Hadamard matrices of order $n$ are conjectured to exist whenever $n$ is $1$, $2$, or a multiple of $4$; a similar conjecture exists for skew Hadamard matrices. We provide constructions covering orders $\le 1208$ of all known Hadamard and skew Hadamard matrices in the open-source software SageMath. This allowed us to verify the correctness of results given in the literature. Within this range, just one order, $292$, of a skew Hadamard matrix claimed to have a known construction, required a fix. We also produce the up to date tables, for $n \le 2999$ (resp. $n\le 999$ for skew case), of the minimum exponents $m$ such that a (skew) Hadamard matrix of order $2^m n$ is known, improving over 100 entries in the previously published sources. We explain how tables' entries are related to Riesel numbers. As a by-product of the latter, we show that the Paley constructions of (skew-)Hadamard matrices do not work for the order $2^m 509203$, for any $m$.

翻译：当阶数$n$为$1$、$2$或$4$的倍数时，哈达玛矩阵被推测总是存在的；对于斜哈达玛矩阵也存在类似的猜想。我们在开源软件SageMath中提供了覆盖所有已知哈达玛矩阵和斜哈达玛矩阵、阶数$\le 1208$的构造。这使得我们能够验证文献中给出的结果的正确性。在此范围内，仅有一个声称已知构造的斜哈达玛矩阵的阶数$292$需要进行修正。我们还制作了截至$n \le 2999$（斜哈达玛矩阵情况为$n\le 999$）的最新表格，记录了使得$2^m n$阶（斜）哈达玛矩阵已知的最小指数$m$，改进了先前已发表文献中超过100个条目。我们解释了表格条目如何与里塞尔数相关联。作为后者的副产品，我们证明了对于任意$m$，阶数为$2^m 509203$的（斜）哈达玛矩阵的佩利构造均不成立。