  Hadamard matrices are $(-1, +1)$ square matrices with mutually orthogonal
rows. The Hadamard conjecture states that Hadamard matrices of order $n$ exist
whenever $n$ is $1$, $2$, or a multiple of $4$. However, no construction is
known that works for all values of $n$, and for some orders no Hadamard matrix
has yet been found. Given the many practical applications of these matrices, it
would be useful to have a way to easily check if a construction for a Hadamard
matrix of order $n$ exists, and in case to create it. This project aimed to
address this, by implementing constructions of Hadamard and skew Hadamard
matrices to cover all known orders less than or equal to $1000$ in SageMath, an
open-source mathematical software. Furthermore, we implemented some additional
mathematical objects, such as complementary difference sets and T-sequences,
which were not present in SageMath but are needed to construct Hadamard
matrices.
  This also allows to verify the correctness of the results given in the
literature; within the $n\leq 1000$ range, just one order, $292$, of a skew
Hadamard matrix claimed to have a known construction, required a fix.

Combinatorics