【Hadamard Matrices: The Magic Square of Information—How to Build a Noise-Resistant World】Hadamard Matrices: The Magic Square of Information—How to Build a Noise-Resistant World

+1 and -1. That square grid woven by just two numbers might at first glance look like a simple numerical puzzle. However, this mathematical structure—the "Hadamard matrix"—is precisely what rescues images of Mars arriving from the harshness of space from a sea of noise, prevents interference in mobile networks where hundreds of millions connect simultaneously, and is now quietly controlling the explosion of computational complexity and errors at the core of cutting-edge technologies like massive Large Language Models (LLMs) and quantum computers.

Why is this "magic square of information," discovered by 19th-century mathematicians out of pure curiosity, able to bring such a powerful order called "orthogonality" to the physical world? Why can a mere sequence of signs act as a shield against the universal law of entropy increase? In this report, we will uncover the full picture and future of the Hadamard matrix, an unsung mathematical hero supporting the foundations of modern civilization, ranging from NASA's Mariner program to the latest generative AI quantization techniques, and even the "unsolved problem of order 668" that continues to baffle mathematicians as of 2025.

Chapter 1 Introduction: The Boundary Between Order and Decay

1.1 The Battle Between the Physical Quantity Called Information and Entropy

There is one inescapable law in the universe we inhabit: the second law of thermodynamics, or the law of entropy increase. Things with form crumble, organized information is buried in noise, and order heads toward chaos. This is equally true in the worlds of communication engineering and information theory; from electrical signals flowing through cables and radio waves flying through space to the delicate superposition states of qubits, all "information" is constantly exposed to the pressure of "decay" known as environmental noise.
To resist this irreversible flow and to accurately transmit, store, and process information, humanity has acquired the weapon of mathematics. Within that arsenal, there exists a tool that is exceptionally simple yet holds powerful magic. That is the "Hadamard matrix."

1.2 The Magic of "Orthogonality" Woven by +1 and -1

The components of a Hadamard matrix are surprisingly simple. There are only two types of elements: +1 and -1. However, when these simple values are arranged within a square matrix grid according to a specific rule, a powerful mathematical order called "orthogonality" emerges.
Geometrically, orthogonality means being "perpendicular." Vectors that are orthogonal to each other do not interfere with each other's components. The fact that each row (or column) of the matrix possesses this property suggests that signals or codes generated by a Hadamard matrix can be completely separated and restored through mathematical operations, no matter how much they are superimposed. This makes it possible to perform a task akin to picking a jewel dropped in muddy water out by distinguishing it from every single particle of mud.

1.3 Structure and Purpose of This Report

This report clarifies, along with technical mechanisms, how this "magic square," born from 19th-century pure mathematical inquiry, supported 20th-century space exploration and communication revolutions, and became embedded in the core of cutting-edge technologies like 21st-century quantum computers and Large Language Models (LLMs).
How did NASA's Mariner program rescue images of Mars from a sea of noise? How do mobile phone networks, where hundreds of millions talk simultaneously, prevent interference? And how does the latest AI suppress the explosion of computational costs? Behind these seemingly unrelated phenomena, the principles of "orthogonality" and "scattering" of the Hadamard matrix are always at work. In this paper, based on facts and theory, we will comprehensively discuss how this mathematical structure rescues the real world from decay.

Chapter 2 Mathematical Origins: The Maximum Determinant Problem and Perfect Harmony

2.1 Sylvester's Pavement: A Discovery Ahead of Its Time

Perhaps it is an irony of history that the first discoverer of the Hadamard matrix was not Jacques Hadamard. In 1867, the British mathematician James Joseph Sylvester published a paper with the strange title "Anallagmatic Pavement"1.
Sylvester focused on the combinatorial beauty possessed by a square array where +1 and -1 were arranged like a checkerboard. He discovered that in this array, constructed according to a certain rule, comparing any two rows revealed that the number of matching elements and non-matching elements were exactly equal. In modern terms, this is nothing other than the discovery of "orthogonality," where the inner product of row vectors becomes zero. However, in the mathematical world of that time, this discovery was regarded merely as a type of pure mathematical puzzle, and it would take a century for its engineering applications to blossom4.

2.2 Hadamard's Maximum Determinant Problem

In 1893, the French mathematician Jacques Hadamard was working on an important problem in analysis. The problem was: "In an $n$-th order square matrix $A$ whose elements are complex numbers within the unit disk (absolute value of 1 or less), what is the maximum value of the absolute value of its determinant, $|\det(A)|$?"
Geometrically, a determinant represents the "volume" of an $n$-dimensional parallelepiped spanned by the column vectors (or row vectors) that make up the matrix. If the lengths of the vectors are limited, how can the volume be maximized? Intuitively, the volume should be maximized when the vectors are perpendicular (orthogonal) to each other. This is the same logic as why the area of a rectangle is larger than that of a parallelogram.
After a rigorous proof, Hadamard derived the following inequality (Hadamard's inequality):
$$|\det(A)| \le n^{n/2}$$
He then identified the following two points as the necessary and sufficient conditions for this equality to hold (= for the volume to be maximized):

The absolute value of every element in the matrix is $1$ (i.e., the elements are $+1$ or $-1$).
Every row (and column) of the matrix is orthogonal to each other.

A matrix that satisfies these conditions is what would later be called a "Hadamard matrix," named after him5.

2.3 Deep Dive into Definitions and Basic Properties

Let us formally describe the mathematical definition of a Hadamard matrix $H_n$ again. $H_n$ is a square matrix of order $n$ that satisfies the following conditions:
$$H_n \in {+1, -1}^{n \times n}$$
$$H_n H_n^T = n I_n$$
Here, $H_n^T$ is the transposed matrix and $I_n$ is the identity matrix. This equation, $H_n H_n^T = n I_n$, is the source of the "magic" of Hadamard matrices.
The fact that the diagonal components are $n$ means that the magnitude (square of the norm) of each row vector is $n$, and the fact that the off-diagonal components are $0$ means that different row vectors are orthogonal to each other1.
Furthermore, Hadamard matrices have a concept called "Normalization." Even if you flip the signs of rows or columns, or swap their order, the properties of the Hadamard matrix (orthogonality) are not lost. Using this operation, a matrix where all elements in the first row and first column are set to $+1$ is called a normalized Hadamard matrix.
In a normalized Hadamard matrix, if $n \ge 4$, any row (excluding the first row) contains exactly $n/2$ of $+1$ and $-1$ each. This "balance" becomes an extremely important property for removing DC offsets (direct current components) in later communication applications.

2.4 Sylvester's Recursive Construction Method

What Sylvester discovered, and what serves as the foundation for generating Hadamard matrices, is the recursive construction method (Sylvester's Construction). This is described using the Kronecker Product4.
The simplest first-order Hadamard matrix $H_1$ is $$.
From here, every time the order is doubled, the matrix is expanded as follows:
$$H_{2n} = \begin{pmatrix} H_n & H_n \ H_n & -H_n \end{pmatrix}$$
Specifically:
$$H_2 = \begin{pmatrix} 1 & 1 \ 1 & -1 \end{pmatrix}$$
$$H_4 = \begin{pmatrix} 1 & 1 & 1 & 1 \ 1 & -1 & 1 & -1 \ 1 & 1 & -1 & -1 \ 1 & -1 & -1 & 1 \end{pmatrix}$$
This fractal-like self-similar structure is not only beautiful but also extremely important from the perspective of computational algorithms. By utilizing this structure, the "Fast Walsh-Hadamard Transform (FWHT)" becomes possible, which calculates matrix-vector products at high speed, dramatically reducing the computational complexity from the usual $O(n^2)$ to $O(n \log n)$. Without this, modern real-time communication and image processing applications would likely have been impossible8.

Chapter 3 Return from Space: Mariner 9 and the Victory of Error Correction

3.1 The Distant Mars and the Wall of Noise

In 1971, NASA's unmanned probe Mariner 9 was attempting to reach the orbit of another planet (Mars) for the first time in human history. The core of the mission was to transmit clear images of the Martian surface back to Earth. However, there was a physical barrier standing in the way. The distance between Mars and Earth is over 100 million kilometers, and the radio waves transmitted are extremely weak. The communication channel was filled with "noise," such as radiation flying through space, solar wind, and thermal noise emitted by receiving antennas on Earth10.
In previous missions up to Mariner 6 and 7, relatively simple encoding methods were used, but with the improvement in image resolution, the required bit error rate (BER) was a strict $10^{-5}$ (one error per 100,000 bits) or less. If raw data were sent as is, bits would be flipped by noise, and the image would be buried in noise like a sandstorm. Here, salvation through "redundancy" and "orthogonalization" of information became necessary10.

3.2 Fusion of Reed-Muller Codes and Hadamard Matrices

To solve this problem, NASA adopted an error-correcting code that applied the properties of Hadamard matrices, specifically a $$ code based on the first-order Reed-Muller code $R(1,)$ (sometimes referred to simply as "Hadamard codes" or "biorthogonal codes" in literature)10.
The mechanism of this encoding is as follows:

Blocking of information: The image data to be transmitted is divided into blocks of 6 bits each. 6 bits of information have $2^6 = 64$ possible patterns (numerical values from 0 to 63).
Assignment of codewords: For each of these 64 patterns, a specific 32-bit bit string (codeword) is assigned.
Use of Hadamard matrices: The codewords used here are a total of 64 vectors, consisting of the row vectors (32) of a 32nd-order Sylvester-type Hadamard matrix $H_{32}$ and vectors (32) with each of their bits flipped.

In a normal binary representation, 6 bits of information would be sent with a length of 6 bits. However, Mariner 9 transmitted them stretched to 32 bits, which is about 5 times the length. This "redundancy," which might seem inefficient at first glance, is the key to creating resistance to noise.

3.3 The Phenomenal Correction Capability of the Code

The parameters of this code $$ have the following meanings.

Code length $n=32$: The number of bits transmitted.
Information points $k=6$: The actual amount of information.
Minimum distance $d=16$: When comparing any two different codewords, the number of differing bits (Hamming distance) is at least 16.

A minimum distance of 16 means that if up to 7 errors (bit flips) occur during communication, the correct codeword can theoretically be identified ($(16-1)/2 = 7.5$).
Since the row vectors of a Hadamard matrix are orthogonal, if we consider them based on elements ($+1/-1$), the number of matching and non-matching positions is equal (16 each). In the binary ($0/1$) world, this corresponds to a Hamming distance of $n/2 = 16$10.

3.4 High-Speed Decoding with the "Green Machine"

Receiving stations on Earth (Deep Space Network) receive extremely weak signals. When receiving 32 bits (which are actually analog values) filled with noise, the receiver calculates the **correlation** between the received signal and all "64 possible codewords that could have been sent."
Here, the properties of the Hadamard matrix shine once again. Instead of calculating the correlation 64 times by brute force, an algorithm called the "Fast Hadamard Transform (FHT)" can be used to dramatically reduce the computational load. At the time, this high-speed processing device was called the "Green Machine"12.
When passed through the FHT, among the 64 outputs, only the output corresponding to the correct transmitted codeword shows a prominently large value (even if there is noise). The other 63 outputs become values close to zero due to orthogonality. Through this "Maximum Likelihood Decoding," NASA succeeded in restoring surprisingly clear images of "Inca City" and Olympus Mons from the Martian sandstorms. This was a monumental moment where mathematical abstraction conquered physical noise10.

Chapter 4: The Communication Revolution: CDMA and the Solution to the "Cocktail Party" Problem

4.1 The Tragedy of the Commons of Radio Waves

At the end of the 20th century, with the explosive spread of mobile phones (cellular communication), the telecommunications industry faced a serious problem: how to share the limited resource of "frequency bandwidth" among many users.
In the first-generation analog systems and the second-generation (2G) TDMA (Time Division) / FDMA (Frequency Division) systems, time or frequency was divided and assigned to each user. However, as the number of users increased, the "slices" became finer, and limits were quickly reached.
The technology that broke through this wall was "CDMA (Code Division Multiple Access)," a foundational technology for third-generation mobile communication systems (3G) centered on the Hadamard matrix16.

4.2 Orthogonal Spreading with Walsh Codes

The principle of CDMA is often compared to a "cocktail party." In a situation where many people are talking at the same time in one room (the same frequency band), how can you hear only a specific person's voice? The answer of CDMA is to "use a different language (code) for each pair."
The "different languages" adopted for this are the "Walsh Codes" generated from each row of the Hadamard matrix16.
For example, 64 types of mutually orthogonal codes are generated from the 64th-order Hadamard matrix $H_{64}$. The base station multiplies (spread-modulates) the data for user A with code $W_A$ and the data for user B with code $W_B$ before transmitting.
$$S_{\text{total}} = (\text{Data}_A \times W_A) + (\text{Data}_B \times W_B) + \dots$$
Although all signals are mixed in space, when user A's terminal multiplies the received signal by its own code $W_A$ and integrates (despreads) it, a miracle occurs.
$$S_{\text{received}} \times W_A = (\text{Data}_A \times W_A \cdot W_A) + (\text{Data}_B \times W_B \cdot W_A) + \dots$$
Here, due to the orthogonality of the Hadamard matrix, $W_B \cdot W_A = 0$, and the signal components of others disappear completely. On the other hand, the autocorrelation $W_A \cdot W_A = 64$, and only one's own signal emerges strongly.
Due to this "Processing Gain," CDMA realized highly secure communication that is extremely resistant to noise and interference from others. IS-95 (cdmaOne), led by Qualcomm, and the subsequent 3G standards were truly the implementation of the orthogonality of the Hadamard matrix into social infrastructure18.

4.3 Transition to 4G/5G and the Inheritance of Orthogonality

Since the 2010s, communication standards have shifted from CDMA to LTE (4G) and then to 5G, and the multiple access method has also become mainstream with OFDM (Orthogonal Frequency Division Multiplexing).
Why did CDMA give up its leading role? The biggest reason is the influence of "multipath." In urban areas, radio waves reflect off buildings, and reflected waves that arrive late cause interference. The orthogonality of codes by the Hadamard matrix is guaranteed only when timing is perfectly synchronized, and that orthogonality is easily broken by delayed waves20.
On the other hand, OFDM uses the orthogonality of "frequency" and has strong resistance to multipath by providing a guard interval (Cyclic Prefix). However, this does not mean that the Hadamard matrix has disappeared from communication. Even within OFDM systems, Walsh-Hadamard sequences and their derivatives are still incorporated as essential elements for reducing the Peak-to-Average Power Ratio (PAPR) and encoding specific control signals (such as ACK/NACK signals). While changing form, the philosophy of "signal separation by orthogonality" continues to flow at the foundation of modern communication20.

Chapter 5: The Magic of Light: Spectroscopic Measurement and Fellgett's Advantage

5.1 From "One by One" to "All at Once"

Hadamard matrices have been causing a quiet revolution not only in the world of communications but also in the field of "spectroscopic measurement," which probes the properties of matter using light.
Conventional spectrometers (dispersive type) use prisms or diffraction gratings to split light into a rainbow, then measure the intensity wavelength by wavelength—"red," "green," "blue"—through a slit. This is called the "scanning method." However, this method has a drawback. At any given moment, all light (energy) other than the wavelength being measured is discarded. Especially in infrared spectroscopy, where signals are weak, there was a problem where the amount of light entering the detector was too low, causing it to be buried in noise23.

5.2 Fellgett's Advantage (Multiplex Advantage)

This is where "Hadamard transform spectroscopy" comes in. In this method, instead of a slit that passes only a single wavelength, a "multi-slit (coded mask)" based on Hadamard matrix patterns (e.g., transmitting +1, blocking -1, or reflecting with a mirror) is used.
This allows light of multiple wavelengths to be "mixed" and guided to the detector simultaneously. For example, when measuring N wavelength bands, approximately half of the light from the wavelength bands is received simultaneously in each of the N measurements.
The measured value y is the product of the original spectrum x and the Hadamard matrix H.
$$y = H x$$
To find the original spectrum x from this, one simply multiplies by the inverse matrix (x = H^-1 y).
What is surprising is the fact that this "mix and measure" method provides a better S/N ratio (signal-to-noise ratio) than measuring one by one. This is called Fellgett's Advantage or multiplex advantage23.
Theoretically, when detector noise is dominant, the S/N ratio improves by approximately sqrt(N)/2 times compared to the scanning method. The larger the number of wavelength divisions N, the greater the benefit.

5.3 Single-Pixel Camera: Seeing the World with One Point

The application of this principle to 2D images is the "single-pixel camera."
While normal digital cameras have sensors with millions of pixels (CCD/CMOS), it is extremely difficult and expensive to create high-pixel sensors for special wavelength bands such as infrared or terahertz waves.
A single-pixel camera has only one light-receiving element (photodiode). Instead, a spatial light modulator such as a DMD (Digital Micromirror Device) is placed in front of the lens, and the total amount of light is measured while rapidly switching Hadamard pattern masks against the scenery26.

Display Hadamard pattern P1 -> Measure light intensity I1
Display Hadamard pattern P2 -> Measure light intensity I2
...

When an inverse Hadamard transform is applied to the data sequence obtained by repeating this, an image is mathematically reconstructed as if it were taken with a camera with millions of pixels. This technology is becoming indispensable for imaging in special environments, such as see-through photography in fog, medical microscopes, and even resource exploration satellites. Texas Instruments' DLP technology and others have commercialized products that apply this principle of Hadamard spectroscopy28.

Chapter 6: Into the Quantum World: Superposition and Grover's Search

6.1 Qubits and Hadamard Gates

In the 21st century, humanity has begun to race toward the realization of a "quantum computer" that applies the strange properties of quantum mechanics to computation. In this cutting-edge field, the Hadamard matrix sits in circuit diagrams as a physical "gate." That is the "Hadamard gate (H gate)"30.
Bits in classical computers can only take the state of either "0" or "1." However, qubits can realize a "superposition state" that takes both states simultaneously.
The operation that creates this superposition is the Hadamard gate.
The Hadamard gate for one qubit is represented by the following matrix.
$$H = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \ 1 & -1 \end{pmatrix}$$
Applying this gate to the initial state |0> = (1 0)^T:
$$H|0\rangle = \frac{1}{\sqrt{2}} (|0\rangle + |1\rangle) \equiv |+\rangle$$
This state |+> becomes "0" with a 50% probability and "1" with a 50% probability when observed. In other words, it "diffuses" determined information into a cloud of possibilities. Applying a Hadamard gate to all n qubits instantly generates a state where all 2^n possible states are equally superimposed. This is the starting point for quantum parallel computing32.

6.2 Grover's Algorithm and Amplitude Amplification

The role of the Hadamard matrix is not just to create superposition. In "Grover's algorithm," famous as a quantum search algorithm, the Hadamard matrix functions as an important component for increasing the probability of a successful search.
When searching for specific data from an unsorted database (N elements), Grover's algorithm can complete the task in approximately sqrt(N) steps, whereas classical methods require an average of N/2 checks.
At the core of this algorithm is the "Diffusion Operator." This operator is defined as follows34.
$$D = H^{\otimes n} (2|0\rangle\langle 0| - I) H^{\otimes n}$$
Here, H^⊗n is the Hadamard transform for n qubits. The role of this operator is called "Inversion about the mean."
Intuitively, it is a process where the probability amplitude of information is "mixed" by the Hadamard transform, only the amplitude of the correct data is amplified (strengthened by interference), and the amplitudes of other incorrect data are canceled out and disappear.
Here too, the principle of "signal separation by diffusion and inverse diffusion" seen in Mariner 9 and CDMA is reproduced at the level of quantum mechanical amplitude. The Hadamard matrix is the conductor's baton for controlling the physical phenomenon of quantum interference36.

Chapter 7: Lightweighting Artificial Intelligence: "Rotation" Technology Supporting LLMs

7.1 The Rise of Massive AI and the Wall of "Outliers"

In the 2020s, the emergence of Large Language Models (LLMs), represented by ChatGPT, has dramatically improved AI capabilities. However, as a trade-off, model sizes have ballooned, requiring massive data centers and enormous amounts of electricity to run models with hundreds of billions of parameters.
To run these models on edge devices like smartphones and laptops, a technology called "Quantization" is attracting attention. This is a technique that reduces the number of bits used to represent parameters from the conventional 16 bits (FP16) to 4 or 3 bits, thereby reducing memory usage and computational load 38.
However, the problem of "Outliers" stands in the way. Inside LLMs, particularly in the outputs of activation functions, there are features (channels) that occasionally take on extremely large values. If the quantization range is widened to accommodate these few outliers, information for the vast majority of smaller values is lost. Conversely, if the range is set to match the smaller values, the outliers are clipped, resulting in large errors (noise) that cause the AI's performance (intelligence) to collapse 40.

7.2 "Energy Leveling" via Randomized Hadamard Transform (RHT)

To solve this dilemma, Hadamard matrices have once again been introduced. In the latest quantization methods (such as QuIP# and QuaRot), a process is performed to apply a "rotation" using a randomized Hadamard matrix to the weights and activation values of the matrix 9.
The mathematical significance of this process is "Incoherence Processing." Multiplying by a Hadamard matrix is equivalent to a coordinate rotation in high-dimensional space.
Energy (outliers) that was protruding on a specific coordinate axis (feature) is dispersed "thinly and widely" across all coordinate axes through rotation. Some researchers call this "democratization."
$$X_{\text{rotated}} = H \cdot X$$
The rotated data $X_{\text{rotated}}$ no longer has protruding peaks and instead has a smooth distribution close to a Gaussian distribution. In this state, even if quantized with low bits (such as 4-bit), information is preserved uniformly, and fatal errors are less likely to occur.
During decoding, the original information can be accurately restored by multiplying by the inverse matrix $H^{-1}$ (which, in the case of a Hadamard matrix, is $H^T$ multiplied by a constant).

7.3 Why Hadamard instead of a random orthogonal matrix?

Theoretically, a similar "leveling" effect can be obtained with any random orthogonal matrix. However, the decisive reason why Hadamard matrices are chosen in the field of AI is "computational speed."
While general dense matrix multiplication has a computational cost of $O(n^2)$, Hadamard matrices can be calculated in $O(n \log n)$ using the Fast Walsh-Hadamard Transform (FWHT). In models with hundreds of billions of parameters, this speed difference is a decisive factor that determines whether or not they can be put into practical use 9.
"Information is more resistant to compression and noise (quantization error) when dispersed across the whole rather than concentrated in one point." This holographic principle has been passed down steadily from the mathematics of 150 years ago to modern generative AI.

Chapter 8 Unsolved Territory: The Hadamard Conjecture and the Mystery of the 668th Order

8.1 The Hadamard Conjecture: A 100-Year-Old Problem

Although Hadamard matrices have advanced so far in engineering applications and support social infrastructure, a massive mystery remains unsolved from the perspective of pure mathematics. This is the "Hadamard Conjecture."
Hadamard Conjecture:
"If a natural number $n$ is 1, 2, or a multiple of 4, then a Hadamard matrix of order $n$ always exists."
It has been proven that for a Hadamard matrix of order $n$ to exist, it is a necessary condition that $n=1, 2$ or $n$ is a multiple of 4. However, whether this is a "sufficient condition"—that is, "can one always be created if it is a multiple of 4"—has not been proven by anyone since it was proposed in 1893 33.

8.2 The Smallest Unsolved Order

Mathematicians have used computers to combine various construction methods (Sylvester's method, Paley's method, Williamson's method, Ito's method, etc.) to prove their "existence" one by one.
In 2005, Hadi Kharaghani and Behruz Tayfeh-Rezaie constructed a Hadamard matrix of order 428, which had been a long-standing issue. This updated the smallest unsolved order 5.
As of 2025, the smallest order for which the existence of a Hadamard matrix has not been confirmed is 668 (in the form $n=4k$).
$668 = 4 \times 167$. 167 is a prime number. Whether this "magic square of order 668" exists is one of the greatest unsolved problems (the Holy Grail) in modern combinatorial mathematics. If it is proven not to exist, the conjecture will collapse, but many mathematicians believe that "it exists, but it just hasn't been found yet" 4.
When this matrix of order 668 is found, or when it is proven not to exist, it may open the door to new coding theory and cryptographic technology.

Conclusion: Mathematics as a Holographic Shield

What has become clear through this investigation is the fact that the Hadamard matrix is not merely a "table of +1s and -1s," but a universal "lens" for manipulating and protecting information.

As a tool for separation: In Mariner 9 and CDMA, orthogonality was used to strictly distinguish between "signals" and "noise (or others)."
As a tool for collection: In spectroscopic measurements and single-pixel cameras, weak signals were superimposed and measured, dramatically improving the S/N ratio.
As a tool for diffusion: In quantum computing and LLM quantization, information is not concentrated in specific bits or parameters but is "delocalized" across the entire system, enabling computational parallelization and error resilience.

Common to all these applications is the philosophy that "information is more resilient to the imperfections of the real world (noise, interference, quantization errors, entropy) when it is distributed and held across the whole rather than kept locally." This is fundamentally linked to the principle of holograms (where the whole image can be reconstructed even if parts are missing).
Discovered in the 19th century by Sylvester as a paving pattern and defined by Hadamard as the limit of a determinant, this mathematical structure continues to function as an "invisible foundation" supporting the massive edifice of our digital society even today in the 21st century. As the mystery of the 668th order shows, the exploration is not yet over. As long as humanity seeks to handle more information, faster and more accurately, the magic of this "magic square of information" will never be lost.

Appendix: Key Data and Comparison Tables

Table 1: Comparison of Orthogonal Transforms in Communication and Signal Processing

Table 2: List of Technical Specifications for Hadamard Matrix Applications

Table 3: Status of Hadamard Matrix Existence (as of 2025)

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