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Error-correcting code
The Hadamard code is an error-correcting code named after the French mathematician Jacques Hadamard that is used for error detection and correction when
Hadamard_code
Mathematics concept
of Hadamard's maximal determinant problem. Certain Hadamard matrices can almost directly be used as an error-correcting code using a Hadamard code (generalized
Hadamard_matrix
Class of error-correcting code
Hadamard code is a [ 2 r , r , 2 r − 1 ] 2 {\displaystyle [2^{r},r,2^{r-1}]_{2}} linear code and is capable of correcting many errors. Hadamard code could
Linear_code
Family of linear error-correcting codes
that the dual code of the Hamming code is the shortened Hadamard code, also known as a Simplex code. The parity-check matrix has the property that any two
Hamming_code
Scheme for controlling errors in data over noisy communication channels
practical interest Goppa code, used in the McEliece cryptosystem Hadamard code Hagelbarger code Hamming code Latin square based code for non-white noise (prevalent
Error_correction_code
Involutive change of basis in linear algebra
The Hadamard transform (also known as the Walsh–Hadamard transform, Hadamard–Rademacher–Walsh transform, Walsh transform, or Walsh–Fourier transform) is
Hadamard_transform
Error-correcting codes used in wireless communication
channels. Reed–Muller codes generalize the Reed–Solomon codes and the Walsh–Hadamard code. Reed–Muller codes are linear block codes that are locally testable
Reed–Muller_code
Type of error-correcting code
Locally decodable codes are especially useful for data transmission over noisy channels. The Hadamard code (a special case of Reed Muller codes) was used in
Locally_decodable_code
Family of error-correcting codes that encode data in blocks
codes, Hadamard codes, Expander codes, Golay codes, Reed–Muller codes and Polar codes. These examples also belong to the class of linear codes, and hence
Block_code
Type of error-correcting code
most famous error-correcting codes, the Hadamard code, is a locally testable code. A codeword x is encoded in the Hadamard code to be the linear function
Locally_testable_code
mutually orthogonal Regular Hadamard matrix Hadamard conjecture Complex Hadamard matrix Butson-type Hadamard matrix Hadamard code – a system used for signal
List of things named after Jacques Hadamard
List_of_things_named_after_Jacques_Hadamard
Binary sequence used in telecommunications
one. Gold codes are used in GPS. The GPS C/A ranging codes are Gold codes of period 1,023. Hadamard code Hamming code JPL code Kasami code Zadoff–Chu
Gold_code
code K {\displaystyle {\mathit {K}}} form a Hadamard exponent. Thus, x K {\displaystyle {\mathit {xK}}} is the standard form of some complex Hadamard
Coding theory approaches to nucleic acid design
Coding_theory_approaches_to_nucleic_acid_design
First spacecraft to enter orbit around Mars (1971–1972)
Instead of using a repetition code, a [32, 6, 16] Hadamard code was used, which is also a 1st-order Reed-Muller code. Errors of up to seven bits per
Mariner_9
Method for encoding data in communications
uses of balanced-weight codes include biphase mark code uses a 1 of 2 code; 6b/8b encoding uses a 4 of 8 code; the Hadamard code is a 2 k − 1 {\displaystyle
Constant-weight_code
System of rules to convert information into another form or representation
Reed–Muller, Walsh–Hadamard, Bose–Chaudhuri–Hochquenghem, Turbo, Golay, algebraic geometry codes, low-density parity-check codes, and space–time codes. Error detecting
Code
Sequence of digital values used for synchronisation
conjecture on circulant Hadamard matrices Schmidt, Kai-Uwe; Willms, Jürgen (2016). "Barker sequences of odd length". Designs, Codes and Cryptography. 80
Barker_code
GOST (hash function) Group coded recording HAS-160 HAS-V HAVAL Hadamard code Hagelbarger code Hamming bound Hamming code Hamming(7,4) Hamming distance
List of algebraic coding theory topics
List_of_algebraic_coding_theory_topics
24-dimensional repeating pattern of points
Voyager probes, as it is much more compact than the previously used Hadamard code. Quantizers, or analog-to-digital converters, can use lattices to minimise
Leech_lattice
Divide-and-conquer algorithm to compute a Hadamard transform
computational mathematics, the Hadamard ordered fast Walsh–Hadamard transform (FWHTh) is an efficient algorithm to compute the Walsh–Hadamard transform (WHT). A naive
Fast_Walsh–Hadamard_transform
Computational theory
the block length of the Walsh-Hadamard code is 2k. An equivalent definition of the long code is as follows: The Long code encoding of j∈[n] is defined
Long_code_(mathematics)
Quantum logic gate
gate with respect to a Hadamard transformed basis { | + ⟩ , | − ⟩ } {\displaystyle \{|+\rangle ,|-\rangle \}} . The Hadamard transformed basis of a one-qubit
Controlled_NOT_gate
In coding theory, concatenated codes form a class of error-correcting codes that are derived by combining an inner code and an outer code. They were conceived
Concatenated error correction code
Concatenated_error_correction_code
Open conjecture that no real circulant Hadamard matrix has order greater than 4
conjecture on circulant Hadamard matrices, also called the circulant Hadamard matrix conjecture, states that no real circulant Hadamard matrix has order greater
Ryser's conjecture on circulant Hadamard matrices
Ryser's_conjecture_on_circulant_Hadamard_matrices
Process in quantum computing
classical repetition code. A phase-flip code is similarly constructed and is equivalent to the bit-flip code up to transversal Hadamard gates. Consider the
Quantum_error_correction
Complex-valued mathematical sequence
transmissions. Zadoff–Chu sequences are an improvement over the Walsh–Hadamard codes used in UMTS because they result in a constant-amplitude output signal
Zadoff–Chu_sequence
American mathematician (1895–1973)
worked mainly in the field of analysis. The Walsh function and the Walsh–Hadamard code are named after him. The Grace–Walsh–Szegő coincidence theorem is important
Joseph_L._Walsh
one-way functions can be viewed as an algorithm for list-decoding the Hadamard code). Goldwasser, S. and Bellare, M. "Lecture Notes on Cryptography" Archived
Hard-core_predicate
Set of quantum operations
generated using Hadamard, S, and CNOT gates, so the Clifford group is sometimes defined as the (finite) group of unitaries generated using Hadamard, S, and CNOT
Clifford_group
Pair of related error-correcting codes
Golay code is 2.M12, where M12 is the Mathieu group M12. The extended ternary Golay code can be constructed as the span of the rows of a Hadamard matrix
Ternary_Golay_code
concatenation of Reed–Solomon codes with the Hadamard code; this concatenation turns out to be an ϵ {\displaystyle \epsilon } -balanced code, which gives rise to
Small-bias_sample_space
Structure in combinatorial mathematics
constructed using the field with 11 elements, and is the Hadamard 2-design associated to the size 12 Hadamard matrix; see Paley construction I. Algebraically this
Block_design
Basic circuit in quantum computing
sometimes included in instruction sets. The Hadamard or Walsh-Hadamard gate, named after Jacques Hadamard (French: [adamaʁ]) and Joseph L. Walsh, acts
Quantum_logic_gate
Orthogonal matrix
derived from the ordering of the Hadamard matrix by first applying the bit-reversal permutation and then the Gray-code permutation: W ( 8 ) = [ 1 1 1 1
Walsh_matrix
Definition of quantum circuits
Gottesman–Knill theorem. The Clifford group is generated by three gates: Hadamard, phase gate S, and CNOT. This set of gates is minimal in the sense that
Clifford_gate
Code used in quantum error correction
In quantum computing, the Shor code or Shor nine qubit code is a foundational code in quantum error correction that protects quantum information against
Shor_code
Multiple access scheme in telecommunications
necessarily for different destinations), to transform these using a Walsh Hadamard transform, followed by an IFFT. A number of alternative possibilities exist
Multi-carrier code-division multiple access
Multi-carrier_code-division_multiple_access
Digital communications term
each user is given a different code. An OVSF code tree is a complete binary tree that reflects the construction of Hadamard matrices. Baud GPS signals Maral
Chip_(CDMA)
Constructions of Hadamard matrices
In mathematics, the Paley construction is a method for constructing Hadamard matrices using finite fields. The construction was described in 1933 by the
Paley_construction
Theorem of quantum circuits
The Clifford group can be generated solely by using the controlled NOT, Hadamard, and phase gates (CNOT, H and S); and therefore stabilizer circuits can
Gottesman–Knill_theorem
Weisstein, Eric W. "Hadamard Matrix". MathWorld. Archived from the original on 11 September 2026. Retrieved 10 September 2026. "Hadamard Matrix of Order 668"
List of mathematical discoveries made with artificial intelligence
List_of_mathematical_discoveries_made_with_artificial_intelligence
Type of pseudorandom binary sequence
to the Hadamard transform. This relationship allows the correlation of an MLS to be computed in a fast algorithm similar to the FFT. Barker code Complementary
Maximum-length_sequence
eight chips. Therefore, it can also be viewed as a form of generalized Hadamard transform encoding. IEEE Std 802.11b-1999, §18.4.6.5 Van Nee, Richard;
Complementary_code_keying
Two-bit quantum communication protocol
H\otimes I} unitary operation on the entangled qubit A. In other words, the Hadamard quantum gate H is only applied to A (see the figure above). If the resultant
Superdense_coding
Mechanism in quantum computing
controlled NOT gate can be decomposed into a Hadamard gate on its target, then a controlled Z gate, then a second Hadamard gate on its target. This decomposition
Phase_kickback
Graphical language for quantum processes
Duncan and Perdrix found the additional Euler decomposition rule for the Hadamard gate, which was used by Backens in 2013 to establish the first completeness
ZX-calculus
Format for digital video content
with transform coding. He examined several transform coding techniques, including the DCT, Hadamard transform, Fourier transform, slant transform, and Karhunen-Loeve
Video_coding_format
Widely used standard for video compression
transform block sizes for the integer transform operation. A secondary Hadamard transform performed on "DC" coefficients of the primary spatial transform
Advanced_Video_Coding
in quantum algorithms such as: Controlled-not with many target qubits, Hadamard operation on many qubits, Phase and controlled phase. Quantum algorithms
Quantum_Computation_Language
Change of basis applied in quantum computing
quantum circuit consisting of only O ( n 2 ) {\displaystyle O(n^{2})} Hadamard gates and controlled phase shift gates, where n {\displaystyle n} is the
Quantum_Fourier_transform
Topics referred to by the same term
to: Fair wear and tear, in government and aviation industries Fast Walsh–Hadamard transform, a mathematical algorithm Fast wavelet transform, a mathematical
FWT
Method in quantum communication
} Perform a Hadamard on qubit four followed by a CNOT from qubit three to qubit four. End by performing a Hadamard on qubit three: [ 1 0 0
Entanglement-assisted stabilizer formalism
Entanglement-assisted_stabilizer_formalism
Compact encoding of digital data
1950. Transform coding dates back to the late 1960s, with the introduction of fast Fourier transform (FFT) coding in 1968 and the Hadamard transform in 1969
Data_compression
transform Discrete sine transform Discrete wavelet transform Hadamard transform (or, Walsh–Hadamard transform) Fast wavelet transform Hankel transform, the
List_of_transforms
Special type of Boolean function
y ∈ V is a Hadamard matrix, and the support of f (or its complement) is a Hadamard difference set in V. This construction gives a Hadamard matrix developed
Bent_function
Concept in mathematics
Walsh functions, the Walsh system, the Walsh series, and the fast Walsh–Hadamard transform are all named after the American mathematician Joseph L. Walsh
Walsh_function
Topics referred to by the same term
Look up wht in Wiktionary, the free dictionary. WHT may stand for: Walsh–Hadamard transform Westmoreland Heritage Trail, a rail trail in Pennsylvania Western
WHT
Quantum states of two qubits
the simplest takes a computational basis as the input, and contains a Hadamard gate and a CNOT gate (see picture). As an example, the pictured quantum
Bell_state
Type of mathematical array
2, 2) if and only if there exists a Hadamard matrix of order 4λ. To proceed in one direction, let H be a Hadamard matrix of order 4m in standardized form
Orthogonal_array
Hadamard codes (code values of 0 or 180 degrees) or binary pseudonoise, CI codes are complex-valued orthogonal codes. In the simplest case, CI code values
Carrier_interferometry
and hence the circulant Hadamard conjecture. Ryser's conjecture on circulant Hadamard matrices: no real circulant Hadamard matrix has order greater than
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Geometric inequality applicable to any closed curve
holds for bounded sets S {\displaystyle S} in Hadamard manifolds, which has become known as the Cartan–Hadamard conjecture. In dimension 2 this had already
Isoperimetric_inequality
Quantum computing algorithm
{\displaystyle n} -qubit operations generated by the gates {H, S, CNOT} (where H is Hadamard and S is [ 1 0 0 i ] {\displaystyle {\begin{bmatrix}1&0\\0&i\end{bmatrix}}}
Magic_state_distillation
Polynomial root-finding algorithm
Lebedev, A. V.; Trubnikov, Yu. V.; Chernyavsky, M. M. (August 2024). "On the Hadamard and Vandermonde Determinants and the Bernoulli–Euler–Lagrange–Aitken Method
Bernoulli's_method
Lattice in 8-dimensional space with special properties
generated by this subgroup and the block diagonal matrix H4⊕H4 where H4 is the Hadamard matrix H 4 = 1 2 [ 1 1 1 1 1 − 1 1 − 1 1 1 − 1 − 1 1 − 1 − 1 1 ] . {\displaystyle
E8_lattice
x ) ) {\displaystyle (g\circ f)(x)=g(f(x))} for every value of x. 2. Hadamard product of matrices: If A and B are two matrices of the same size, then
Glossary of mathematical symbols
Glossary_of_mathematical_symbols
Mathematical operation on matrices
{B} )+(\mathbf {A} \mathbf {C} )\otimes [\mathbf {B} ,\mathbf {D} ]} . Hadamard product (element-wise multiplication): The mixed-product property also
Kronecker_product
Concept in quantum information theory
sciences, such as SIC-POVMs, finite projective/affine planes, complex Hadamard matrices and more [see section: Related problems]. MUBs are important for
Mutually_unbiased_bases
Inability to picture something in one's mind
the form of the corresponding printed words. As paraphrased by Jacques Hadamard, The first discovery of this by Ribot was the case of a man whom he mentions
Aphantasia
Technique for comparing quantum states
protocol is | 0 , ϕ , ψ ⟩ {\displaystyle |0,\phi ,\psi \rangle } . After the Hadamard gate, the state of the system is 1 2 ( | 0 , ϕ , ψ ⟩ + | 1 , ϕ , ψ ⟩ )
Swap_test
Symmetric arrangement of finite sets
is built around balanced incomplete block designs (BIBDs), Hadamard matrices and Hadamard designs, symmetric BIBDs, Latin squares, resolvable BIBDs, difference
Combinatorial_design
Russian mathematician (1935–2017)
systematic codes", Doklady Akademii Nauk SSSR, 131 (5): 1011–1014 V.I. Levenshtein, Application of Hadamard matrices to a problem in coding theory, Problems
Vladimir_Levenshtein
Block cipher
Twofish borrows some elements from other designs; for example, the pseudo-Hadamard transform (PHT) from the SAFER family of ciphers. Twofish has a Feistel
Twofish
Amateur telecommunications protocol
functions (or vectors of a Hadamard matrix). The receiver again measures the amplitudes for all 64 vectors (here comes the Hadamard Transform) and chooses
Olivia_MFSK
Deterministic quantum algorithm
|0\rangle } and the final bit is | 1 ⟩ {\displaystyle |1\rangle } . A Hadamard gate is applied to each bit to obtain the state 1 2 n + 1 ∑ x = 0 2 n −
Deutsch–Jozsa_algorithm
Theorem in quantum information science
factors. For example, one might use the controlled NOT gate and the Walsh–Hadamard gate to entangle two qubits without violating the no-cloning theorem, as
No-cloning_theorem
Discrete Fourier transform algorithm
called interaction algorithm, which provided efficient computation of Hadamard and Walsh transforms. Yates' algorithm is still used in the field of statistical
Fast_Fourier_transform
named after Alexander Grothendieck List of things named after Jacques Hadamard List of things named after William Rowan Hamilton List of things named
Lists_of_mathematics_topics
Natural number
{\displaystyle I} is the identity matrix and H {\displaystyle H} is a 24 by 24 Hadamard matrix (Z/23Z ∪ ∞) with a = 2 and b = 3, and entries X(∞) = 1 and X(0)
23_(number)
video compression. It works by taking a frequency transform, usually a Hadamard transform, of the differences between the pixels in the original block
Sum of absolute transformed differences
Sum_of_absolute_transformed_differences
Raster graphics file format
compressed in a 4×4 pixel sub-block with a discrete cosine transform or a Walsh–Hadamard transform. Both transforms are done with fixed-point arithmetic to avoid
WebP
Term in combinatorics
Codes and Cryptography. 56 (1): 79–84. doi:10.1007/s10623-009-9344-5. Feng, Tao; Leung, Ka Hin; Schmidt, Bernhard; Smith, Ken W. (2014). "Hadamard difference
Difference_set
Computer hardware technology that uses quantum mechanics
basis measurement, they behave differently under operations such as the Hadamard gate—which maps |0⟩↔|+⟩ and |1⟩↔|−⟩—demonstrating that relative phase differences
Quantum_computing
2008 open and royalty-free video coding format
and VP7. The discrete cosine transform (DCT) on 4×4 blocks and the Walsh–Hadamard transform (WHT) serve as basic frequency transforms. A maximum of three
VP8
Codes and Cryptography. 46 (3): 343–352. doi:10.1007/s10623-007-9163-5. MR 2372843. Kline, Jeffery (26 July 2019). "Geometric search for Hadamard matrices"
Williamson_conjecture
Sporadic simple group
elements. The double cover 2.M12 is the automorphism group of any 12×12 Hadamard matrix. M12 centralizes an element of order 11 in the monster group, as
Mathieu_group_M12
unstructured search problems Hadamard test: circuit technique for estimating real or imaginary parts of expectation values Hadamard transform: transform used
List_of_algorithms
Optimization algorithm for artificial neural networks
matrix of the derivative on each node. This is often represented as the Hadamard product with the vector of derivatives, denoted by (fl)′⊙, which is mathematically
Backpropagation
British-American mathematician (born 1939)
Theory of Error-Correcting Codes, Elsevier/North-Holland, Amsterdam, 1977. M. Harwit and Neil James Alexander Sloane, Hadamard Transform Optics, Academic
Neil_Sloane
Quantum algorithm for integer factorization
Shor and both are implemented by creating a superposition through using Hadamard gates, followed by implementing f as a quantum transform, followed finally
Shor's_algorithm
Product of numbers from 1 to n
MR 1376175. Hadamard, J. (1968) [1894]. "Sur l'expression du produit 1·2·3· · · · ·(n−1) par une fonction entière" (PDF). Œuvres de Jacques Hadamard (in French)
Factorial
Quantum key distribution protocol
{\displaystyle a_{i}} is encoded in (either in the computational basis or the Hadamard basis). The qubits are now in states that are not mutually orthogonal,
BB84
Integral expressing the amount of overlap of one function as it is shifted over another
where ∙ is face-splitting product, ⊗ denotes Kronecker product, ∘ denotes Hadamard product (this result is an evolving of count sketch properties). This can
Convolution
Reduction of image size to save storage and transmission costs
1952. Transform coding dates back to the late 1960s, with the introduction of fast Fourier transform (FFT) coding in 1968 and the Hadamard transform in 1969
Image_compression
Mathematical models of changing DNA
vary across branches and the Hadamard transform can even provide evidence that the data do not fit a tree. The Hadamard transform can also be combined
Models_of_DNA_evolution
Cloud quantum computing platform
automatically to |00000>) creg c[5]; // allocate 5 classical bits h q[0]; // Hadamard-transform qubit 0 cx q[0], q[1]; // conditional pauli X-transform (ie.
IBM_Quantum_Platform
Correspondence between quantum channels and quantum states
be used to teleport unitary universal gate sets. Consider the popular Hadamard gate H {\displaystyle H} , phase gate S {\displaystyle S} , the so-called
Choi–Jamiołkowski_isomorphism
Theory in statistics
(over the real or complex numbers) both for the matrix product and the Hadamard (entrywise) product. The algebra formed with the matrix product is called
Association_scheme
Golomb coding Golomb ruler Graeco-Latin square Gray code Hadamard matrices Complex Hadamard matrices Butson-type Hadamard matrices Generalized Hadamard matrices
Index of combinatorics articles
Index_of_combinatorics_articles
1996 Stakenbridge B. G. W. Manning · 9.6 km MPC · JPL 10382 Hadamard 1996 RJ3 Hadamard September 15, 1996 Prescott P. G. Comba · 4.1 km MPC · JPL 10383
List of minor planets: 10001–11000
List_of_minor_planets:_10001–11000
travel, tourism, insurance
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