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

TL;DR

This paper explores algebraic hypersurfaces over finite fields that produce minimal, multi-weight linear codes, enabling the construction of democratic secret sharing schemes with easily describable access structures.

Contribution

It introduces a class of algebraic hypersurfaces leading to q-divisible, minimal linear codes with at most five weights, and characterizes their associated secret sharing schemes.

Findings

Codes are q-divisible with at most 5 weights.

For odd q, codes are minimal and yield democratic secret sharing schemes.

Access structures are explicitly characterized and easily described.

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.