Abstract 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.

Abstract We provide a geometric construction of $$[n,9,n-9]_q$$ [ n , 9 , n - 9 ] q near-MDS codes arising from elliptic curves with n $${\mathbb {F}}_q$$ F q -rational points. Furthermore, we show that in some cases these codes cannot be extended to longer near-MDS codes.