Instead of using a repetition code, a 6, 16 Hadamard code was used.

Compared to a 5-repetition code, the error correcting properties of this Hadamard code are much better, yet its rate is comparable.

It was used to communicate with the Voyager probes, as it is much more compact than the previously-used Hadamard code.

* This Golay code is only triple-error correcting, but it could be transmitted at a much higher data rate than the Hadamard code that was used during the Mariner mission.

Certain Hadamard matrices can almost directly be used as an error-correcting code using a Hadamard code ( generalized in Reed – Muller codes ), and are also used in balanced repeated replication ( BRR ), used by statisticians to estimate the variance of a parameter estimator.

This construction demonstrates that the rows of the Hadamard matrix can be viewed as a length linear error-correcting code of rank n, and minimum distance with generating matrix

The Hadamard code, by contrast, is constructed from the Hadamard matrix by a slightly different procedure.

Then the bit flip code from above can recover by transforming into the Hadamard basis before and after transmission through.

List-decoding ( describes list decoding ; the core of the Goldreich-Levin construction of hard-core predicates from one-way functions can be viewed as an algorithm for list-decoding the Hadamard code ).

The sequency ordering of the rows of the Walsh matrix can be derived from the ordering of the Hadamard matrix by first applying the bit-reversal permutation and then the Gray code permutation.

In coding theory, the Walsh – Hadamard code, named after the American mathematician Joseph Leonard Walsh and the French mathematician Jacques Hadamard, is an example of a linear code over a binary alphabet that maps messages of length to codewords of length.

The Walsh – Hadamard code is unique in that each non-zero codeword has Hamming weight of exactly, which implies that the distance of the code is also.

In standard coding theory notation, this means that the Walsh – Hadamard code is a-code.

The Hadamard code can be seen as a slightly improved version of the Walsh – Hadamard code as it achieves the same block length and minimum distance with a message length of, that is, it can transmit one more bit of information per codeword, but this improvement comes at the expense of a slightly more complicated construction.

After the Dreyfus affair, which involved him personally because his wife was related to Dreyfus, Hadamard became politically active and a staunch supporter of Jewish causes though he professed to be an atheist in his religion .< ref name = " atheist ">

Castelin demanded that proceedings should be instituted against the accomplices of the traitor, among whom he named Dreyfus ' father-in-law Hadamard, the naval officer Emile Weyl, and Bernard Lazare.

The order of a Hadamard matrix must be 1, 2, or a multiple of 4.

Let H be a Hadamard matrix of order n. Then the partitioned matrix

It can be shown by induction that the image of the Hadamard matrix under the above homomorphism is given by

The Hadamard conjecture should probably be attributed to Paley.

Two Hadamard matrices are considered equivalent if one can be obtained from the other by negating rows or columns, or by interchanging rows or columns.

Another generalization defines a complex Hadamard matrix to be a matrix in which the entries are complex numbers of unit modulus and which satisfies H H < sup >*</ sup >= n I < sub > n </ sub > where H < sup >*</ sup > is the conjugate transpose of H. Complex Hadamard matrices arise in the study of operator algebras and the theory of quantum computation.

Butson-type Hadamard matrices are complex Hadamard matrices in which the entries are taken to be q < sup > th </ sup > roots of unity.

A necessary condition on the existence of a regular n × n Hadamard matrix is that n be a perfect square.

A circulant matrix is manifestly regular, and therefore a circulant Hadamard matrix would have to be of perfect square order.

The Hadamard transform can be regarded as being built out of size-2 discrete Fourier transforms ( DFTs ), and is in fact equivalent to a multidimensional DFT of size.

The Hadamard transform can be defined in two ways: recursively, or by using the binary ( base-2 ) representation of the indices n and k.

Similar techniques can be applied for multiplications by matrices such as Hadamard matrix and the Walsh matrix.

Examples of Hadamard matrices were actually first constructed by James Joseph Sylvester in 1867.

In this manner, Sylvester constructed Hadamard matrices of order 2 < sup > k </ sup > for every non-negative integer k.

Hadamard matrices of orders 12 and 20 were subsequently constructed by Hadamard ( in 1893 ).

Regular Hadamard matrices are real Hadamard matrices whose row and column sums are all equal.

In mathematics, Hadamard's inequality, first published by Jacques Hadamard in 1893, is a bound on the determinant of a matrix whose entries are complex numbers in terms of the lengths of its column vectors.

* Regular Hadamard matrix, a Hadamard matrix whose row and column sums are all equal.

The theory of nonlinear functionals was continued by students of Hadamard, in particular Fréchet and Lévy.

Hadamard also founded the modern school of linear functional analysis further developed by Riesz and the group of Polish mathematicians around Stefan Banach.

The US philosopher Charles Sanders Peirce praised Cantor's set theory, and, following public lectures delivered by Cantor at the first International Congress of Mathematicians, held in Zurich in 1897, Hurwitz and Hadamard also both expressed their admiration.

Extending these deep ideas of Riemann, two proofs of the asymptotic law of the distribution of prime numbers were obtained independently by Jacques Hadamard and Charles Jean de la Vallée-Poussin and appeared in the same year ( 1896 ).

While the original proofs of Hadamard and de la Vallée-Poussin are long and elaborate, and later proofs have introduced various simplifications through the use of Tauberian theorems but remained difficult to digest, a surprisingly short proof was discovered in 1980 by American mathematician Donald J. Newman.

Further developments have produced improved estimation methods for the same stability measure, the variance / deviation of frequency, but these are known by separate names such as the Hadamard variance, modified Hadamard variance, the total variance, modified total variance and the Theo variance.

Hadamard described the process as having steps ( i ) preparation, ( ii ) incubation, ( iv ) illumination, and ( v ) verification of the five-step Graham Wallas creative-process model, leaving out ( iii ) intimation, with the first three cited by Hadamard as also having been put forth by Helmholtz:

The mathematical term well-posed problem stems from a definition given by Jacques Hadamard.

The main extra idea needed is an argument closely related to the theorem of Hadamard and de la Vallée Poussin, used by Deligne to show that various L-series do not have zeros with real part 1.

Inspired by the work of on Morse theory, found another proof, using Deligne's l-adic Fourier transform, which allowed him to simplify Deligne's proof by avoiding the use of the method of Hadamard and de la Vallée Poussin.

Using Riemann's ideas and by getting more information on the zeros of the zeta function, Jacques Hadamard and Charles Jean de la Vallée-Poussin managed to complete the proof of Gauss's conjecture.

Maurice attended the secondary school Lycée Buffon in Paris where he was taught mathematics by Jacques Hadamard.

The prime number theorem was first proved in 1896 by Jacques Hadamard and by Charles de la Vallée Poussin independently, using properties of the Riemann zeta function introduced by Riemann in 1859.

The n-dimensional parallelotope spanned by the rows of an n × n Hadamard matrix has the maximum possible n-dimensional volume among parallelotopes spanned by vectors whose entries are bounded in absolute value by 1.