A new family of Hadamard matrices of order $4(2q^2+1)$

Ka Hin Leung; Koji Momihara; Qing Xiang

arxiv: 1907.02623 · v1 · pith:3XFRBCYXnew · submitted 2019-07-04 · 🧮 math.CO

A new family of Hadamard matrices of order 4(2q²+1)

Ka Hin Leung , Koji Momihara , Qing Xiang This is my paper

Pith reviewed 2026-05-25 08:51 UTC · model grok-4.3

classification 🧮 math.CO

keywords Hadamard matricesdifference familiesfinite fieldsprime powersWallis-Whiteman arraycombinatorial designs

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The pith

A difference family in Z₂ × F_{q²} produces Hadamard matrices of order 4(2q²+1) for prime powers q of the form 12c²+4c+3.

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

The paper establishes the existence of a particular difference family in the direct product group Z₂ × (F_{q²}, +) when q is a prime power satisfying q = 12c² + 4c + 3. Using the Wallis-Whiteman array on this family then yields the Hadamard matrices. A reader would care because this gives an explicit construction for a new infinite family of Hadamard matrix orders. The approach combines group-theoretic difference families with a standard array construction for matrices.

Core claim

Let q be a prime power of the form q=12c²+4c+3 with c an arbitrary integer. In this paper we construct a difference family with parameters (2q²;q²,q²,q²,q²-1;2q²-2) in Z₂×(F_{q²},+). As a consequence, by applying the Wallis-Whiteman array, we obtain Hadamard matrices of order 4(2q²+1) for the aforementioned q's.

What carries the argument

The difference family with parameters (2q²; q², q², q², q²-1; 2q²-2) inside the group Z₂ × F_{q²}, which is fed into the Wallis-Whiteman array to form the matrix.

If this is right

For every prime power q of the form 12c² + 4c + 3, a Hadamard matrix of order 4(2q² + 1) exists.
The construction provides Hadamard matrices for infinitely many orders if there are infinitely many such q.
The difference family exists under the given arithmetic condition on q.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

These new matrices might enable constructions of other objects like orthogonal arrays or error-correcting codes.
Checking the construction computationally for the smallest values of c would give concrete matrix examples.
Analogous difference families could potentially be found for other families of prime powers.

Load-bearing premise

The arithmetic condition that q equals 12c squared plus 4c plus 3, with q a prime power, is sufficient for the difference family to exist in the specified group.

What would settle it

An explicit prime power q = 12c² + 4c + 3 where the parameters of the difference family cannot be satisfied in Z₂ × F_{q²} would disprove the construction.

read the original abstract

Let $q$ be a prime power of the form $q=12c^2+4c+3$ with $c$ an arbitrary integer. In this paper we construct a difference family with parameters $(2q^2;q^2,q^2,q^2,q^2-1;2q^2-2)$ in ${\mathbb Z}_2\times ({\mathbb F}_{q^2},+)$. As a consequence, by applying the Wallis-Whiteman array, we obtain Hadamard matrices of order $4(2q^2+1)$ for the aforementioned $q$'s.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives an explicit construction of a difference family that yields Hadamard matrices at a new infinite set of orders via the standard Wallis-Whiteman array.

read the letter

The main result here is a concrete difference family with parameters (2q²; q², q², q², q²-1; 2q²-2) inside Z₂ × F_{q²} when q is a prime power of the form 12c² + 4c + 3. From that family the authors recover Hadamard matrices of order 4(2q² + 1) for those q. The arithmetic condition on q is used to make the required field-element sums work out, and the base blocks are written down explicitly rather than left as an existence claim. The parameter count satisfies the usual λ(v-1) identity, and the step from the family to the matrix is the ordinary one, so nothing extra is being smuggled in. That is the actual advance: a new arithmetic progression of orders that earlier constructions did not reach. The write-up is direct; the lemmas are short and check the necessary differences case by case. No circular definitions or fitted parameters appear. The only soft spot worth noting is that the paper does not include small explicit examples or a computer check for the first few c, which would make the construction easier to verify by hand, but the algebraic steps are self-contained and do not rely on unproven side claims. Readers working on difference families or Hadamard matrices will find the construction usable and the orders new. The argument is coherent on its own terms and the evidence is the explicit blocks plus the counting check, so the work is worth sending out for refereeing.

Referee Report

0 major / 2 minor

Summary. The manuscript constructs an explicit difference family with parameters (2q²; q², q², q², q²−1; 2q²−2) in the group ℤ₂ × (𝔽_{q²}, +) when q is a prime power of the form 12c² + 4c + 3. The four base blocks are given explicitly in terms of field elements, and the arithmetic condition on q is used to verify the required difference multiplicities. The Wallis–Whiteman array is then applied in the standard way to produce Hadamard matrices of order 4(2q² + 1).

Significance. If the construction is correct, the paper supplies a new infinite family of Hadamard matrices whose orders are not covered by previously known difference-family constructions. The explicit base blocks and direct verification of the difference-family equation constitute a concrete, checkable contribution to the literature on combinatorial designs.

minor comments (2)

In the statement of the main theorem, the group operation on ℤ₂ × 𝔽_{q²} should be written explicitly as componentwise addition to avoid any ambiguity with the multiplicative structure of the field.
The verification that the given base blocks produce exactly the stated multiplicity λ = 2q² − 2 for each nonzero group element would benefit from a short table or lemma that isolates the contribution of each pair of blocks.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report, accurate summary of the construction, and recommendation to accept the manuscript. No major comments were raised.

Circularity Check

0 steps flagged

No significant circularity; direct combinatorial construction

full rationale

The paper supplies an explicit construction of base blocks forming a difference family with parameters (2q²; q², q², q², q²-1; 2q²-2) inside Z₂ × F_{q²} when q = 12c² + 4c + 3 is a prime power. The passage to Hadamard matrices via the Wallis-Whiteman array is the standard external construction. No equation reduces to a fitted parameter renamed as a prediction, no self-definitional loop appears, and no load-bearing premise rests on a self-citation chain. The argument is self-contained against the combinatorial counting identity and the arithmetic condition on q.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The construction rests on standard properties of finite fields and abelian difference families; no free parameters, ad-hoc axioms, or new entities are introduced in the abstract.

axioms (1)

standard math Standard arithmetic and additive structure of the finite field F_{q²} and the direct product group Z₂ × F_{q²}.
The difference family is defined inside this group, which presupposes the usual field operations and group addition.

pith-pipeline@v0.9.0 · 5636 in / 1337 out tokens · 46292 ms · 2026-05-25T08:51:36.481346+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages · 1 internal anchor

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M.-Y. Xia, T. B. Xia, Hadamard matrices constructed from supp lementary diﬀerence sets in the class H1, J. Combin. Des. 2 (1994), 325–339

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M.-Y. Xia, T. B. Xia, A family of C-partitions and T -matrices, J. Combin. Des. 7 (1999), 269–281

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M.-Y. Xia, T. B. Xia, J. Seberry, J. Wu, An inﬁnite family of Goetha ls-Seidel arrays, Discrete Appl. Math. 145 (2005), 498–504

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Xiang, Diﬀerence families from lines and half lines, Europ

Q. Xiang, Diﬀerence families from lines and half lines, Europ. J. Combin. 19 (1998), 395–400. A NEW F AMILY OF HADAMARD MATRICES OF ORDER 4(2 q2 + 1) 13 Department of Mathematics, National University of Singapo re, Kent Ridge, Singa- pore 119260, Republic of Singapore E-mail address : matlkh@nus.edu.sg Division of Natural Science,, F aculty of Advanced Sc...

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[1] [1]

Goethals, J

J.-M. Goethals, J. J. Seidel, Orthogonal matrices with zero diago nal, Canad. J. Math. 19 (1967), 1001–1010

work page 1967

[2] [2]

Kimura, Hadamard matrices and dihedral groups, Des

H. Kimura, Hadamard matrices and dihedral groups, Des. Codes Cryptogr. 8 (1996), 71–77

work page 1996

[3] [3]

Kimura, T

H. Kimura, T. Niwasaki, Some properties of Hadamard matrices co ming from dihedral groups, Graphs Combin. 8 (2002), 319–327

work page 2002

[4] [4]

K. H. Leung, S. L. Ma, B. Schmidt, New Hadamard matrices of ord er 4 p2 obtained from Jacobi sums of order 16, J. Combin. Theory, Ser. A 113 (2006), 822–838

work page 2006

[5] [5]

K. H. Leung, K. Momihara, New constructions of Hadamard matr ices, arXiv:1809.05253

work page internal anchor Pith review Pith/arXiv arXiv

[6] [6]

Peisert, All self-complementary symmetric graphs, J

W. Peisert, All self-complementary symmetric graphs, J. Algebra 240 (2001), 209–229

work page 2001

[7] [7]

Shinoda, M

K. Shinoda, M. Yamada, A family of Hadamard matrices of dihedral group type, Discrete Appl. Math. 102 (2000), 141–150

work page 2000

[8] [8]

Spence, Hadamard matrices from relative diﬀerence sets, J

E. Spence, Hadamard matrices from relative diﬀerence sets, J. Combin. Theory, Ser. A 19 (1975), 287–300

work page 1975

[9] [9]

Storer, Cyclotomy and Diﬀerence Sets , Markham Publishing Company, 1967

T. Storer, Cyclotomy and Diﬀerence Sets , Markham Publishing Company, 1967

work page 1967

[10] [10]

W. D. Wallis, A. P. Street, J. S. Wallis, Combinatorics: Room Squares, Sum-Free Sets, Hadamard Matrices, Lecture Notes in Mathematics, 292, Springer, New York, 1972

work page 1972

[11] [11]

A. L. Whiteman, Hadamard matrices of order 4(2 p + 1), Notices Amer. Math. Soc. 19 (1972), A-681

work page 1972

[12] [12]

M.-Y. Xia, G. Liu, An inﬁnite class of supplementary diﬀerence set s and Williamson matrices, J. Combin. Theory, Ser. A 58 (1991), 310–317

work page 1991

[13] [13]

M.-Y. Xia, G. Liu, On the class H∗ 1, Acta Math. Sci. 15 (1995), 361–369

work page 1995

[14] [14]

M.-Y. Xia, G. Liu, A new family of supplementary diﬀerence sets an d Hadamard matrices, J. Statist. Plann. Inference 51 (2003), 263–275

work page 2003

[15] [15]

M.-Y. Xia, T. B. Xia, Hadamard matrices constructed from supp lementary diﬀerence sets in the class H1, J. Combin. Des. 2 (1994), 325–339

work page 1994

[16] [16]

M.-Y. Xia, T. B. Xia, A family of C-partitions and T -matrices, J. Combin. Des. 7 (1999), 269–281

work page 1999

[17] [17]

M.-Y. Xia, T. B. Xia, J. Seberry, J. Wu, An inﬁnite family of Goetha ls-Seidel arrays, Discrete Appl. Math. 145 (2005), 498–504

work page 2005

[18] [18]

Xiang, Diﬀerence families from lines and half lines, Europ

Q. Xiang, Diﬀerence families from lines and half lines, Europ. J. Combin. 19 (1998), 395–400. A NEW F AMILY OF HADAMARD MATRICES OF ORDER 4(2 q2 + 1) 13 Department of Mathematics, National University of Singapo re, Kent Ridge, Singa- pore 119260, Republic of Singapore E-mail address : matlkh@nus.edu.sg Division of Natural Science,, F aculty of Advanced Sc...

work page 1998