Skew Hadamard designs and their codes

Skew Hadamard designs (4n−1, 2n−1, n−1) are associated to order 4n skew Hadamard matrices in the natural way. We study the codes spanned by their incidence matrices A and by I +A and show that they are self-dual after extension (resp. extension and augmentation) over fields of characteristic dividing n. Quadratic Residues codes are obtained in the case of the Paley matrix. Results on the p−rank of skew Hadamard designs are rederived in that way. Codes from skew Hadamard designs are classified. A new optimal self-dual code over F5 is constructed in length 20. Six new inequivalent [56, 28, 16] self-dual codes over F7 are obtained from skew Hadamard matrices of order 56, improving the only known quadratic double circulant code of length 56 over F7.