace codes","One theorem unifies Hermitian and norm-trace code weights"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The formula's validity depends on the prior Gröbner basis result that the vanishing ideal of the curve is generated by $\\operatorname{Tr}(y) - x^u$ and $x^{u(q-1)+1} - x$, with initial ideal $(y^{q^{s-1}}, x^{u(q-1)+1})$ under the weighted degree lexicographic order; if that description fails for some allowed parameters $q, s, u$, the footprint bound on supports is not justified.","fun_headline_variants_meta":{"raw":{"variants":["Formula reveals all Hamming weights of norm-trace codes","Norm-trace code weight hierarchy solved in one theorem","From monomial sets to quantum codes: weight hierarchy solved","Exact weight hierarchy for decreasing norm-trace codes","One theorem unifies Hermitian and norm-trace code weights"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000266,"raw_usage":{"total_tokens":1717,"prompt_tokens":1156,"completion_tokens":561,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":772,"completion_tokens_details":{"reasoning_tokens":481}},"tokens_in":772,"tokens_out":561,"duration_ms":5853,"temperature":1.0,"reasoning_tokens":481,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T16:29:49.171383+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the smallest non-trivial case, for example $q=3$, $s=2$, $u=2$, and exhaustively compute the generalized Hamming weights of a small decreasing code $\\operatorname{ev}(M)$ by enumerating all subcodes and comparing each $d_r$ with $|X_u| - \\max|\\Delta^*(N)|$; any mismatch would refute Theorem 3.2. A more direct check is to compute a Gröbner basis of $I(X_u)$ for such parameters: if the initial ideal is not $(y^{q^{s-1}}, x^{u(q-1)+1})$, then Inequality (4) and everything built on it collapses.","supporting_citations":[{"cited_title":"Carvalho, H","cited_arxiv_id":null,"evidence_quote":"Supplies the vanishing ideal and initial ideal of the extended norm-trace curve used to define $\\Delta^*$, the critical prior result the footprint computations rest on."},{"cited_title":"Beelen and M","cited_arxiv_id":null,"evidence_quote":"Provides the generalized Hamming weight formula for affine Cartesian codes that is compared to, and matched by, the Reed-Muller-type codes in Theorem 4.3, and the maximization principle for the sets $M'(r)$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the weight hierarchy of one-point Hermitian codes, which the authors recover as the special case $s=2$, $u=q+1$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces generalized Hamming weights and their monotonicity and Singleton-type bounds, the definitions and properties the paper computes."},{"cited_title":"Geil and T","cited_arxiv_id":null,"evidence_quote":"Provides the footprint bound $|V(f)| \\le |\\Delta(I,f)|$ that is the starting inequality the paper makes tight."},{"cited_title":"Heijnen and R","cited_arxiv_id":null,"evidence_quote":"Earlier order-bound method for generalized Hamming weights of one-point AG codes; the paper's footprint approach supersedes it for norm-trace codes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Prior computation of relative generalized Hamming weights of one-point AG codes via the Feng-Rao bound, which the paper tightens for norm-trace code pairs."},{"cited_title":"Jaramillo-Velez, H","cited_arxiv_id":null,"evidence_quote":"Supplies the relative footprint formula for relative generalized Hamming weights of evaluation codes used in Theorem 5.2."}],"review_version":1}