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Minimal Linear Codes Violating the Ashikhmin-Barg Condition from Arbitrary Projective Linear Codes

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Pith's one-line read This paper proves that every projective linear code can be converted, through a simplex complementary construction followed by a generator-block padding step, into a minimal code violating the Ashikhmin-Barg condition while preserving its…

desk verdict A genuinely useful toolkit for building AB-violating minimal codes, but Theorem 2.1 is false as stated for k=1; fix that edge case and the table errors and this deserves refereeing. read the letter →

arxiv 2505.07130 v2 pith:FGGXHV5G submitted 2025-05-11 cs.IT math.IT

Hao Chen , Yaqi Chen , Conghui Xie , Huimin Lao This is my paper · ORCID
classification cs.ITmath.IT MSC 94B0594B65
keywords minimallinearcodesAshikhmin-Bargconditionprojectivesimplexcomplementaryweightdistributionself-orthogonaloptimal
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

Minimal linear codes are codes in which no nonzero codeword's support contains another's except by scalar multiplication, and the Ashikhmin-Barg condition $w_{\min}/w_{\max} > (q-1)/q$ is a well-known sufficient test for minimality that is not necessary. The paper proves a general transformation: given any $q$-ary linear code satisfying that condition, append a block of $n' = \lceil q w_{\min}/(q-1)\rceil - w_{\max}$ nonzero coordinates to a maximum-weight generator row and pad all other rows with zeros; the new code is still minimal, keeps minimum weight $w_{\min}$, and has weight ratio at most $(q-1)/q$, so it violates the condition. It then shows that the simplex complementary code of any projective linear code satisfies the condition once enough extra coordinate dimensions are added. Together these two steps turn arbitrary projective linear codes into minimal codes that violate the Ashikhmin-Barg condition, replacing a collection of special constructions with a general pipeline.

What carries the argument

The load-bearing object is the padded generator matrix $G'$ with first row $(a, r_1)$, where $a$ is a block of $n' = \lceil q w_{\min}/(q-1)\rceil - w_{\max}$ nonzero coordinates and $r_1$ is a maximum-weight codeword of $C$, while every other row has zeros in the first $n'$ positions. A codeword using the first row has full weight in the padding block and a codeword of $C$ in the remaining coordinates, so its weight is at least $w_{\min} + n'$; a codeword avoiding the first row is just a codeword of the old code. Support containment then forces the two codewords to be scalar multiples, preserving minimality, while the ratio $w_{\min}/(w_{\max}+n')$ drops to $(q-1)/q$ or below. The complementary ingredient is the simplex complementary code of a projective code: delete the columns of the projective code from a simplex generator matrix and lift the dimension by appending zero coordinates, yielding a code whose asserted minimum distance is $q^{k+h-1}-w$ and maximum weight $q^{k+h-1}$.

What would settle it

Take a small projective code such as the binary $[4,3,2]_2$ code, delete its four columns from the binary simplex $[2^{k+h}-1,\ k+h,\ 2^{k+h-1}]_2$ generator matrix for $h=1$ and $h=2$, and compute the true minimum distance and maximum weight by exhaustive search; if either differs from $q^{k+h-1}-w$ or $q^{k+h-1}$, Theorem 3.1's parameter claim is false and the arbitrary-projective-code conversion fails at that step.

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Extended reading notes

Core claim

Theorem 2.1 states that from any $q$-ary linear code $C$ with $w_{\min}/w_{\\max} > (q-1)/q$, the explicit generator matrix whose first row is $(a, r_1)$, where $a$ is a block of $n' = \lceil q w_{\min}/(q-1)\rceil - w_{\max}$ nonzero field elements and $r_1$ is a maximum-weight codeword, and whose remaining rows are $(0, r_i)$, produces a minimal code $C'$ with minimum weight $w_{\min}$ and $w_{\min}(C')/w_{\max}(C') \le (q-1)/q$. Theorem 3.1 adds that the simplex complementary code of a projective $[n,k]_q$ code with maximum weight $w$ has parameters $[(q^{k+h}-1)/(q-1)-n,\ k+h,\ q^{k+h-1}-w]_q$, maximum weight $q^{k+h-1}$, and satisfies the Ashikhmin-Barg condition when $h > \log_q w - k + 2$. Consequently, every projective linear code can be transformed, in two steps, into a minimal linear code violating the Ashikhmin-Barg condition.

Load-bearing premise

The entire construction leans on the imported parameter formula for the simplex complementary code—that its minimum distance is exactly $q^{k+h-1}-w$ and its maximum weight exactly $q^{k+h-1}$—so if that formula fails for some projective input code, the claimed conversion collapses, and the present paper does not independently prove the formula.

Editorial extensions

If this is right

  • Any $q$-ary linear code meeting the Ashikhmin-Barg condition can be run through the padding transform, so minimal codes violating the condition no longer depend on special Boolean functions, partial difference sets, or few-weight constructions.
  • The padding step preserves the minimum weight exactly, so codes whose minimum distance is already near optimal or best known can be converted into Ashikhmin-Barg-violating codes with only a modest increase in length.
  • Because Theorem 3.1 produces an Ashikhmin-Barg-satisfying minimal code from any projective linear code, every projective code, including multi-weight and non-minimal codes, feeds the construction; the paper demonstrates this on best-known table codes.
  • When the input code is self-orthogonal, such as a doubly even binary code, the output can be made self-orthogonal as well, yielding infinite families of binary self-orthogonal minimal codes violating the condition.
  • The weight distribution of the output is completely determined from the input weight distribution and the distribution of the subcode generated by all but the first generator row, so the spectra of the new minimal codes are explicit.

Reading between the lines

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

  • If the imported simplex-complementary parameter formula is sound, the two-step pipeline suggests that Ashikhmin-Barg-violating minimal codes are a generic phenomenon, not a rare special construction; one should expect them across most parameter ranges.
  • The construction saturates the boundary by achieving weight ratio at most $(q-1)/q$, which raises the natural dual question of how much padding is necessary to reach a given ratio and whether every minimal AB-violating code can be traced back to an AB-satisfying ancestor.
  • The self-orthogonal variant points toward possible applications in secret-sharing and secure computation, where minimal codes with small weight ratios are used to shape access structures, though connecting these specific parameters to access-structure properties would require further analysis.
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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 / 7 minor

Summary. The paper develops a method to convert minimal q-ary linear codes satisfying the Ashikhmin-Barg condition wmin/wmax > (q-1)/q into minimal codes violating it. The transformation takes a generator matrix whose first row is a maximum-weight codeword, prepends a constant nonzero block of length n' = ceil(q wmin/(q-1)) - wmax to that row, and zero-pads the other rows; Theorem 2.1 claims the resulting code is minimal, has minimum weight wmin and maximum weight at least wmax + n', so that wmin/wmax(C') <= (q-1)/q. The paper then recalls, from the authors' earlier work [9], the simplex complementary code of a projective [n,k]_q code, which has parameters [(q^{k+h}-1)/(q-1)-n, k+h, q^{k+h-1}-w]_q and maximum weight q^{k+h-1}, and is AB-satisfying when h > log_q w - k + 2. Combining the two results, the paper claims that every projective linear code can be transformed into a minimal AB-violating code. It gives several infinite families, including binary self-orthogonal examples, and determines weight distributions of the transformed codes.

Significance. The transformation is an attractive, explicit tool: it produces AB-violating minimal codes from any AB-satisfying code with one extra parameter, and the weight-distribution formula in Theorem 6.1 is precise and easy to apply; some examples are checked by Magma. If the claims stand, the paper provides a very broad class of AB-violating minimal codes and many near-optimal parameters. The main obstacle is that Theorem 2.1 is false for k=1, and the proof of the central minimality claim, as well as the self-orthogonal variant, need to be completed. These are repairable, so the overall program is defensible but not yet fully established.

major comments (3)
  1. [Theorem 2.1 (Section 2)] Theorem 2.1 is false as stated for k = 1, since the construction requires a second generator row of minimum weight. Taking C = [1,1,1]_q, we have wmin = wmax = 1 and AB is satisfied; the construction gives n' = 1 and C' = {00, 11} (up to scalar normalization), whose minimum and maximum weights are both 2, so wmin(C')/wmax(C') = 1 > (q-1)/q and wmin(C') is not equal to wmin. The theorem needs an explicit hypothesis k >= 2 (or a separate treatment of k = 1), and the proof should justify that for k >= 2 a minimum-weight row r2 can be chosen linearly independent of the maximum-weight row r1.
  2. [Section 2, proof of Theorem 2.1, Case 2] The minimality argument in Case 2 is incomplete. From supp(x'_2) subset of supp(x'_1) and minimality of C one obtains x'_2 = lambda' x'_1, but this alone does not imply that x2 cannot lie in the subspace spanned by r'_2,...,r'_k; one must use that the projection y1 of x1 has a nonzero r1-component and hence is not in C* = span(r2,...,rk), so lambda' y1 cannot occur as the last block of a codeword with zero first block. Without this rank/span argument the assertion 'Therefore, x2 cannot be...' does not follow. The proof also states wmin(C') = wmin without proving the lower bound; this bound should be stated explicitly, since any codeword either has weight at least wmin from C*, or weight at least n' + wmin from the first block.
  3. [Corollary 2.1 and Section 5] The self-orthogonal version is not proved. The proof of Corollary 2.1 is a single sentence, and no condition on the prepended block a is derived. For binary codes, (a,r1)·(a,r1) = wt(a) + wt(r1) mod 2; if C is doubly even, wt(r1) = 0 mod 4, so one needs wt(a) even. When a is the all-one vector of length n', this requires n' even, whereas the corollary states length n+n'+1, suggesting an extra coordinate is used; Section 5's examples use length n+n' with no extra coordinate. The statement needs an explicit construction and a proof that the chosen a (and extra coordinate, if any) preserves self-orthogonality, since all the infinite self-orthogonal families in Section 5 depend on this result.
minor comments (7)
  1. [Abstract and headings] The text contains typos such as 'Ashikhamin-Barg', 'codndition', and 'simplx'; these should be corrected throughout.
  2. [Theorem 3.1, proof] The proof writes h >= log_q w - k + 2, but the strict inequality h > log_q w - k + 2 is required for the strict Ashikhmin-Barg comparison; the proof should be aligned with the theorem statement.
  3. [Theorem 3.1 and reference [9]] The complementary-code parameters are imported from [9] without a specific statement or theorem number; please cite the exact result in [9] so the reader can verify that the hypotheses (projectivity, the length condition, and the embedding dimension) match the present setting.
  4. [Theorem 6.1] The theorem describes the 'set of nonzero weights' as a union; if a weight w_i occurs only in C* or only outside C*, one of the two corresponding multiplicities is zero, so the displayed union is the set of possible weights rather than necessarily the exact support. This should be stated more precisely.
  5. [Table 2] The row '[292, 8, 96]_2 -> [256, 8, 96]_2' is suspicious because Theorem 2.1 should increase the length; please check the parameters and the underlying computation for that row.
  6. [Section 5] The assertion that the simplex complementary code of a doubly even binary code is again doubly even is used without proof or reference; it is true but should be justified.
  7. [Terminology] Some examples refer to 'optimal' minimum distances when the cited tables give the best known lower bound; the terminology should be made consistent.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the AB-violating ratio is produced by the explicit choice of n', with weight distributions and minimality proved from the input code; cited complementary-code parameters are independent prior results.

full rationale

Theorem 2.1's construction is not circular: n' is explicitly defined as ceil(q * w_min / (q - 1)) - w_max, and the inequality w_min / (w_max + n') <= (q - 1) / q is an identity following from that definition, not a fitted parameter relabeled as a prediction. The minimality of C' is argued from the minimality of C, and the minimum weight w_min(C') = w_min is witnessed by the second generator row (when k >= 2), so the asserted AB violation is a property of the explicitly constructed code rather than an imported conclusion. Theorem 6.1's weight-distribution formulas are honest bookkeeping from the distributions of C and its subcode C*; the weights of C' are computed, not fitted. Theorem 3.1 imports the simplex-complementary-code parameters from the authors' earlier paper [9]; this is a self-citation and is load-bearing for the 'arbitrary projective code' pipeline, but the cited result is a parameter-free published statement whose assumptions do not include the target AB-violating conclusion, so under the review rules it constitutes independent support rather than circularity. The same holds for the specific complementary-code parameters cited from the submitted companion [10]. No step reduces to its own input by construction. A separate correctness concern is that Theorem 2.1's proof needs a second generator row and therefore fails for k = 1 (e.g., C = [1,1,1]_q gives C' with ratio 1, not an AB violation); this is a domain error, not a circular derivation, and does not affect the circularity score.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The construction's free parameters are limited to the extension dimension h and the choice of input code; n' is determined by w_min and w_max. The key imported premise is the simplex complementary code parameter formula from [9], plus the trusted codetable [16] for optimality claims. No new entities are postulated.

free parameters (2)
  • h = h > log_q w - k + 2 (any such integer)
    Extension dimension in Theorem 3.1; the Ashikhmin-Barg condition for the simplex complementary code forces h above this threshold. It is a construction knob, not fitted to data.
  • input projective code C (with maximum weight w) = arbitrary [n,k]_q projective code
    The construction is parameterized by an arbitrary projective code, whose weight distribution feeds into the output weight distribution of Theorem 6.1.
assumptions (5)
  • standard math Ashikhmin-Barg theorem: w_min/w_max > (q-1)/q implies minimality of a q-ary linear code
    Cited from [3]; used in Theorems 2.1 and 3.1 to conclude minimality of the intermediate codes.
  • standard math The q-ary simplex code has parameters [(q^K-1)/(q-1), K, q^{K-1}] with all nonzero weights equal to q^{K-1}
    Background for the simplex complementary construction in Section 3 and for the families in Section 6.
  • domain assumption Simplex complementary code parameters: deleting the n columns of a projective code with maximum weight w yields a code of minimum distance q^{K-1} - w and maximum weight q^{K-1}
    Imported from reference [9] by the same authors; Theorem 3.1 consists of substituting these values plus the AB inequality, with no proof in this paper.
  • standard math Doubly even binary linear codes are self-orthogonal, and the simplex complementary code of a doubly even code is doubly even
    Used in Section 5; follows from weight divisibility by four, but stated in the text without proof.
  • domain assumption The input few-weight, cyclic, and best-known codes from [6, 12, 13, 14, 16, 20, 27, 41] have the stated parameters and weight distributions
    All Section 4 and Section 5 families use published code parameters or codetable entries as inputs; Table 9 optimality claims trust [16].

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Pith. "Pith review of Minimal Linear Codes Violating the Ashikhmin-Barg Condition from Arbitrary Projective Linear Codes." pith.science (2026). https://pith.science/paper/FGGXHV5G

@misc{pith2026250507130,
  author       = {Pith},
  title        = {Pith review of: Minimal Linear Codes Violating the Ashikhmin-Barg Condition from Arbitrary Projective Linear Codes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FGGXHV5G}},
  note         = {Machine review of arXiv:2505.07130}
}
read the original abstract

In recent years, there have been many constructions of minimal linear codes violating the Ashikhmin-Barg condition from Boolean functions, linear codes with few nonzero weights or partial difference sets. In this paper, we first give a general method to transform a minimal code satisfying the Ashikhmin-Barg condition to a minimal code violating the Ashikhmin-Barg condition. Then we give a construction of a minimal code satisfying the Ashikhmin-Barg condition from an arbitrary projective linear code. Hence an arbitrary projective linear code can be transformed to a minimal codes violating the Ashikhmin-Barg condition. Then we give infinite many families of minimal codes violating the Ashikhamin-Barg condition. Weight distributions of constructed minimal codes violating the Ashikhmin-Barg condition in this paper are determined. Many minimal linear codes violating the Ashikhmin-Barg condition with their minimum weights close to the optimal or the best known minimum weights of linear codes are constructed in this paper. Moreover, many infinite families of self-orthogonal binary minimal codes violating the Ashikhmin-Barg condition are also given.

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