Skew Hadamard Conjecture Overview

• Skew Hadamard Conjecture is a dual concept in design and matrix theory, linking abelian difference sets with skew Hadamard matrices.
• In abelian design, counterexamples to the classical Paley uniqueness are provided by non-Paley constructions using Dickson polynomials.
• The matrix existence problem remains open, with computational evidence and symplectic ETF links guiding current research.

The expression Skew Hadamard Conjecture has two established meanings in the literature. In abelian design theory, it refers to the classical assertion that, up to equivalence, the only skew Hadamard difference sets in abelian groups are the Paley difference sets. In matrix theory, it refers to the existence conjecture for skew Hadamard matrices. Both usages are organized around skew decompositions—either a finite abelian group as $D\cup(-D)\cup\{0\}$, or a Hadamard matrix as $H=I_n+S$ with $S^\top=-S$—but their current status is different: the abelian uniqueness conjecture has been disproved by several infinite families of non-Paley examples, while the matrix existence conjecture remains open and has been linked to real symplectic equiangular tight frames (Momihara, 2013, Ding et al., 2013, Cati et al., 2024, Fallon, 17 Sep 2025).

1. Definitions and scope

In the difference-set setting, let $(G,+)$ be a finite group of order $|G|=v$. A $k$-subset $D\subseteq G$ is a $(v,k,\lambda)$ difference set if every nonzero $g\in G$ can be written in exactly $\lambda$ ways as $H=I_n+S$0 with $H=I_n+S$1. Two difference sets $H=I_n+S$2 with the same parameters in an abelian group $H=I_n+S$3 are called equivalent if there is an automorphism $H=I_n+S$4 and an element $H=I_n+S$5 such that $H=I_n+S$6. A difference set is called skew Hadamard when $H=I_n+S$7 is the disjoint union

$H=I_n+S$8

with $H=I_n+S$9 (Momihara, 2013).

In the matrix setting, a real $S^\top=-S$0 matrix $S^\top=-S$1 with entries in $S^\top=-S$2 is a Hadamard matrix if

$S^\top=-S$3

It is skew Hadamard if, in addition,

$S^\top=-S$4

equivalently,

$S^\top=-S$5

A well-known counting argument implies that a skew Hadamard matrix can exist only for $S^\top=-S$6 or $S^\top=-S$7 (Cati et al., 2023, Fallon, 17 Sep 2025).

Usage	Statement	Status in the cited literature
Abelian difference sets	Up to equivalence, only the Paley difference sets occur	Disproved
Skew Hadamard matrices	A skew Hadamard matrix exists for every $S^\top=-S$8 or $S^\top=-S$9	Open

The terminological overlap is substantive rather than accidental. Recent work shows that the matrix conjecture is equivalent to a symplectic ETF existence conjecture, while the difference-set literature provides constructions, invariants, and inequivalence criteria that clarify why the abelian conjecture failed in its original form (Fallon, 17 Sep 2025, Momihara, 2013).

2. Paley difference sets and the classical abelian conjecture

The Paley construction is the basic template in the abelian theory. Let $(G,+)$0 be a prime power, $(G,+)$1, and

$(G,+)$2

Then $(G,+)$3 is a skew Hadamard difference set with parameters

$(G,+)$4

This family supplied, for a long time, the only known infinite family in abelian groups (Momihara, 2013, Ding et al., 2013).

Two related conjectures were formulated in the abelian case. One states that if an abelian group $(G,+)$5 admits a skew Hadamard difference set, then $(G,+)$6 must be elementary abelian. The other, described as the classical Skew Hadamard Conjecture, states that up to equivalence the only skew Hadamard difference sets in abelian groups are the Paley difference sets (Momihara, 2013).

The Paley family also provides a diagnostic benchmark. In that case, counting arguments show that the triple intersection numbers

$(G,+)$7

take only two distinct values as one varies the exponent $(G,+)$8. In particular, the multiset

$(G,+)$9

has size at most $|G|=v$0. This very small range became a useful test for distinguishing new skew Hadamard difference sets from Paley examples (Momihara, 2013).

3. Refutation by non-Paley constructions

The classical abelian uniqueness conjecture was disproved in 2006 by Ding and Yuan. For every odd $|G|=v$1 and every $|G|=v$2, they constructed

$|G|=v$3

as a skew Hadamard difference set in $|G|=v$4, where $|G|=v$5 is the first-kind Dickson polynomial of order $|G|=v$6. The key fact was that $|G|=v$7 is a planar function on $|G|=v$8 for $|G|=v$9 odd; planar functions over fields of characteristic $k$0 directly yield skew Hadamard difference sets by taking their image sets (Ding et al., 2013).

A further family arises from Dickson polynomials of order $k$1. For $k$2 and $k$3,

$k$4

If $k$5 is odd and $k$6, then for every $k$7,

$k$8

is a skew Hadamard difference set in $k$9 with

$D\subseteq G$0

Here the proof is explicitly different from the $D\subseteq G$1-case because $D\subseteq G$2 is not planar in $D\subseteq G$3 (Ding et al., 2013).

The nonplanar proof proceeds through additive-character sums, Gauss sums, and Stickelberger’s theorem. One rewrites

$D\subseteq G$4

introduces the quadratic character $D\subseteq G$5, expands in multiplicative characters, and applies $D\subseteq G$6-adic valuation estimates via Stickelberger’s theorem. The paper’s technical core is a pair of digit-sum inequalities in ternary expansion, together with carry-analysis lemmas, that force the required congruences (Ding et al., 2013).

The same work states that these sets are inequivalent to all existing ones for $D\subseteq G$7 by comparing triple intersection numbers, and that the construction gives the third infinite family of skew Hadamard difference sets in abelian groups. It also proves that every $D\subseteq G$8 is equivalent to exactly one of

$D\subseteq G$9

This establishes that non-Paley skew Hadamard difference sets are not isolated anomalies but belong to systematic families (Ding et al., 2013).

4. Triple-intersection invariants and infinite inequivalence results

A decisive refinement in the inequivalence theory is the use of triple intersection numbers modulo a prime. Fix a prime $(v,k,\lambda)$0, let $(v,k,\lambda)$1 be a primitive element of $(v,k,\lambda)$2, let $(v,k,\lambda)$3 be skew Hadamard, and fix a non-square $(v,k,\lambda)$4 with $(v,k,\lambda)$5. For each exponent $(v,k,\lambda)$6 with $(v,k,\lambda)$7, define

$(v,k,\lambda)$8

If $(v,k,\lambda)$9 is equivalent to $g\in G$0, then the multiset

$g\in G$1

coincides modulo $g\in G$2 with

$g\in G$3

Thus the residue-class multiset mod $g\in G$4 is an equivalence invariant (Momihara, 2013).

For cyclotomic constructions, the invariant is computed via character sums. If $g\in G$5 is a multiplicative character of order $g\in G$6 on $g\in G$7 and $g\in G$8 is a union of $g\in G$9th-order cyclotomic classes, then the indicator function of $\lambda$0 is expanded in multiplicative characters, and the triple-intersection size becomes a sum involving terms of the form

$\lambda$1

Weil’s bound and Davenport–Hasse lifting are then used to control these sums and reduce them modulo $\lambda$2 (Momihara, 2013).

The main lifting statements are formulated for Feng–Xiang skew Hadamard difference sets. If $\lambda$3 is such a set in $\lambda$4, $\lambda$5 is its lift to $\lambda$6, $\lambda$7 is the odd prime dividing the cyclotomic order $\lambda$8, and $\lambda$9 is an odd prime with $H=I_n+S$00, then Theorem 7 states that the number of distinct residues in

$H=I_n+S$01

is preserved by lifting to $H=I_n+S$02. Theorem 11 extends this to arbitrary odd $H=I_n+S$03 via base-$H=I_n+S$04 reduction to an odd $H=I_n+S$05. In both cases, one obtains lift-invariance of the number of distinct residue classes in the triple-intersection multiset (Momihara, 2013).

The concrete example in the paper takes $H=I_n+S$06, $H=I_n+S$07, cyclotomic order $H=I_n+S$08, and

$H=I_n+S$09

For $H=I_n+S$10, a computer check gives

$H=I_n+S$11

so modulo $H=I_n+S$12 these values collapse to at least three distinct residues. Because Paley difference sets always give at most two residues modulo any prime $H=I_n+S$13, this example is inequivalent to Paley, and by Theorems 7 and 11 the same remains true for its odd-prime lifts and, after the appropriate base-$H=I_n+S$14 reduction, for every odd $H=I_n+S$15 of arbitrary size. The paper concludes that there are infinitely many skew Hadamard difference sets in elementary abelian groups that are not equivalent to Paley difference sets (Momihara, 2013).

5. The matrix existence conjecture and constructive evidence

In matrix theory, the Skew Hadamard Conjecture asserts that a skew Hadamard matrix of order $H=I_n+S$16 exists if and only if $H=I_n+S$17 or $H=I_n+S$18; another formulation includes the order $H=I_n+S$19 case explicitly as admissible (Cati et al., 2024, Cati et al., 2023, Fallon, 17 Sep 2025). Unlike the classical abelian difference-set conjecture, this existence statement remains unresolved.

The constructive record has, however, become extensive. A 2023 SageMath implementation of Hadamard-matrix constructions reported that all known constructions produce skew Hadamard matrices in every multiple of $H=I_n+S$20 up to $H=I_n+S$21 except $H=I_n+S$22 exceptional orders, and that within the $H=I_n+S$23 range the order $H=I_n+S$24, long claimed to be known, required a fix because the published reference actually gave a non-skew construction (Cati et al., 2023).

A later database paper substantially sharpened that picture. It provides constructions covering orders $H=I_n+S$25 of all known Hadamard and skew Hadamard matrices in SageMath, verifies the correctness of results given in the literature, and states that within this range just one order, $H=I_n+S$26, of a skew Hadamard matrix claimed to have a known construction, required a fix. By combining all constructions in the literature, it gives for every odd $H=I_n+S$27 a concrete skew Hadamard matrix of order $H=I_n+S$28, where the exponent $H=I_n+S$29 is minimal and tabulated, and concludes that up to order $H=I_n+S$30 every admissible order $H=I_n+S$31 is covered by at least one explicit construction (Cati et al., 2024).

The same source explains the role of the minimal exponent $H=I_n+S$32 through Paley-type constructions and its relation to Riesel numbers. If, for a given odd $H=I_n+S$33, one can solve

$H=I_n+S$34

with $H=I_n+S$35 a prime or prime power, then Paley I yields a skew Hadamard matrix of order $H=I_n+S$36. Conversely, if $H=I_n+S$37 is a Riesel number, then $H=I_n+S$38 is composite for every $H=I_n+S$39, so no Paley-I construction can cover the orders $H=I_n+S$40. As a specific by-product, the paper shows that the Paley constructions of skew-Hadamard matrices do not work for the order $H=I_n+S$41 for any $H=I_n+S$42 (Cati et al., 2024).

These database results do not resolve the conjecture, but they delimit it sharply: the matrix existence problem is still open in general, while the constructive frontier through explicit algorithms and software verification is already broad and systematically organized.

6. Symplectic reformulation and current research directions

A recent reformulation places the matrix conjecture inside symplectic frame theory. Let $H=I_n+S$43 denote $H=I_n+S$44 equipped with the standard symplectic form

$H=I_n+S$45

A finite sequence $H=I_n+S$46 is a frame if its frame operator $H=I_n+S$47 is invertible; it is $H=I_n+S$48-tight when $H=I_n+S$49; and it is equiangular if its Gram matrix satisfies $H=I_n+S$50 for $H=I_n+S$51. A frame that is both tight and equiangular is an ETF (Fallon, 17 Sep 2025).

In this setting, the symplectic Gerzon bound states that if $H=I_n+S$52 is equiangular of size $H=I_n+S$53, then $H=I_n+S$54. The admissible pairs $H=I_n+S$55 for symplectic ETFs are

$H=I_n+S$56

This yields the Symplectic Zauner Conjecture, asserting that a $H=I_n+S$57 ETF in $H=I_n+S$58 exists exactly for those pairs (Fallon, 17 Sep 2025).

The central equivalence theorem is that, for $H=I_n+S$59, there exists a $H=I_n+S$60 ETF in $H=I_n+S$61 if and only if there is a skew Hadamard matrix of order $H=I_n+S$62, and there exists a $H=I_n+S$63 ETF in $H=I_n+S$64 if and only if there is a skew Hadamard matrix of order $H=I_n+S$65. In the $H=I_n+S$66 case, the Gram matrix becomes a skew conference matrix; in the $H=I_n+S$67 case, the argument passes through tournaments, Seidel matrices, and counts of induced $H=I_n+S$68-vertex subgraphs called diamonds. Saturation of the sharp upper bound

$H=I_n+S$69

forces the tournament to be switching-equivalent to a doubly-regular tournament, after which a block-matrix construction yields the corresponding skew conference matrix and hence the desired Hadamard matrix (Fallon, 17 Sep 2025).

Several open directions remain active. In the difference-set literature, one problem is to prove inequivalence for the “starter” Feng–Xiang sets by purely theoretical means, without computer enumeration of intersection numbers. Another is to find other skew Hadamard difference-set constructions with the lifting property beyond the Paley and Feng–Xiang families. A third is to decide equivalence or inequivalence for other recently discovered constructions, such as those of Ding–Pott–Wang and Muzychuk, using similar invariants (Momihara, 2013). On the matrix and frame side, future directions include numerical and combinatorial searches for symplectic ETFs, for example via frame-potential optimization, and the exploration of complex-to-symplectic “shadow” constructions relating complex SIC-POVMs to real symplectic packing problems (Fallon, 17 Sep 2025).

Taken together, these developments produce a bifurcated but coherent picture. The classical abelian uniqueness conjecture has failed decisively, through explicit non-Paley families and through invariants that remain stable under lifting. The matrix existence conjecture, by contrast, survives, now supported by large-scale constructive evidence, software verification, and an exact reformulation in symplectic frame theory.