  A classification of Hadamard matrices of order $2p+2$ with an automorphism of
order $p$ is given for $p=29$ and $31$. The ternary self-dual codes spanned by
the newly found Hadamard matrices of order $60$ with an automorphism of order
$29$ are computed, as well as the binary doubly even self-dual codes of length
$120$ with generator matrices defined by related Hadamard designs. Several new
ternary near-extremal self-dual codes, as well as binary near-extremal doubly
even self-dual codes with previously unknown weight enumerators are found.
