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The weight hierarchy of decreasing norm-trace codes

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2411.13375 v1 pith:25D2O2J5 submitted 2024-11-20 cs.IT math.ACmath.AGmath.IT

Eduardo Camps-Moreno , Hiram H. López , Gretchen L. Matthews , Rodrigo San-José This is my paper · ORCID
classification cs.ITmath.ACmath.AGmath.IT MSC 94B0511T7113P2581P70
keywords generalizedHammingweightsweighthierarchynorm-tracecodesdecreasingmonomialevaluationfootprintboundquantumHermitian
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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 $s$.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

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.

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 (3)
  1. [Section 3, Theorem 3.2, case (2.2), page 10] 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.
  2. [Section 3, Theorem 3.2, case (1.2.2), page 9] 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.
  3. [Section 4, Lemma 4.2 and Theorem 4.3] 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.
minor comments (5)
  1. [Theorem 5.3, page 17] 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.
  2. [Table 1, caption] The caption says 'with lambda_2 < lambda_2'; this should clearly be 'lambda_2 < lambda_1'.
  3. [Table 1, row (8,7) for q=3, s=2, u=2] 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.
  4. [Section 4, Proposition 4.5 and Corollary 4.7] 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.
  5. [Example 5.6, page 18] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the weight-hierarchy formula is derived by a two-sided footprint bound with explicit polynomial constructions; overlapping-author citations are independent, parameter-free prior facts and do not contain the target result.

full rationale

The central claim is the identity d_r(ev(M)) = |X_u| - max{|Delta*(N)| : N subset of M, |N|=r}. This is proved by a genuine two-sided argument, not by definition or by fitting. Inequality (4) supplies the lower bound d_r >= |X_u| - max|Delta*(N)| for every r-dimensional subcode via the footprint/initial-ideal estimate. Theorem 3.2 then supplies the matching upper bound by explicitly constructing, for each admissible monomial set N, polynomials F with Fin = N and counting the common zeros on X_u directly, e.g. in Case (1.1) the count a1 q^{s-1} + b_r(v-a1) + sum (a_{i+1}-a_i)(b_i-b_r) is obtained by a combinatorial zero-count, not assumed to equal the footprint. The bound (3) itself is derived in the paper from the S-polynomial computation, using only the prior initial-ideal fact in(I_Xu) = (y^{q^{s-1}}, x^{u(q-1)+1}). That fact is quoted from [10, Prop. 2.2], an earlier paper by overlapping authors, but it is a parameter-free algebraic statement about the curve and its vanishing ideal; it does not state the generalized Hamming weights of the decreasing norm-trace codes and is independently checkable, so it is real evidence rather than a circular premise. Similarly, Theorem 5.2 imports the relative-footprint characterization from [30] and the Cartesian-code comparisons in Section 4 use [3] only as an external benchmark; neither contains the norm-trace weight-hierarchy formula being derived. The only notable weakness is that Case (2.2) of Theorem 3.2 is sketched in one sentence, and the edge case a1=0 in Case (1.2.2) is not fully spelled out. These are exposition/completeness gaps, not circular reductions: the proof still aims to establish equality by construction rather than by identifying the conclusion with an input. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported to force the choice, and no known hierarchy is merely relabeled.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No constants are fitted to data; q, s, and u are parameters of the code family, not free parameters adjusted to force the result. The proof imports two prior results as black boxes: the Groebner basis description of I(Xu) from [10] and the affine Cartesian code hierarchy from [3]. No new particles, forces, dimensions, or other entities are introduced.

assumptions (4)
  • domain assumption The vanishing ideal of Xu is I(Xu) = (Tr(y) - x^u, x^{u(q-1)+1} - x), and these generators form a Groebner basis with initial ideal (y^{q^{s-1}}, x^{u(q-1)+1}) for the weighted degree lex order.
    Cited to [10, Prop. 2.2]; used implicitly in Inequality (3) and Lemma 3.1 to define Delta*.
  • domain assumption Affine Cartesian code weight hierarchies and the maximizing property of lexicographically first monomial sets from [3, Prop. 3.8].
    Used in Lemma 4.2 and Theorem 4.3 to identify the maximizing monomial set and to relate dr(ev(M<=d)) to dr(Cart_u(d)).
  • standard math Rational point counts and fiber structure of the extended norm-trace curve: |A0| = q^{s-1}, |A_gamma| = u q^{s-1} for gamma in F_q^*, with fibers under trace and u-th power described in Lemma 2.5.
    Used in the zero-counting constructions of Theorem 3.2 and in the examples.
  • standard math Footprint bound |V(I)| <= |Delta(I)| for zero-dimensional ideals, with equality for radical ideals.
    Standard commutative algebra result, referenced to [12], used to pass from the footprint Delta*(F) to |V_Xu(F)|.

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Pith. "Pith review of The weight hierarchy of decreasing norm-trace codes." pith.science (2026). https://pith.science/paper/25D2O2J5

@misc{pith2026241113375,
  author       = {Pith},
  title        = {Pith review of: The weight hierarchy of decreasing norm-trace codes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/25D2O2J5}},
  note         = {Machine review of arXiv:2411.13375}
}
abstract

The Generalized Hamming weights and their relative version, which generalize the minimum distance of a linear code, are relevant to numerous applications, including coding on the wire-tap channel of type II, $t$-resilient functions, bounding the cardinality of the output in list decoding algorithms, ramp secret sharing schemes, and quantum error correction. The generalized Hamming weights have been determined for some families of codes, including Cartesian codes and Hermitian one-point codes. In this paper, we determine the generalized Hamming weights of decreasing norm-trace codes, which are linear codes defined by evaluating monomials that are closed under divisibility on the rational points of the extended norm-trace curve given by $x^{u} = y^{q^{s - 1}} + y^{q^{s - 2}} + \cdots + y$ over the finite field of cardinality $q^s$, where $u$ is a positive divisor of $\frac{q^s - 1}{q - 1}$. As a particular case, we obtain the weight hierarchy of one-point norm-trace codes and recover the result of Barbero and Munuera (2001) giving the weight hierarchy of one-point Hermitian codes. We also study the relative generalized Hamming weights for these codes and use them to construct impure quantum codes with excellent parameters.

Figures

Figures reproduced from arXiv: 2411.13375 by the authors.

Figure 1
Figure 1. Example of the decomposition of the footprint used in the proof of Lemma 3.1 with r = 3. We present now the main result of this section, in which we show how to construct sets of polynomials attaining the footprint-like bound in Inequality (3). Theorem 3.2. Let M ⊂ ∆(y q s−1 , xu(q−1)+1) be a decreasing set and 1 ≤ r ≤ |M|. Let Nr := {N = {M1, . . . , Mr} ⊂ M : Mi ̸= Mj for i ̸= j}. Then, the r th general￾ized Hammi… view at source ↗

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