Non Classical Structures and Linear Codes

*Surdive Atamewoue Tsafack*

## **Abstract**

This chapter present some new perspectives in the field of coding theory. In fact notions of fuzzy sets and hyperstructures which are consider here as non classical structures are use in the construction of linear codes as it is doing for fields and rings. We study the properties of these classes of codes using well known notions like the orthogonal of a code, generating matrix, parity check matrix and polynomials. In some cases particularly for linear codes construct on a Krasner hyperfield we compare them with those construct on finite field called here classical structures, and we obtain that linear codes construct on a Krasner hyperfield have more codes words with the same parameters.

**Keywords:** Linear codes, Fuzzy set, Krasner hyperstructures, Fuzzy logic, Algebraic hyperstructures

### **1. Introduction**

In mathematics, non classical structures as fuzzy sets and algebraic hyperstructures approach better many well known real life situation, and represent natural extension of classical algebraic structures.

Regarding fuzzy sets theory (fuzzy logic), this was introduced in the middle of 1960 by Lotfi Zadeh [1]. This concept is considered today as one of the most important of the second half of twentieth century, this in view of its applications in technological sciences and the impressive quantities of paper and book related to it.

As for algebraic hyperstructures, they were introduced by a french mathematician F. Marty [2] in 1934. Since then, more than a thousand papers and several book have been written on this topic. A well known type of algebraic hyperstructures is due to Krasner [3], who used as a technical tool in a study of his on the approximation of valued fields. In the literature they are called Krasner hyperrings and Krasner hyperfields.

Transmission on coding theory always suppose to encode its information and decode the received information, this is what the classical coding theory introduced in 1948 by C. Shannon [4] deals with. It should ne noted that the handling information are certains. So how can we do if the informations are uncertain? Thus as a new perspective for the algebraic coding, we present below a connection between fuzzy sets, Krasner hyperstructures and linear codes, and we find out how they can bring more in classical coding theory.