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sing a backtracking algorithm along with an essential change to the rows of representatives of known $13710027$ equivalence classes of Hadamard matrices of order $32$, we make an exhaustive computer search feasible and show that there are exactly $6662$ inequivalent skew-Hadamard matrices of order $32$. Two skew-Hadamard matrices are considered {\sf SH}-equivalent if they are similar by a signed permutation matrix. We determine that there are precisely $7227$ skew-Hadamard matrices of order $32$ up to {\sf SH}-equivalence. This partly settles a problem posed by Kim and Sol\'{e}. As a consequence, we provide the classification of association schemes of order $31$.
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