Some Applications of Hyperstructures in Coding Theory

Abstract

The notion of hyperrings (hyperfields) is a generalization of the notion of rings (fields) where the additive composition ” + ” and/or the multiplicative composition ” · ” are changed to a hyperoperation. Similarly, there is the notion of hypervector space and hypermodule where internal and/or external compositions on the classical form have been generalized. In this paper, we define linear codes and cyclic codes over a finite Krasner hyperfield and we characterize these codes by their generator matrices and parity check matrices. We also demonstrate that codes over finite Krasner hyperfields are more interesting for code theory than codes over classical finite fields.