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.
翻译:线性误差校正码可用于构建秘密共享计划;然而,一般而言,发现这些秘密共享计划的接入结构,特别是确定高效接入结构是困难的。在这里,我们调查了某些代数超表层相对于有限面积的特性,其相交数与任何超高空飞机的相交数仅需要几个数值;这些品种产生了可互换的线性代码,其重量最多为5美元。此外,对于奇数来说,这些代码是最低的,我们根据它们的双重代码来描述秘密共享计划的接入结构。 事实上,由此获得的秘密共享计划是民主的,每个参与者都属于同样数量的最起码的接入数据集,并且很容易被描述。