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Augmented trace codes yield four-weight self-orthogonal codes, LCD codes, and AMDS quantum codes

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2026-07-09 17:57 UTC pith:O2ZMGLQL

load-bearing objection Four new families of four-weight self-orthogonal codes from augmented defining-set constructions; two of the four 'four-weight' claims lack frequency verification. the 2 major comments →

arxiv 2607.07181 v1 pith:O2ZMGLQL submitted 2026-07-08 cs.IT math.IT

Four classes of few-weight self-orthogonal codes and their applications for LCD codes and quantum codes

Yue Huang , Zhonghao Liang , Chenlu Jia , Yongkang Wan , Qunying Liao This is my paper
classification cs.IT math.IT
keywords codesclassesself-orthogonalquantumclassapplicationsdualfew-weight
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper constructs four classes of self-orthogonal linear codes over finite fields by augmenting trace-based defining-set codes. The central mechanism is the augmented code construction: starting from a defining set D ⊆ F_{p^m}² carved out by a trace condition Tr(αxy + βy + x) = γ, one forms the code whose codewords are (Tr(ax + by))_{(x,y)∈D} + c·1, where a, b range over F_{p^m} and c ranges over F_p. The addition of the scalar-shift parameter c is what makes the resulting code p-divisible and hence self-orthogonal (by a sufficient condition of Li and Heng). For two specific defining sets D₁ (full plane) and D₂ (restricting x ≠ 0), the authors compute the weight distributions via Gauss sums and character sums, showing that when γ takes a distinguished value (γ = Tr(−β/α) for D₁, γ = 0 for D₂), the augmented code has exactly four distinct nonzero weights with fully explicit frequencies. They then determine dual-code parameters for two of these families, apply a matrix-extension lemma to produce LCD (linear complementary dual) codes, and use the self-orthogonal containment C̄_{D₂} ⊆ C̄_{D₂}^⊥ to build a quantum code meeting the almost-MDS (AMDS) bound when m = 2.

Core claim

The paper's main result is that for the defining sets D₁ = {(x,y) ∈ F_{p^m}² : Tr(αxy + βy + x) = γ} and D₂ = {(x,y) ∈ F*_{p^m} × F_{p^m} : Tr(αxy + x) = γ}, the augmented codes C̄_{D₁} and C̄_{D₂} are four-weight self-orthogonal linear codes with parameters [p^{m−1}(p^m + p − 1), 2m+1, p^{2m−2}(p−1)]_p (for D₁ with γ = Tr(−β/α)) and [p^{m−1}(p^m − 1), 2m+1, p^{m−1}(p^m − p^{m−1} − p + 1)]_p (for D₂ with γ = 0, which is also projective). The weight distributions are computed in full for these two cases via quadratic Gauss sum evaluations. The dual of C̄_{D₁} has minimum distance 2, and the dual of C̄_{D₂} has minimum distance 3. Appending an identity block to the generator matrix yields LCD码

What carries the argument

The augmented code construction C̄_D = {(Tr(ax+by))_{(x,y)∈D} + c·1 : a,b ∈ F_{p^m}, c ∈ F_p}, combined with the sufficient condition of Li and Heng (a p-divisible code containing the all-ones vector is self-orthogonal), quadratic Gauss sum evaluations (Lemma II.4) to resolve the character sums arising in weight computation, Pless power moments to determine dual minimum distances, the LCD construction lemma (appending [I_k : G] to a self-orthogonal code's generator matrix), and the CSS-type quantum code construction via nested self-orthogonal containment C₁^⊥ ⊆ C₁ ⊆ C₂ (Lemma II.8).

Load-bearing premise

For the two cases where γ does not take its distinguished value (Theorems III.2 and III.5), the paper claims the codes are 'four-weight' but does not compute the weight frequencies, explicitly stating that certain character-sum cardinalities are 'not easy to compute.' Without these frequencies, one cannot confirm that all four candidate weight values are actually attained, so the 'four-weight' label for those two subfamilies is unverified.

What would settle it

If, for γ ≠ Tr(−β/α) in the D₁ case or γ ≠ 0 in the D₂ case, one of the four candidate weight values has zero frequency (i.e., no codeword achieves that weight), the corresponding code would be three-weight or fewer, contradicting the 'four-weight' claim of Theorems III.2 and III.5. This could be checked by exhaustive computation for small p and m.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The LCD code family C₃ (Theorem V.3, with m = 2) has a dual that is almost optimal by the sphere-packing bound, meaning no code with the same length and one higher dimension at distance 3 can exist — providing concrete near-optimal LCD parameters for cryptographic applications.
  • The quantum code [[p(p²−1), p(p²−1)−6, 3]]_p (m = 2) meets the quantum Singleton bound's AMDS threshold, adding a new infinite family of distance-3 quantum codes over odd prime fields.
  • The projective property of C̄_{D₂} (dual distance ≥ 3) means its dual is free of coordinate repetitions, making it suitable for secret-sharing schemes where each coordinate carries independent information.
  • The method of augmenting trace codes with a scalar shift c ∈ F_p to force p-divisibility — and hence self-orthogonality — is demonstrated to be effective for quadratic-defining-set trace codes, suggesting it may extend to other defining-set families with similar algebraic structure.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 8 minor

Summary. This manuscript constructs four classes of four-weight self-orthogonal linear codes over finite fields using augmented codes derived from specific defining sets. The defining sets are D1 = {(x,y) ∈ F_{p^m}² : Tr(αxy+βy+x) = γ} and D2 = {(x,y) ∈ F*_{p^m} × F_{p^m} : Tr(αxy+x) = γ}. For the cases γ = Tr(-β/α) (Theorem III.1) and γ = 0 (Theorem III.4), complete weight distributions are computed (Tables I, II), self-orthogonality is established via p-divisibility and Lemma II.6, and dual code parameters are determined using Pless power moments. As applications, the authors construct LCD codes (Theorems V.1–V.3) and quantum codes (Theorem V.7), showing that one class of LCD dual codes is almost optimal by the sphere packing bound and that the quantum codes are AMDS for m = 2. The proofs for Theorems III.1 and III.4 are detailed and verified by Magma computations (Table III).

Significance. The construction of few-weight self-orthogonal codes is an active area in coding theory with applications in secret sharing, authentication, and quantum coding. The paper's approach of using augmented codes from defining sets follows a recent line of work (Li-Heng [29], Heng et al. [30]). The complete weight distributions for Theorems III.1 and III.4, the determination of dual code distances, and the downstream LCD and quantum code constructions are concrete contributions. The Magma verification for multiple parameter sets (Table III) and the comparison tables (Tables IV, V) in the appendix strengthen the novelty claims. The almost-optimal LCD result (Theorem V.3) and the AMDS quantum code result (Theorem V.7) are specific and checkable.

major comments (2)
  1. Theorems III.2 and III.5 claim that C̄_{D1} (when γ ≠ Tr(-β/α)) and C̄_{D2} (when γ ≠ 0) are 'four-weight' codes, but Remarks III.3 and III.6 explicitly state that the weight distributions (frequencies) cannot be computed. The proofs derive four candidate nonzero weights and show that wt(c) must be one of them, establishing that at most four weights occur. However, without computing the frequencies, the paper does not verify that all four candidate weights are actually achieved by some codeword with positive frequency. A code where one candidate weight is never attained would be three-weight or fewer, contradicting the theorem statement. The Magma examples in Table III (p=3, m=2,3,4) do show four distinct weights for these cases, but no general argument is given. This is a load-bearing gap for two of the four claimed 'four-weight' classes. The authors should either (a) compute the weight
  2. distributions for Theorems III.2 and III.5 to verify all four weights have positive frequency, or (b) restate Theorems III.2 and III.5 as 'at most four-weight' codes and adjust the title and abstract accordingly. Note that this concern does not affect Theorems III.1 and III.4 (which have complete weight distributions with verified positive frequencies), the self-orthogonality claims (which follow from p-divisibility), or the downstream LCD and quantum code constructions (which use only Theorems III.1 and III.4).
minor comments (8)
  1. Introduction: The paper attributes the defining-set construction to 'Ding et al.[10] in 2007' but reference [10] is a 2005 paper. Reference [9] (Ding-Niederreiter, 2007) appears to be the correct citation for cyclotomic linear codes. Please correct the citation.
  2. Introduction: 'introduced by Ding et al.[10] in 2007' — reference [10] is dated 2005 and concerns authentication codes, not the defining-set construction per se. Reference [9] seems more appropriate. Please clarify.
  3. The notation for the augmented code C̄_{D_i} (Eq. I.4) uses c ∈ F_p for the augmentation parameter, while the original code C_D uses (a,b) ∈ F_{p^m}². This mixing of field sizes is correct but could be stated more explicitly to improve readability.
  4. §III.B, Proposition III.8: The case analysis is lengthy. A brief summary table of the values of (t₁, t₂, t₃) and the corresponding Ω₃ values, before diving into cases, would aid the reader.
  5. Table III: For Theorem III.2 and Theorem III.5 with p=3, m=2, the weight enumerators are identical (1 + 104z^18 + 112z^15 + 24z^12 + 2z^24). This may be correct but should be noted or explained, as it could appear to be a copy error.
  6. §V.A, Theorem V.3, proof part (3): The inequality p^{n-(p(p²-1)+2)} < 1 + n(p-1) is stated without derivation. A brief justification or reference would help, as this is the key step for the almost-optimal claim.
  7. Several references have 2025–2026 dates (e.g., [8], [16], [32], [50], [51], [55], [57]) and may be preprints or forthcoming. Please verify availability and update citations to final published versions where applicable.
  8. The abstract states 'three classes of four-weight self-orthogonal codes' in addition to the projective one, but as noted in the major comments, two of these (Theorems III.2, III.5) lack verified weight distributions. The abstract should accurately reflect the actual proven results.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained with external tools

full rationale

The paper's central derivation chain is self-contained. The weight distributions in Theorems III.1 and III.4 are computed from first principles using Gauss sums (Lemma II.4, from Lidl-Niederreiter [33]), additive character sums (Lemma II.1), and the Pless power moments (Lemma II.5, from Huffman-Pless [19]) — all external results. The self-orthogonality follows from the p-divisibility criterion in Lemma II.6 (from Li-Heng [29]), which is an independent sufficient condition. The LCD code construction (Lemma II.7, from [55]) and quantum code construction (Lemma II.8, from Ling-Luo-Xing [34]) are standard external results applied to the constructed self-orthogonal codes. The augmented code construction (Eq. I.4) is a direct definition, not a fitted ansatz. The defining sets D1 and D2 (Eqs. I.2-I.3) are explicit algebraic sets, not defined in terms of the output code properties. No step in the derivation chain reduces to its own inputs by construction. The two self-citations to Li-Heng [29] for the augmented code framework and self-orthogonality criterion are not load-bearing in a circular sense: [29] provides a general sufficient condition (p-divisibility implies self-orthogonality) that is applied here to independently constructed codes, not a result that is re-derived or assumed. The weight computations proceed by direct character sum evaluation. The downstream LCD and quantum code constructions use only Theorems III.1 and III.4 (which have complete, verified weight distributions), not the incomplete Theorems III.2 and III.5. The concern about Theorems III.2/III.5 not verifying that all four weights are achieved (since frequencies are not computed) is a correctness/completeness gap, not a circularity issue — the candidate weights are derived from independent computation, not from assuming the conclusion. Score 2 reflects the minor self-citation to [29] for the augmented code framework, which is not circular but provides the construction paradigm.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 0 invented entities

No new mathematical entities are introduced. The constructions use standard objects: finite fields, trace functions, defining sets, augmented codes, Gauss sums, and character sums.

free parameters (3)
  • p = odd prime, parameter of construction
    The prime characteristic of the finite field F_{p^m}. Not fitted to data; it is a free parameter of the construction that ranges over all odd primes.
  • m = integer ≥ 2
    The extension degree of F_{p^m} over F_p. A free parameter of the construction.
  • α, β, γ = α ∈ F*_{p^m}, β ∈ F_{p^m}, γ ∈ F_p
    Parameters defining the sets D1 and D2. They are not fitted to data but are structural parameters of the construction. Different choices yield different code families (e.g., γ = Tr(-β/α) vs. γ ≠ Tr(-β/α)).
axioms (5)
  • domain assumption Lemma II.6 (Li-Heng [29], Theorem 1): If 1_n ∈ C and C is p-divisible, then C is self-orthogonal.
    This is the key sufficient condition for self-orthogonality used in all four main theorems. It is invoked at the end of each proof (Section IV-A) to conclude self-orthogonality from p-divisibility. The paper verifies p|w_i for all computed weights.
  • domain assumption Lemma II.7 ([55], Lemma 5.9): If C is self-orthogonal with generator matrix G, then [I_k : G] generates an LCD code.
    Used in Theorems V.1–V.3 to construct LCD codes from the self-orthogonal codes. This is a standard construction lemma from prior work.
  • domain assumption Lemma II.8 ([34], Theorem 2.6): CSS-like construction of quantum codes from nested self-orthogonal codes.
    Used in Theorem V.7 to construct quantum codes from C̄_{D2}^⊥ and its supercode. Standard quantum code construction lemma.
  • standard math Standard properties of Gauss sums, quadratic characters, and additive characters over finite fields (Lemmas II.1–II.4).
    These are well-known results from finite field theory (Lidl-Niederreiter [33]) used throughout the weight distribution computations in Propositions III.7–III.10 and the main theorem proofs.
  • standard math Pless power moments (Lemma II.5) relate the weight distribution of a code to that of its dual.
    Used in Section IV-A to determine dual code distances (d(C̄_{D1}^⊥) = 2, d(C̄_{D2}^⊥) = 3) from the weight distributions of the primal codes.

pith-pipeline@v1.1.0-glm · 31605 in / 5311 out tokens · 422351 ms · 2026-07-09T17:57:55.995410+00:00 · methodology

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read the original abstract

Since self-orthogonal codes, few-weight codes, linear complementary dual codes(LCD codes, for short) and quantum codes have nice applications in coding theory and cryptography, they have received continuous attention. In 2024, by introducing the notion of the augment code, Heng et al.[30] constructed several classes of few-weight self-orthogonal codes basing on defining sets, which are introduced by Ding et al.[10] in 2007. In this manuscript, for two classes of defining sets, we consider the corresponding augmented codes, construct a class of projective four-weight self-orthogonal codes and three classes of four-weight self-orthogonal codes. And for two classes of these four-weight self-orthogonal linear codes, we determine the parameters of their dual codes. As applications, we construct two classes of LCD codes and a class of quantum codes. In particular, we prove that there exists a class of these LCD codes whose dual codes are almost optimal LCD codes according to the sphere packing bound, and a class of quantum codes are AMDS according to the quantum Singleton bound.

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