Hadamard Test-Based Construction Methods

• Hadamard Test-Based Construction is a method that leverages norm conditions, base sequences, and algebraic tests to systematically produce Hadamard matrices with guaranteed orthogonality.
• It employs combinatorial techniques such as the Goethals–Seidel array and Yang’s multiplication theorems to generate extensive catalogs of nonequivalent matrices.
• Test-based approaches also use polynomial ideals and equivalence transformations to certify orthogonality, with direct implications for error-correcting codes and quantum algorithms.

Hadamard Test-Based Construction refers to a family of methods that use combinatorial or algebraic "test" conditions—often in the form of autocorrelation, orthogonality, or spectral requirements—to systematically construct Hadamard matrices or related objects, such as Hadamard states or codes. Central to these constructions is the use of test sequences, base sequences, or test-based algebraic criteria to generate matrices meeting strict Hadamard properties. These methods are fundamental in combinatorics, coding theory, quantum information, and mathematical physics.

1. The Base Sequence Framework and Goethals–Seidel Construction

A cornerstone of Hadamard test-based construction is the use of base sequences. A quadruple of binary sequences $(A; B; C; D) \in BS(m, n)$, with $A, B$ of length $m$ and $C, D$ of length $n$, is a base sequence if it satisfies a prescribed "norm" condition. The construction proceeds as follows:

• Generate new base sequences in $BS(m+n, m+n)$ by the mapping

$(A; B; C; D) \mapsto (A, C; A, -C; B, D; B, -D),$

where $A, C$ denotes concatenation.

• To form a Hadamard matrix $H$ of order $4d$ (where $d = m+n$), encode $A, B, C, D$ as circulant matrices $Z_0, Z_1, Z_2, Z_3$ and assemble them in the Goethals–Seidel array:

$$H = \begin{pmatrix} Z_0 & Z_1R & Z_2R & Z_3R \ -Z_1R & Z_0 & -Z_3R & Z_2R \ -Z_2R & Z_3R & Z_0 & -Z_1R \ -Z_3R & -Z_2R & Z_1R & Z_0 \end{pmatrix}$$

where $R$ is the back-diagonal permutation matrix. Orthogonality and the Hadamard property $HH^\top = (order)I$ are guaranteed by the norm condition imposed on the base sequences.

In a key specialization, using base sequences from $BS(n+1, n)$ yields Hadamard matrices of order $8n+4$. Theoretical completeness of $BS(8,7)$ classification allows explicit enumeration at order 60: 1012 inequivalent Hadamard matrices are constructed directly, and further expanded to 1759 classes using the transposition map (Djokovic, 2010).

2. Yang’s Multiplication Theorems and Sequence Algebra

Yang’s multiplication theorems enable the systematic combinatorial production of large base sequences or T-sequences by combining normal, near-normal, or base sequences through structured block operations:

• The four theorems produce mappings such as
  • $NS(n) \times BS(s,t) \to TS(d)$,
  • $NS(n) \times BS(s,t) \to BS(d,d)$,
  • $NN(n) \times BS(s,t) \to TS(d)$,
  • $BS(m+1,m) \times BS(n+1,n) \to BS(d,d)$,
  • where $d$ is a function of the input parameters.
• The key property is that algebraic compositions preserve the autocorrelation or norm conditions required for Hadamard construction: e.g., concatenating and interleaving parameterized blocks, so as to produce quadruples of sequences satisfying required test properties (see equations such as

$$X_k = \left[ \frac{-f_kA}{0_{s+t}}, \frac{g_kC + h_kD}{0_{s+t}}, \frac{f_kB}{-h_kC + g_kD}, \dots \right]$$

with coefficients $f_k, g_k, h_k$ determined algorithmically).

• Constructions using Yang’s theorems can be cascaded: output base sequences can be fed back as inputs to further multiplication steps, yielding an extensive catalog of nonequivalent Hadamard matrices (Djokovic, 2010).

3. Algebraic Test-Based Approaches and Polynomial Ideals

A complementary algebraic test-based approach encodes the construction of special subclasses (such as cocyclic Hadamard matrices) as the solution set to a system of polynomial equations—the Hadamard ideal. For a group $G$ of order $4t$:

• Define multivariable polynomials in variables $x_{i,j}$ subject to constraints $x_{i,j}^2=1$, cocycle conditions

$$p_{i,j,k}(X) = x_{i,j} - x_{ij,k} x_{j,k} x_{i,jk},$$

and the cocyclic Hadamard row test

$\sum_{j \in G} x_{i,j} = 0\quad \forall i\neq 1.$

The common zeros of this system in $\{\pm1\}$ correspond one-to-one with the set of cocyclic Hadamard matrices (Álvarez et al., 2016).

• By reparameterizing the variables as group-cocycle coordinates, the dimension of the algebraic system is reduced, and Gröbner basis techniques enable explicit enumeration and algorithmic construction up to large orders (e.g., order 124).
• This methodology is a direct formalization of the "Hadamard test": the validity of a candidate set of cocycles or difference sets is certified by the vanishing of the defining polynomials.

4. Test-Based Procedures in Balancedly Splittable and Symmetric Structures

A further systematic test-based paradigm emerges in the construction of balancedly splittable Hadamard matrices (BSHM). Here, the dot products in a designated submatrix are restricted to exactly two values:

• Given a Hadamard matrix $H$ of order $n$, partition $H$ into submatrices $H_1$ and $H_2$ such that

$H_1^{\top} H_1 = \ell I_n + aA + b(J_n - A - I_n),$

with $A$ a $(0,1)$-matrix. The combinatorial feasibility is governed by the test equation

$n(\ell - a^2) = \ell^2 - a^2,$

together with parity and eigenvalue constraints (Jedwab et al., 2022).

• Only parameter sets passing these "test" constraints admit BSHM constructions; further, strongly regular graph (SRG) structure and partial difference sets (PDS) in abelian $2$-groups are used as combinatorial certifying tools.

For symmetric Hadamard matrices, variations such as the Propus Construction employ circulant or block matrices $A, B = C, D$ satisfying

$A A^\top + 2 B B^\top + D D^\top = 4n I_n,$

and embed these in carefully designed arrays (propus or generalized propus arrays) to achieve symmetry and orthogonality (Seberry et al., 2015, Balonin et al., 2017).

5. The Role of the Transposition Map and Equivalence Classification

A salient feature of test-based construction is the systematic exploitation of equivalence transformations to enlarge the catalog of Hadamard matrices:

• The transposition map $H \mapsto H^\top$ generates new nonequivalent matrices because, in arrangements such as the Goethals–Seidel array, odd permutations of the circulant blocks yield Hadamard matrices not equivalent under signed permutations or simultaneous row/column operations.
• In explicit enumeration (e.g., order 60), starting from a set of $1012$ constructed inequivalence classes, transposing produces an additional $747$ inequivalent classes, boosting the total to $1759$ (Djokovic, 2010).
• The fact that equivalence classes can be broadened by operations as basic as transposition emphasizes both the combinatorial richness and the residual search space left even after systematic test-based generation.

6. Significance, Applications, and Broader Implications

Hadamard test-based constructions—combinatorial and algebraic—enable large-scale, explicit enumeration of nonequivalent Hadamard matrices, with direct ramifications for theory and practice:

• In combinatorial design, test-based methods supply vast families of matrices for constructions of balanced incomplete block designs, difference sets, and related objects.
• In error-correcting coding and signal processing, the diversity of constructed Hadamard matrices allows optimization for spectral properties and robustness.
• The algebraic test methods (as in Gröbner-basis–driven cocyclic Hadamard matrix construction) link group cohomology, polynomial systems, and combinatorial matrix theory for both investigation and practical code generation.
• The adaptability of these constructions (incorporating both sequence-based and algebraic test criteria) suggests their utility in emerging application areas such as random I/O codes and quantum algorithms, where orthogonality, symmetry, and combinatorial diversity are critical.

In conclusion, Hadamard test-based construction encompasses a diverse set of techniques—norm-satisfying base sequences, algebraic polynomial systems, sequence multiplications, group-theoretic difference sets—which are unified by their reliance on verifiable combinatorial or algebraic "test" conditions. These methods have established the practical feasibility of constructing large catalogs of Hadamard matrices and have been instrumental in exploring their applications in design theory, coding, and computational mathematics.