Some hypersurfaces over finite fields, minimal codes and secret sharing schemes

Linear error-correcting codes can be used for constructing secret sharing schemes; however,
finding in general the access structures of these secret sharing schemes and, in particular,
determining efficient access structures is difficult. Here we investigate the properties of certain
algebraic hypersurfaces over finite fields, whose intersection numbers with any hyperplane
only takes a few values; these varieties give rise to q-divisible linear codes with at most 5
weights. Furthermore, for q odd, these codes turn out to be minimal and we characterize the
access structures of the secret sharing schemes based on their dual codes. Indeed, the secret
sharing schemes thus obtained are democratic, that is each participant belongs to the same
number of minimal access sets and can easily be described