Partial Hadamard Matrices: Theory & Applications

• Partial Hadamard matrices are defined as M×N matrices with unit modulus entries whose rows are orthogonal, generalizing classical square Hadamard matrices.
• They are constructed using combinatorial models, Fourier truncation, and balanced multi-splittable techniques, enabling applications in design theory, coding, and quantum algebra.
• Research addresses extension, isolation, and regularity properties, with challenges in completability and classification driving advances in high-dimensional geometry and compressed sensing.

A partial Hadamard matrix is an $M\times N$ matrix, typically over $\{+1,-1\}$ or the unit circle $\mathbb T$, whose $M$ rows are pairwise orthogonal, i.e., $H H^* = N I_M$ where $H^*$ is the conjugate transpose. Partial Hadamard matrices generalize the classical Hadamard structure from square matrices to rectangular settings and arise in combinatorics, design theory, quantum algebra, coding, and high-dimensional geometry. Their study encompasses construction, classification, applications, and the connections to design-theoretic structures and quantum symmetries.

1. Formal Definitions and Equivalences

Let $H\in M_{M\times N}(\mathbb T)$, i.e., $M$ rows and $N$ columns with entries of unit modulus. The defining orthogonality condition is

$H H^* = N I_M$

equivalently, for distinct rows $R_i$ and $R_j$,

$$\langle R_i, R_j \rangle = 0,\quad \|R_i\|^2 = N \quad(\forall i)$$

In the real case ($H\in\{\pm1\}^{M\times N}$) the same formula applies, and the rows are orthogonal in $\mathbb R^N$. For square $M=N$ the standard Hadamard matrix is recovered; for $M<N$ one has a truncation. Column-negation equivalence states that flipping any subset of columns preserves the partial Hadamard status (Launey et al., 2010).

Completeness problems ask when a partial Hadamard can be extended by additional, mutually orthogonal, unit-modulus rows to a full Hadamard matrix. In the case $M=N-1$, a minor condition holds: $|\det H(j)|$ for the $(N-1)\times(N-1)$ minors must be independent of $j$ and equal $N^{N/2-1}$ for completability (Banica et al., 2013).

2. Construction Methodologies and Combinatorial Models

Many construction techniques are based on combinatorial and algebraic motifs:

• Ito’s Hadamard Graph Model: Vertices are signed vectors in $\{\pm1\}^{4t}$, orthogonality induces the edge structure, and cliques correspond to sets of mutually orthogonal vectors. The induced subgraph $G_t$ (those orthogonal to $R_1, R_2, R_3$ from a reference Hadamard) is used; maximal cliques yield large $(m+3)\times 4t$ partial Hadamard matrices (Álvarez et al., 2012).
• Fourier Matrix Truncations: Arbitrary $M\times N$ submatrices of a complex Hadamard (e.g., Fourier) retain the orthogonality properties, and primitive-set conditions determine when the submatrix itself is complex Hadamard (Herr et al., 2019).
• Balancedly Multi-Splittable Structures: There is an equivalence between the existence of a projective plane of order $p\equiv3\pmod4$ and a balancedly multi-splittable, embeddable $p^2\times p(p+1)$ partial Hadamard matrix. Explicit constructions use quadratic character sums indexed over points and slopes in $GF(p)^2$ (Kharaghani et al., 2023).
• Partitioning Hadamard Vectors: The set of all Hadamard vectors in $\mathbb R^m$ (up to sign) can be partitioned into Hadamard matrices if and only if $m=2^n$; for other $m$ only partial partitions are possible (Casazza et al., 2016).

Algorithmically, both exhaustive and heuristic clique-search methods have been employed, including local search, genetic algorithm adaptations, and block-wise heuristics (Álvarez et al., 2012). Asymptotic bounds exist: for large $t$ there are partial matrices of size approximately $(4t)/3 \times 4t$ with polynomial-time construction (Álvarez et al., 2012).

3. Extension, Isolation, and Regularity Properties

The extension problem asks when a partial Hadamard can be completed to a full Hadamard matrix. For real matrices with $N-M\leq7$, extension is always possible, though not always for general complex matrices (Banica et al., 2013). In quantum information, the existence of unextendible maximally entangled bases (UMEBs) in $\mathbb C^d\otimes\mathbb C^d$ corresponds to the non-completability of certain partial Hadamard matrices; $(d-1)\times d$ matrices can always be completed, dimensions $d=4n+1$ admit explicit non-completable construction (Wang et al., 2016).

Isolation is characterized by defect: the algebraic manifold $\mathcal C_{M,N}$ of partial Hadamards admits deformations. If $H$ admits only $M+N-1$ trivial deformations, it is isolated—there are no infinitesimal non-equivalent neighbors (Banica et al., 2017). Master Hadamard matrices with separable structure $H_{ij}=\lambda_i^{n_j}$ have explicit defect formulas; only Fourier matrices for $N$ prime are genuinely isolated (Banica et al., 2017).

Regularity refers to formal vanishing sums of roots of unity: a partial Hadamard is regular if all row-orthogonality relations decompose as formal sums of cycles. The Butson class—matrices where all entries are roots of unity—is conjectured to be regular, and regular matrices are conjectured to be affine deformations of Butsons (Banica et al., 2017, Banica et al., 2013).

4. Applications in Design Theory, Coding, and Quantum Algebra

Partial Hadamard matrices are pervasive in

• Design Theory: They manifest in block designs, orthogonal arrays, and underpin the geometry of finite projective planes. Balancedly multi-splittable matrices correspond exactly to existence conditions for finite projective planes of certain orders (Kharaghani et al., 2023).
• Coding Theory: Partitioning Hadamard vectors into full matrices provides constructions for Hadamard codes and frames used in signal sets (low cross-correlation) and tight $2$-frames (Casazza et al., 2016).
• Quantum Algebra: The quantum algebraic viewpoint associates to any $H\in M_{M\times N}(\mathbb T)$ a quantum semigroup of partial permutations. The submagic matrix $P_{ij}=\mathrm{Proj}(R_i/R_j)$ encapsulates the algebraic symmetries, and completion problems in this context correspond to questions of extension to magic unitaries and quantum groups (Banica et al., 2013, Banica et al., 2017).

In high-dimensional geometry, embeddable partial Hadamard matrices yield maximal systems of equiangular lines, often achieving asymptotic optimality relative to the Gerzon bound (Kharaghani et al., 2023).

5. Entropic and Sensing Applications

Partial Hadamard matrices are exploited in compressed sensing:

• Entropy Preserving Sensing: For i.i.d. discrete sources, a deterministic partial Hadamard ($H_S$—a small subset of rows from a full Hadamard) suffices to preserve entropy nearly losslessly even with vanishing sampling rate ($|S|/N\to0$) (Haghighatshoar et al., 2012). Row selection is guided by conditional entropy under the polar transform.
• Continuous Sources: No nontrivial dimensionality reduction is possible for continuous distributions; the measurement rate must tend to $1$ for reliable reconstruction (Haghighatshoar et al., 2012). Theoretical support is via discrete entropy power inequalities and martingale absorption arguments.

High computational efficiency ensues, with $O(N\log N)$ encoding and decoding via the fast Hadamard transform, and the chosen submatrices being well-conditioned for robust numerical reconstruction.

6. Classification, Parameterization, and Compatibility Criteria

Classification of submatrices that preserve the Hadamard property is governed by primitive-set compatibility:

• Primitive Sets and Cyclotomic Divisibility: Given row and column selections $J,K$, $H_{J,K}$ is Hadamard if and only if the cyclotomic polynomials $\Phi_s(z)$ for all nontrivial primitives of $J$ divide the column polynomial $K(z)$ (Herr et al., 2019).
• Compatibility Graphs: The collection of all submatrices of a Fourier matrix partitions into equivalence classes indexed by unordered primitive set pairs $(\mathcal P_m(J), \mathcal P_m(K))$. Compatibility graphs $G(m,n)$ encode feasible matches; all submatrices in the same class share Hadamard status, facilitating combinatorial enumeration (Herr et al., 2019).

For $n=2,3$ explicit $p$-adic order tests are available. Larger dimensions currently lack closed-form parameterizations, and the systematic study is an open field.

7. Open Problems, Conjectures, and Future Directions

Outstanding open questions include:

• Extension and Non-completable Cases: Comprehensive characterization of non-completable partial Hadamards in complex settings remains incomplete, though the UMEB correspondences are suggestive (Wang et al., 2016).
• Regularity and Affine Butson Conjecture: The conjecture that every regular partial Hadamard is an affine deformation of a Butson matrix is open (Banica et al., 2017, Banica et al., 2013).
• Classification for Large Parameters: The full primitive-set compatibility graph for large $n$ is not parameterized (Herr et al., 2019).
• Embeddability and Projective Planes: For $q\equiv1\pmod{4}$ and composite $n$, the existence of balancedly multi-splittable partial Hadamard matrices is linked to unsolved questions in finite geometry (Kharaghani et al., 2023).
• Quantum Semigroup Deformations: The classification and isolation in the context of submagic matrices and their associated quantum semigroups is open (Banica et al., 2017, Banica et al., 2013).

The study of partial Hadamard matrices thus encapsulates deep connections between combinatorics, algebra, geometry, quantum symmetries, and information theory, with vivid interplay between structure, randomness, and rigidity across mathematics and its applications.