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REVIEW 3 major objections 4 minor 9 references

Additive codes from linear codes

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Two constructions turn linear codes into additive codes with guaranteed minimum distance; one family is distance-optimal.

desk verdict Theorem 3 is false as stated at q=2, s=h=1, but the fix is a one-line condition and the rest of the construction is solid. read the letter →

arxiv 2506.03805 v1 pith:UR5LLAHD submitted 2025-06-04 cs.IT math.COmath.IT

Simeon Ball , Tabriz Popatia This is my paper · ORCID
classification cs.ITmath.COmath.IT MSC 94B0594B6511T71
keywords additivecodeslineargeneralisedHammingweightGriesmerboundminimumdistancetrace-normconstructionpartialsemifield
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

Additive codes over a finite field are subspaces that are linear over a smaller field, and they are studied partly because they include quantum stabilizer codes and can outperform ordinary linear codes. The paper offers two general constructions that start with a linear code over $\mathbb{F}_q$ and produce an additive code over $\mathbb{F}_{q^h}$ with a guaranteed minimum distance. The first construction uses partial semifields and ties the distance to a generalized Hamming weight of the original code; the second uses a trace-and-norm argument to build an infinite family, namely a $[q^{st}-1, (s(t+1)+1)/h, d]^h_q$ code with $d \ge q^{st}-1 - \frac{q^{st}-1}{q^s-1} q^{s-h}$ whenever $h \le s$ and $t \ge 2$. In the special case $s=h$ and $t=2$, the same construction attains the additive Griesmer bound, so its distance $q^{2h}-q^h-1$ is the maximum possible. The upshot is a general recipe for turning known linear codes into additive codes whose distance behaviour is certified rather than merely observed.

What carries the argument

The mechanism is a set $\mathcal{X}$ of rank-$h$ $\mathbb{F}_q$-subspaces of the vector space $\mathbb{F}_q \times \mathbb{F}_{q^s} \times \mathbb{F}_{q^{st}}$, viewed as columns of a generator matrix; a codeword has a zero coordinate exactly when the corresponding subspace lies in a hyperplane. To control the number of zeros, the authors choose $\lambda_1,\dots,\lambda_h \in \mathbb{F}_{q^s}$ and coefficients $\theta_j$ satisfying the moment conditions $\sum_{j=1}^h \theta_j \lambda_j^{q^{s-1-i}}=0$ for $i=0,\dots,h-2$ and $\sum_{j=1}^h \theta_j \lambda_j^{q^{s-h}}\ne 0$. Multiplying the trace equations by $\theta_j$ and summing collapses them into a single polynomial in $x$ of degree $B=\frac{q^{st}-1}{q^s-1}q^{s-h}$, so each hyperplane contains at most $B$ of the subspaces; this yields both the rank and the distance bound. The first construction instead uses a partial semifield, a set of $k\times k$ matrices whose nonzero linear combinations are nonsingular, to lift a linear code to an additive code whose distance is at least the $h$-th generalized Hamming weight of the original code.

What would settle it

Check the parameter set $q=2$, $s=h=1$, $t=2$ in Theorem 3: the theorem asserts a $[3,4,d]^1_2$ additive code, i.e. a $4$-dimensional $\mathbb{F}_2$-subspace of $\mathbb{F}_2^3$, which cannot exist; equivalently, the degree bound equals the length, so the rank argument cannot deliver the stated dimension. That single parameter set settles whether the theorem as stated is correct.

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Extended reading notes

Core claim

The paper's central claim is that additive codes with strong distance guarantees can be manufactured from linear codes. Theorem 3 asserts that for $h \le s$ and $t \ge 2$ there exists a $[q^{st}-1, (s(t+1)+1)/h, d]^h_q$ additive code with $d \ge q^{st}-1 - \frac{q^{st}-1}{q^s-1} q^{s-h}$. The construction indexes columns of a generator matrix by nonzero elements of $\mathbb{F}_{q^{st}}$, each column being a rank-$h$ $\mathbb{F}_q$-subspace, and proves via trace functions and the norm $N(x)=x^{(q^{st}-1)/(q^s-1)}$ that every hyperplane contains few of these subspaces. Theorem 4 shows that for $s=h$ and $t=2$ no additive code with length $q^{2h}-1$ and dimension $(3h+1)/h$ can have distance larger than $q^{2h}-q^h-1$, so the constructed code is optimal.

Load-bearing premise

The load-bearing premise is that the polynomial degree bound $B=\frac{q^{st}-1}{q^s-1}q^{s-h}$ is strictly smaller than the code length $n=q^{st}-1$; if $B=n$, a nonzero coefficient vector can produce the zero codeword and the claimed dimension fails, as it does when $q=2$, $s=h=1$, $t=2$.

Editorial extensions

If this is right

  • Theorem 3 gives an infinite family of additive codes with explicit distance lower bounds for every prime power $q$, every $h \le s$, and every $t \ge 2$.
  • For $s=h$ and $t=2$, the family hits the additive Griesmer bound, so the parameters $[q^{2h}-1,(3h+1)/h, q^{2h}-q^h-1]^h_q$ are optimal.
  • The construction generalizes the earlier quaternary $[63,5,45]_2^2$ example and explains why that code exists even though the cyclic-code hypothesis behind it was not satisfied.
  • The codes are shown not to be equivalent to linear codes in all cases checked by computer ($s \le 8$), so they add to the supply of additive codes that can outperform linear codes; for example, no linear $[63,5,45]_4$ code is known.

Reading between the lines

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

  • Because the construction needs only the moment conditions on the $\lambda_j$, varying those choices should produce many inequivalent additive codes with the same certified distance; a computational search over $\mathbb{F}_{q^s}$ could map out the full family.
  • For $s=h$, $t=2$ the distance is settled, but the paper notes that for $h<s$ or $t\ge3$ the additive Griesmer bound does not rule out larger distances; testing those ranges is a natural next step.
  • The non-equivalence to linear codes is verified only for $s\le8$; if the rank obstruction can be proved unconditionally, the construction would give an infinite family of additive codes with no linear counterpart.
  • Because additive codes over $\mathbb{F}_{q^h}$ correspond to subgroups of the Pauli group, a stabilizer construction from these codes could inherit the proven distance bound; checking which parameters yield self-orthogonal or dual-containing codes would make this concrete.
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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 / 4 minor

Summary. The paper presents two constructions of additive codes over finite fields, starting from linear codes. The first construction (Theorem 1) uses a partial semifield and yields an additive code whose minimum distance is bounded below by the h-th generalized Hamming weight of the original linear code; Theorem 2 then converts this into a Griesmer-type lower bound. The second construction (Theorem 3) builds a small-dimensional additive code from the norm and trace functions over extensions, producing an infinite family with an explicit distance lower bound. Section 5 claims optimality for the special case s=h, t=2 by comparing with the additive Griesmer bound from the authors' previous work, and Section 6 discusses non-equivalence to linear codes and known additive codes that outperform linear codes.

Significance. If the main construction is correct, Theorems 1 and 2 are a clean and potentially useful method for turning any linear code with good generalized Hamming weights into an additive code with explicitly controlled distance. The norm-trace construction in Theorem 3 is also an interesting idea that generalizes the quaternary example of Guan, Li, Liu, and Ma. The paper correctly makes use of external published results, including the authors' own prior Griesmer bound, as tools rather than as circular inputs. However, the current statement of Theorem 3 is false in an exceptional but nonempty family, and the optimality claims in Section 5 rest on an arithmetic error. These issues are repairable, but they are load-bearing for the paper's central claims.

major comments (3)
  1. [Theorem 3, §4] Theorem 3 is false as stated for q=2, s=h=1. The proof concludes that no non-zero (a1,a2,a3) gives the zero codeword, but this requires the root bound B = ((q^{st}-1)/(q^s-1)) q^{s-h} to be strictly less than the length n=q^{st}-1. For q=2, s=h=1 one has B=n, so the root bound does not exclude an all-zero codeword. Indeed, taking (a1,a2,a3)=(1,1,0) gives, for every x in F_{2^t}^*, the coordinate value 1 + tr(1*N(x)) + Tr(0) = 1+1 = 0. Hence the constructed code has F_2-rank at most t+1, not the claimed t+2, and for t=2 the claimed [3,4,d]_2 code is impossible by dimension. The theorem can be repaired by adding a hypothesis such as h<s or q^s>2, equivalently q^{s-h}<q^s-1, but as written the infinite family is not established.
  2. [§5, first paragraph] The lower bound reported for s=h, t=2 is arithmetically incorrect. Theorem 3 gives d ≥ q^{2h}-1 - ((q^{2h}-1)/(q^h-1)) q^0 = q^{2h}-q^h-2, not q^{2h}-q^h-1 as asserted in Section 5. The stronger bound is actually false: for q=3, h=1, the codeword corresponding to (a1,a2,a3)=(-1,1,0) has weight 4 = q^{2h}-q^h-2, so the constructed code has distance exactly 4, not at least 5. Therefore the text's claim that Theorem 3 implies d ≥ q^{2h}-q^h-1 is unsupported and contradicted by this example. Section 5 must be rewritten, either by weakening the optimality claim to match the correct lower bound or by supplying a new argument that closes the gap of one.
  3. [Theorem 4 proof, §5] The proof of Theorem 4 applies [2, Theorem 9] with m=3. This choice requires q^{h+1} < d = q^{2h}-q^h, equivalently q^h > q+1, which fails for h=1. Thus the proof as written does not cover h=1. The conclusion of Theorem 4 is true for h=1 by the Singleton bound, but that argument is not the one given in the paper; a separate case or a different choice of m is needed. As it stands the proof is incomplete for part of the parameter range claimed in the theorem.
minor comments (4)
  1. [Theorem 2 proof] The summation in the proof is written as 'j−1X j=0' and should be a sum over i from 0 to j−1; the repeated use of j is confusing.
  2. [Theorem 3 proof] The proof states 'the F_q rank of the code is st+t+1', but the dimension claimed in the theorem is st+s+1; this appears to be a typo and should be corrected.
  3. [Notation, §2] The notation [n,k/h,d]^h_q is defined only informally; for clarity the authors should state explicitly that k is the F_q-dimension and that k/h is allowed to be non-integral.
  4. [Section 6] The statement that non-equivalence to linear codes was 'verified by computer for all s ≤ 8' is not reproducible without code or a detailed verification protocol; please include the computational data or a script.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found; self-citations are used as external tools and do not force the conclusions.

full rationale

The paper's constructions are self-contained. Theorem 1 derives the additive code's distance bound from the generalized Hamming weight of the input linear code via the partial semifield property; Theorem 2 applies the classical Griesmer bound; Theorem 3 constructs explicit rank-h subspaces and bounds the number of zeros by a degree argument. Theorem 4 uses [2, Theorem 9] as an externally published Griesmer-type bound for additive codes. Although [2] shares authors with the present paper, it is a general theorem whose statement does not include the target result, and it is applied as a proof tool rather than as an input that determines the construction. Similarly, Section 6 invokes [1, Lemma 3.6] as an external linear-equivalence criterion and verifies the non-equivalence computationally for small parameters. No equation in the paper reduces to an earlier result by construction, no fitted parameter is relabeled as a prediction, and no uniqueness claim is smuggled in through self-citation. The identified counterexample to Theorem 3 at q=2, s=h=1 is a correctness gap in the root-bound argument, not a circular dependence, and therefore does not affect the circularity score.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central constructions rely on standard facts about finite fields and codes, on the authors' own published Griesmer bound for additive codes, and on a geometric existence assertion for the elements θ_j. No new entities are postulated and no numerical parameters are fitted to data; the only choices are the λ_j, which are shown to exist for the relevant parameter range.

assumptions (6)
  • standard math The Griesmer bound for generalized Hamming weights of linear codes: for a j-dimensional subcode, |Support(D)| ≥ sum_{i=0}^{j-1} ceil(d/q^i).
    Used in the proof of Theorem 2 to bound d_h(C).
  • domain assumption The additive Griesmer bound from [2, Theorem 9]: a [n, r/h, d]^h_q code must satisfy n ≥ ceil(r/h) + d - m + ceil(d/f(q,m)) with the stated parameters.
    Self-cited but published prior work; used in Theorem 4 to prove optimality. Treating it as independent since it is a parameter-free bound from a peer-reviewed paper.
  • domain assumption Existence of partial semifields: sets of k×k matrices over F_q such that any nonzero F_q-linear combination is nonsingular, for the parameters used in Theorem 1.
    Invoked in Theorem 1; the paper cites semifield literature [8,9] for existence.
  • standard math The geometric fact that for h ≤ s one can choose h points in AG(s,q) spanning an (h-1)-dimensional affine subspace not containing the zero vector.
    Used in the proof of Theorem 3 to ensure the matrix Λ is non-singular and the vector θ exists.
  • standard math A nonzero polynomial of degree D over a finite field has at most D roots.
    Used in the proof of Theorem 3 to bound the number of zeros of the trace equation.
  • domain assumption Equivalence criterion: an additive code is equivalent to a linear code iff the associated subspace set is contained in a Desarguesian spread ([1, Lemma 3.6]).
    Used in Section 6 to discuss non-equivalence.

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Cite this review

Pith. "Pith review of Additive codes from linear codes." pith.science (2026). https://pith.science/paper/UR5LLAHD

@misc{pith2026250603805,
  author       = {Pith},
  title        = {Pith review of: Additive codes from linear codes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UR5LLAHD}},
  note         = {Machine review of arXiv:2506.03805}
}
abstract

We introduce two constructions of additive codes over finite fields. Both constructions start with a linear code over a field with $q$ elements and give additive codes over the field with $q^h$ elements whose minimum distance is demonstrably good.

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Reference graph

Works this paper leans on

9 extracted references · 8 canonical work pages

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