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Asymmetric SICs over finite fields

T0 review · 1 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper proves a Hadamard-matrix construction that yields infinitely many new SICs over finite fields and exhibits the first SICs with trivial automorphism groups.

desk verdict A clean construction of finite-field SICs from modular Hadamard matrices, with a solid main theorem and an asymmetric-symmetry claim that rests on reproducible computation the paper does not quite ship. read the letter →

arxiv 2506.20778 v1 pith:FJJ4IRIT submitted 2025-06-25 math.MG math.CO

Joseph W. Iverson , Dustin G. Mixon This is my paper · ORCID
classification math.MGmath.CO MSC 05B2011T30
keywords equiangularlinesSIC-POVMfinitefieldsmodularHadamardmatricesautomorphismgroupsWeyl–HeisenbergsymmetrytotallyasymmetricSICZauner'sconjecture
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

The paper proves a general recipe for building symmetric, informationally complete sets of vectors, known as SICs, over finite fields. Starting from any modular Hadamard matrix of order $d$ with $d \equiv 8$ modulo the ground-field characteristic, it forms $d^2$ vectors by perturbing each column with the same one-coordinate factor, and these vectors always form a SIC with the same fixed parameters. Because Hadamard matrices are abundant, the construction yields infinitely many new SICs over every field $\mathbb{F}_{p^2}$ with $p \equiv 3 \pmod{4}$, and at least two of them, in dimension 36 over $\mathbb{F}_{49}$, have no nontrivial automorphism at all. This matters because every previously known SIC, over any field, carried Weyl–Heisenberg symmetry; the paper's examples show that symmetry is not a necessary feature of SICs.

What carries the argument

The load-bearing object is the modular Hadamard matrix together with the deformation $x_{ij} = h_j \circ (1 + z e_i)$, in which the $j$-th column $h_j$ is pointwise multiplied by a vector that is $1+z$ in one coordinate and 1 everywhere else. This deformation converts the row and column orthogonality of $H$ into exact inner-product conditions: any two distinct constructed vectors have cross inner-product product 16, the diagonal inner product is 12, and the resolution identity is $96I$. A second object, the block matrix $\widetilde{H}$ whose $(i,k)$ block is $r_k r_i^T$, mediates the symmetry analysis: symmetries of the SIC are trapped between symmetries of $H$ and symmetries of $\widetilde{H}$.

What would settle it

Independently recompute the Gram matrix for the vectors produced from the two order-36 Hadamard matrices in Example 23 and search for any index permutation preserving all inner products up to the equivalence allowed in Definition 13; finding even one nontrivial permutation would destroy the totally asymmetric example. For the infinite-family claim, verifying that a Paley-constructed Hadamard matrix of order $d$ satisfies $H^T H = dI$ over the relevant prime and that $d \equiv 8 \pmod{p}$ would confirm or refute each individual dimension.

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Extended reading notes

Core claim

The central claim is Theorem 2: for a field $K$ with characteristic not 2, a dimension $d \equiv 8 \pmod{\operatorname{char} K}$, and any modular Hadamard matrix $H \in K^{d \times d}$ (entries with square 1 and $H^T H = dI$), the vectors $x_{ij} = h_j \circ (1 + z e_i)$ with $z = -2(1+i)$ form a $(12,16,96)$-SIC. The proof computes the Gram matrix entry by entry: diagonal inner products equal 12, off-diagonal products of mutually conjugate inner products equal 16, and the sum of the rank-one projectors is $96I$. Feeding in known families of Hadamard matrices gives infinitely many dimensions for each prime $p \equiv 3 \pmod{4}$, and feeding in two order-36 matrices from a published catalog gives SICs over $\mathbb{F}_{49}$ with trivial strong and weak automorphism groups. The paper also proves a sandwich theorem showing that automorphisms of the SIC always sit between automorphisms of the Hadamard matrix and automorphisms of a related block matrix.

Load-bearing premise

The strongest concrete claim of a totally asymmetric SIC rests on the computer-based automorphism-group computation for the two order-36 SICs in Example 23 being correct; if those computations or the underlying catalog entries are wrong, only Conjecture 24 remains, while the infinite-dimension claim separately relies on Paley's Hadamard constructions and Dirichlet's theorem.

Editorial extensions

If this is right

  • For every prime $p \equiv 3 \pmod{4}$, SICs exist in infinitely many dimensions over $\mathbb{F}_{p^2}$, extending the previously known case $p = 3$.
  • Conditional on the modular Hadamard conjecture, SICs over some finite field exist for almost every dimension $d$.
  • For $p = 3$, the construction gives SICs in every dimension $d \equiv 2 \pmod{6}$, a much larger family than the earlier one with $d = 2^{2k+1}$.
  • The order-36 examples over $\mathbb{F}_{49}$ are the first SICs with trivial automorphism groups, so finite-field SICs can be totally asymmetric.
  • The construction turns any modular Hadamard matrix of suitable order into a SIC, so future progress on modular Hadamard matrices automatically produces more SIC dimensions.

Reading between the lines

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

  • Beyond the paper, the same deformation could be tested with values of $z$ other than $-2(1+i)$; if any other choice works in positive characteristic, the fixed parameter triple $(12,16,96)$ is not special to the construction.
  • Beyond the paper, the sandwich theorem suggests a concrete search strategy for complex SICs without Weyl–Heisenberg symmetry: take a complex Hadamard matrix with a small automorphism group and try the same deformation, since the paper only shows why this fails below dimension 8.
  • Beyond the paper, if totally asymmetric SICs are truly abundant over finite fields, asymmetry itself is not an obstruction to informationally complete equiangular systems, which weakens the heuristic that complex SICs should be sought only among symmetric examples.
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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

1 major / 5 minor

Summary. This paper constructs SICs (symmetric, informationally complete sets of lines) over finite fields from modular Hadamard matrices. Theorem 2 states that if K has characteristic p, d ≡ 8 mod p, and H ∈ K^{d×d} satisfies H_ij^2 = 1 and H^T H = dI, then the vectors x_ij = h_j ∘ (1 + z e_i) with z = -2(1+i) form a (12,16,96)-SIC in L^d, where L = K(i). The proof is an explicit Gram-matrix and frame-operator computation. From this, Corollary 6 gives infinitely many SICs over F_{p^2} for each prime p ≡ 3 mod 4, using Paley's Hadamard constructions and Dirichlet's theorem, and Corollary 10 gives, conditionally on the p-modular Hadamard conjecture, SICs over some finite field for almost every dimension. Section 3 analyzes symmetries, establishing an automorphism-group sandwich (Theorem 15) and a graph criterion (Proposition 17) for computing SIC automorphism groups. Section 4 gives computational examples, including Example 23, which claims totally asymmetric SICs over F_49 from Spence–Turyn Hadamard matrices #23 and #24, and states Conjectures 22 and 24.

Significance. Assuming the computational claims are correct, the paper makes two notable contributions. Corollary 6 is an unconditional infinite-family result for finite-field SICs over each F_{p^2} with p ≡ 3 mod 4, extending previous results that were limited to p = 3. Example 23 provides the first claimed finite-field SICs with trivial automorphism group, directly addressing the question of whether SICs can exist without Weyl–Heisenberg symmetry. The central construction is elementary and self-contained: Theorem 2 is proven by a clean Gram-matrix computation with no fitted parameters, the infinite families follow from standard Paley/Dirichlet arguments, and the conditional statement in Corollary 10 is explicitly labelled as resting on Conjecture 9. The automorphism-group sandwich and the graph criterion are useful tools that reduce SIC symmetry computations to finite graph-isomorphism problems, and the paper is careful to distinguish theorems, computational observations, and conjectures.

major comments (1)
  1. [§4, Example 23] The claim that matrices #23 and #24 from the Spence–Turyn catalog [27] produce totally asymmetric SICs is the only evidence for the abstract's assertion that 'some of our new SICs exhibit trivial automorphism groups.' The manuscript does not include the matrices, the GAP/GRAPE scripts, or the outputs, and the claim depends on a correct transcription of the external catalog and a correct encoding of the graphs Γ_ω in Proposition 17. If any of these steps is wrong, the exhibited examples of total asymmetry disappear and only Conjecture 24 remains. Please provide the matrices (or an explicit construction), the computational verification scripts, and the resulting automorphism-group output, or an independent certificate of the claimed triviality.
minor comments (5)
  1. [§4, after Conjecture 22] The sentence 'We omit our proof for the sake of brevity' concerns a nontrivial containment PΓL(2,q) ≤ Aut_s(\tilde H); since this is presented as progress on Conjecture 22, the proof should be supplied in an appendix or the statement should be explicitly labelled as a computational observation.
  2. [§4, Example 23] The phrase 'three have trivial weak automorphism group' is ambiguous, because matrix #3's SIC is then said to have weak automorphism group of order 2; clarify whether the first phrase refers to the weak automorphism group of the Hadamard matrix or of the SIC.
  3. [§2, Example 7] The p = 3 classification of modular Hadamard matrices is cited to [20], whose title concerns Hadamard matrices modulo 5; please verify the citation, which may need to be [19] or a different source.
  4. [§3, Definition 13] The condition on the field automorphism γ is typeset incorrectly; it should state that γ commutes with conjugation, i.e., γ(ᾱ) = overline{γ(α)} for every α ∈ L.
  5. [Throughout] The congruence 'd ≡ 8 mod char K' is nonstandard when char K = 0; a parenthetical clarification that this means d = 8 in the characteristic-zero case would help the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 2 derives the SIC conditions directly from the Gram matrix of an explicitly defined vector family, and Corollary 6 follows from Paley and Dirichlet without recycling the conclusion.

full rationale

The main result (Theorem 2) is a direct construction: for any modular Hadamard matrix H, the vectors x_ij = h_j ∘ (1 + z e_i) are defined, and Lemma 3 computes their Gram matrix explicitly from H_ij^2 = 1 and H^T H = dI. The proof of Theorem 2 then verifies each SIC axiom by substitution; no parameter is fitted and no target consequence is used as an input. Corollary 5 and Corollary 6 are unconditional consequences that invoke external mathematical facts (Paley's Hadamard constructions and Dirichlet's theorem), not the conclusion being derived. The symmetry analysis in Theorem 15 and Proposition 17 is also proved from the Gram-matrix structure; the sandwich ı(Aut_w(H)) ≤ Aut_s({x_ij}) ≤ Aut_w({x_ij}) ≤ Aut_s(˜H) follows from the paper's own lemmas, with no reliance on a self-citation to force the result. Self-citations to [6], [12], [13], and [14] provide background or context (Gerzon bound, tight projective designs, roux discussion, 2-transitive SICs) and are not load-bearing for the new construction. The only external dependence of consequence is Example 23's use of Spence and Turyn's catalog matrices (#3, #23, #24) and GAP/GRAPE computations to assert trivial automorphism groups; that is an empirical reproducibility dependency, not a circular derivation, since the assertion is not smuggled in through the paper's own definitions. Consequently, the derivation chain is self-contained conditional on the supplied Hadamard matrix, and no circular step can be exhibited.

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

The paper introduces no fitted parameters: z = -2(1+i) and the SIC values (12,16,96) follow from the Gram-matrix algebra. Background assumptions are Paley's Hadamard constructions, Dirichlet's theorem, the conditional p-modular Hadamard conjecture, and the reliability of the Spence-Turyn catalog and GAP computations. No new entities are postulated.

assumptions (6)
  • standard math Paley's construction I yields a Hadamard matrix of order q+1 for every prime power q ≡ 3 mod 4.
    Used in Corollary 6, Case I, to obtain Hadamard matrices of order d = q+1.
  • standard math Paley's construction II yields a Hadamard matrix of order 2(q+1) for every prime power q ≡ 1 mod 4.
    Used in Corollary 6, Case II (p=7), to obtain order d = 2(q+1).
  • standard math Dirichlet's theorem on arithmetic progressions: there are infinitely many primes in any residue class a mod m with gcd(a,m)=1.
    Used in Corollary 6 to guarantee infinitely many primes q in the specified congruence classes.
  • domain assumption The p-modular Hadamard conjecture (Kuperberg 2016, Conjecture 5.1) that for odd prime p, d×d p-modular Hadamard matrices exist for all but finitely many d satisfying the necessary parity/quadratic-residue condition.
    Assumed only for the conditional density result Corollary 10, which is explicitly labeled as conditioned on this conjecture.
  • domain assumption The Spence-Turyn catalog of 24 Goethals-Seidel Hadamard matrices of order 36 (file gs.36) is accurate, and the GAP/GRAPE computations of their automorphism groups are correct.
    Supports Example 23's claim of totally asymmetric SICs; this is external computational data.
  • domain assumption The field K has characteristic not 2 and t^2+1 does not split in K, so for K=F_p with p ≡ 3 mod 4 the extension L=F_{p^2} is used.
    Basic setting throughout, from Definition 1 and Section 1.

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Pith. "Pith review of Asymmetric SICs over finite fields." pith.science (2026). https://pith.science/paper/FJJ4IRIT

@misc{pith2026250620778,
  author       = {Pith},
  title        = {Pith review of: Asymmetric SICs over finite fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FJJ4IRIT}},
  note         = {Machine review of arXiv:2506.20778}
}
abstract

Zauner's conjecture concerns the existence of $d^2$ equiangular lines in $\mathbb{C}^d$; such a system of lines is known as a SIC. In this paper, we construct infinitely many new SICs over finite fields. While all previously known SICs exhibit Weyl--Heisenberg symmetry, some of our new SICs exhibit trivial automorphism groups. We conjecture that such \textit{totally asymmetric} SICs exist in infinitely many dimensions in the finite field setting.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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