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The Star Product of Uniformly Random Codes

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Pith's one-line read The star product of two uniformly random linear codes almost always attains the maximum possible dimension min{k1k2, n}, asymptotically as the field size grows, and under an admissibility condition also as the code dimensions grow.

desk verdict Solid q→∞ half with a clean kernel formula; Section 5's dimension-growth claim doesn't follow — the puncturing step is invalid and Corollary 5.12 needs concentration beyond E|ker|→2. read the letter →

arxiv 2511.17236 v2 pith:VWNJWFOS submitted 2025-11-21 cs.IT math.IT

Johan Vester Dinesen , Ragnar Freij-Hollanti , Camilla Hollanti , Benjamin Jany , Alberto Ravagnani This is my paper · ORCID
classification cs.ITmath.IT MSC 94B0594B6515A69
keywords starproductSchurHadamardrandomlinearcodesbilinearformsexpecteddimensionasymptoticmaximalitycode-basedcryptography
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

Two random linear codes, when multiplied coordinatewise, produce a third code whose dimension is usually the largest possible: the smaller of the product of the two dimensions and the ambient length. The paper proves this asymptotic maximality in two regimes: as the field size q tends to infinity, the star-product dimension equals min{k1k2, n} with probability tending to 1; and as the code dimensions grow, the same conclusion holds provided the second dimension does not grow too quickly relative to the first. The proof converts the star product into the image of a linear map that evaluates bilinear forms on random column pairs, reducing the question to counting zero-diagonal matrices of fixed rank. The result matters because star-product dimension controls the efficiency of private information retrieval and secure distributed matrix multiplication, and because deviations from maximal dimension are exactly what code-based distinguishers look for.

What carries the argument

The evaluation map ψ_{G1,G2} from the space of bilinear forms on F_q^{k1} × F_q^{k2} to F_q^n, which sends a form to its values on the column pairs of systematic generator matrices; its image is exactly C1⋆C2, so |ker ψ| determines the dimension defect. Counting |ker ψ| means counting k1×k2 matrices with zero diagonal of each rank r, via the sets S_r^{k1,k2} and q-binomial identities. Jensen's inequality converts the exact expectation of log_q |ker ψ| into a lower bound on the expected dimension; Markov's inequality converts the bounded kernel expectation into high probability of full dimension.

What would settle it

Fix q=2, k1=k2=3, n=7 (so k1k2=9 > n). If Theorem 5.11 were true, P(dim(C1⋆C2)=7) would be at least 1−((2q−1)/q^2)^t = 1−(3/4)^t for large t, tending to 1. A direct computation over F_2^7, either by enumeration or by sampling many random systematic pairs and computing the star-product dimension, could check whether the probability approaches 1; a limiting value strictly below 1 would refute the claimed implication. More directly, verify whether an extended pair with appended random columns can have a full 9-dimensional star product while its 7-coordinate punctured star product has dimension 6;

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Extended reading notes

Core claim

The paper's central claim is that for independent uniformly random linear codes C1,C2 ≤ F_q^n of dimensions k1,k2, the expected star-product dimension is asymptotically maximal: lim_{q→∞} P(dim(C1⋆C2)=min{k1k2,n})=1 (Corollary 4.2), and under an admissibility condition on monotone dimension functions, E[dim(C1⋆C2)] = min{k1(t)k2(t),n(t)}+o(1) (Corollary 5.12). The identification of C1⋆C2 with the image of the evaluation map ψ on bilinear forms gives dim(C1⋆C2) = k1k2 − log_q |ker ψ|, and the expected kernel size is computed exactly by enumerating zero-diagonal k1×k2 matrices of each rank. From this expression, lower bounds on the expected dimension, and concentration through Markov's inequal

Load-bearing premise

The load-bearing premise is that in the regime where the ambient length is smaller than the product of the code dimensions, a full-dimensional star product of the punctured codes guarantees a full-dimensional star product after randomly extending the codes; the proof of Theorem 5.11 asserts this but only establishes a bound that is automatically true, so the n(t) < k1k2 part of the growth result is unsupported as written.

Editorial extensions

If this is right

  • As q→∞, for any fixed n,k1,k2, the star product of two uniformly random codes fills F_q^n when k1k2 ≥ n, and otherwise has dimension exactly k1k2, with probability approaching 1.
  • Under the admissibility condition (k2 not growing too fast relative to k1), the same maximality holds as the code dimensions grow: E[dim(C1⋆C2)] = min{k1k2,n} + o(1).
  • The explicit lower bound of Corollary 3.11 gives a certified estimate for all finite parameters, not just asymptotics, and matches Monte Carlo simulations even at small q and n.
  • For PIR/SDMM, random code pairs yield an upper bound on the PIR rate of 1−min{k1k2,n}/n and a lower bound on the SDMM recovery threshold of min{k1k2,N}, so random codes cannot beat structured MDS constructions in these metrics.
  • Random binary codes of sufficiently large dimension give no high-distance binary CSS-T quantum codes, because their star-square fills the ambient space and leaves no room for the T-gate code C2.

Reading between the lines

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

  • The proof of the n(t)<k1k2 case (Corollary 5.12 via Theorem 5.11) rests on the unproven implication that a full-dimensional punctured star product forces a full-dimensional extended star product; the provided argument only embeds the dual vector, which yields a vacuous bound. If this implication cannot be repaired, the dimension-growth theorem remains established only for n(t) ≥ k1k2.
  • The same bilinear-form counting strategy should extend to star products of three or more random codes, giving analogous maximal dimension results for iterated Schur products; the paper only treats pairs.
  • In code-based cryptography, the near-maximality of random star products gives a clean quantitative baseline: a candidate code pair whose star-product dimension falls noticeably below min{k1k2,n} is statistically distinguishable from random, which could sharpen existing square-based distinguishers.
  • The abstract promises explicit asymptotic upper bounds on the variance of the star-product dimension, but the included text does not appear to contain that analysis; pinning down the variance would let practitioners convert the Markov-based concentration into tail bounds with explicit constants.
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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 the expected dimension of the star product C1⋆C2 of two independent uniformly random linear codes of dimensions k1 and k2 in F_q^n. Using the correspondence between star products and evaluations of bilinear forms, the authors obtain an exact expression for E|ker ψ| (Theorem 3.9), deduce a Jensen lower bound on E[dim C1⋆C2] (Corollary 3.11), and show that as q→∞ the star product has the maximal possible dimension min{k1k2,n} with probability tending to 1 (Corollary 4.2). They then address the regime in which k1,k2 grow with t under an 'admissibility' condition, claiming in Theorem 5.7 that E|ker ψ|→2 and in Theorems 5.9/5.11 and Corollary 5.12 that E[dim C1⋆C2]=min{k1k2,n}+o(1). The final section discusses applications to PIR, SDMM, CSS-T codes, and cryptanalysis.

Significance. If the dimension-asymptotic results were correct, the paper would give a clean two-parameter extension of the square-of-random-code analysis of Cascudo et al. [18]. The exact enumeration in Theorem 3.9 is parameter-free, the lower bound in Corollary 3.11 is checked against Monte Carlo estimates in Table 1, and the q-asymptotic Corollary 4.2 is a solid, self-contained contribution. The manuscript is honest about the technical nature of its admissibility condition and about the overlap with [26]. However, the proofs supporting the dimension-growth claim contain serious gaps, especially in Theorem 5.11 and in the passage from Theorem 5.7 to Corollary 5.12, so the paper's headline dimension-asymptotic result is currently not established. The q-asymptotic part alone would still be a useful contribution.

major comments (3)
  1. [Theorem 5.11] The step 'M implies N' is not proved and is in fact unsupported. Let m=k1k2 and let t be as in the proof. The assumed failure dim(C1⋆C2)<n yields a nonzero v∈(C1⋆C2)⊥; embedding it as ṽ=(v,0,...,0) shows ṽ∈(C̃1⋆C̃2)⊥. But this only implies dim(C̃1⋆C̃2) ≤ m+t−1, which is vacuous because C̃1⋆C̃2 is spanned by m products and hence already has dimension at most m. It does not imply dim(C̃1⋆C̃2)<m, so M^c does not imply N^c. Consequently P(M)≥P(N) does not follow, and the entire n(t)<m(t) case of Corollary 5.12 lacks support.
  2. [Corollary 5.12 (n=m case)] In the regime n(t)=m(t)=k1(t)k2(t), Theorem 5.7 gives only E|ker ψ|=2+o(1). By Corollary 5.10, P(dim≤m−ℓ−1) ≤ (2+ε)/q^{ℓ+1}; summing over ℓ yields E[m−dim] ≤ (2+ε)/(q−1), a positive constant for fixed q, not o(1). The fact that |ker| is a power of q does not close the gap: a distribution with P(|ker|=q)=1/q and P(|ker|=1)=1−1/q has expectation 2 and positive expected deficit. A second-moment or concentration argument is required. The same obstruction applies whenever n(t)−m(t) is bounded, since Theorem 5.9 then gives only a constant lower bound. Thus the claimed o(1) in Corollary 5.12 is unproven.
  3. [Theorem 5.9] The proof as printed contains a sign / law-of-total-probability error. It derives P(N) ≤ ((2q−1)/q^2)^{n−m} (E|ker ψ'|−1), which is an upper bound on the success probability, whereas the theorem claims P(N) ≥ 1−((2q−1)/q^2)^{n−m}. The union bound should be applied to the complement: P(N^c|E_j) ≤ (j−1)((2q−1)/q^2)^{n−m}, hence P(N^c) ≤ ((2q−1)/q^2)^{n−m}(E|ker ψ'|−1). Unless this is a typographical error and the text is corrected, the proof of Theorem 5.9 is invalid.
minor comments (5)
  1. [Definition 3.8] The 'uniformly random code' model samples only codes whose first k coordinates form an identity prefix, i.e., complements of a fixed coordinate subspace. The text argues robustness across models, but a formal statement or a short proof of equivalence to the uniform subspace model would strengthen the paper.
  2. [Theorem 3.9 / Eq. (3.5)] The multiple summations are very hard to parse. Please define all summation ranges explicitly and consider simplifying the displayed formula, e.g., by introducing auxiliary functions for the q-binomial factors.
  3. [Table 1 and surrounding text] The sentence 'our bound is appears to be good' is ungrammatical. Also, report the Monte Carlo standard errors and state the number of samples in the caption (currently given only in the text).
  4. [Remark 5.8] The conjecture lim_{t→∞} E|ker ψ| = exp((q−1)k2(t)(k1(t)−1)/q^{k1(t)})+1 has t-dependent quantities inside the claimed limit. Please clarify the intended asymptotic notation.
  5. [Corollary 5.12] State explicitly that o(1) is as t→∞ with q fixed, and specify which additional growth conditions on n(t)−m(t) are needed for the conclusion to follow from the proved estimates.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main estimates come from new counting arguments and external published counting lemmas; the self-citations are not load-bearing in a circular sense, and the proof gaps identified are correctness issues rather than reductions by construction.

full rationale

No step in the claimed derivation reduces a prediction to its own inputs. Theorem 3.9's expected-kernel formula is derived from the bilinear-form correspondence (proved in Theorem 3.2) and from the enumeration of zero-diagonal matrices in Lemma 3.5. Lemma 3.5's first identity is quoted from [22, Corollary 60]; although [22] shares an author with the present paper, it is a published, parameter-free enumeration result whose stated assumptions do not include the target theorem, so it counts as independent evidence rather than circularity. Corollary 3.11 uses Jensen's inequality in the standard direction and is not an equality-by-construction with Corollary 4.2. Corollary 4.2 follows from the q→∞ limit of E[|ker|] plus Markov's inequality, with no fitted parameters or hidden dependence on the conclusion. The dimension-growth results rest on an explicit admissibility condition (Definition 5.6), candidly described as a technical limitation; this is an assumption, not a circular import. Remark 5.13 explicitly acknowledges the overlapping prior work [26] and compares parameter ranges rather than renaming it. The relevant vulnerabilities are mathematical gaps rather than circularity: Theorem 5.11's puncturing step does not establish the implication needed for P(M)≥P(N), because a nonzero vector in the extended orthogonal complement only gives dim(C~1⋆C~2)≤m+t−1, which is vacuous given that the extended star product is already spanned by m products; and Corollary 5.12's o(1) gap does not follow from the first-moment estimate E|ker|→2 in the n=m regime. These are gaps in the proof of the claimed asymptotic result, not instances of a result being equivalent to its input by construction. Accordingly, the circularity score is 0.

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

The paper introduces no free parameters and no new entities. It relies on standard probabilistic and enumerative tools, plus one explicit technical restriction (admissibility). The main external dependency is [22]'s matrix enumeration, which is a co-author's prior result but not constructed for this paper.

assumptions (5)
  • standard math The enumeration of k1×k2 matrices over F_q of rank r with zero diagonal (Lemma 3.5) is as stated in [22, Corollary 60].
    The expectation formula in Theorem 3.9 and the Jensen lower bound (Corollary 3.11) rest on this count. The paper cites it to prior published work by one of the present authors and does not re-prove it.
  • domain assumption The systematic-generator model (Definition 3.8) is representative of uniform random codes.
    All theorems are stated for codes sampled by uniform random systematic generator matrices. The paper claims robustness to other random models by analogy with [18, Remark 2.1] without giving a proof.
  • ad hoc to paper Admissibility (Definition 5.6): k1(t), k2(t) strictly increasing, k1 ≤ k2, and k1k2/q^{α k1} → 0 for all α < 2 − log_q(2q−1).
    Imposed in Section 5 to make the kernel-counting bounds converge. The authors explicitly call it a technical limitation rather than an intrinsic necessity.
  • standard math q-binomial bounds from [25]: q^{k(n−k)} ≤ [n choose k]_q ≤ (7/2) q^{k(n−k)}.
    Used in Proposition 5.5 to prove E[dim(C1∩C2)] → 0 when n(t) = k1k2.
  • standard math Jensen's inequality applied to log_q of the kernel expectation (Corollary 3.11).
    Converts the exact expectation of |ker ψ| into a lower bound on the expected star-product dimension.

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Cite this review

Pith. "Pith review of The Star Product of Uniformly Random Codes." pith.science (2026). https://pith.science/paper/VWNJWFOS

@misc{pith2026251117236,
  author       = {Pith},
  title        = {Pith review of: The Star Product of Uniformly Random Codes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VWNJWFOS}},
  note         = {Machine review of arXiv:2511.17236}
}
read the original abstract

We consider the problem of determining the expected dimension of the star product of two uniformly random linear codes that are not necessarily of the same dimension. We use a correspondence between the star product and the evaluation of bilinear forms to provide an explicit lower bound on the expected star product dimension. We prove that the expected dimension asymptotically reaches its maximum possible value as the field size increases. Furthermore, we show that the same maximal dimension is achieved asymptotically as the code dimensions increase, subject to a condition bounding their relative growth rates. We also analyze the variance of the star product dimension, providing explicit asymptotic upper bounds. Finally, we discuss the implications of these results for private information retrieval, secure distributed matrix multiplication, quantum error correction, and cryptanalysis.

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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.