**4. Linear codes over Krasner hyperfields**

#### **4.1 Preliminaries**

This section recall notions and results that are required in the sequel. All of this can also be check on [3, 20–22].

Let <sup>H</sup> be a non-empty set and <sup>P</sup><sup>∗</sup> ð Þ <sup>H</sup> be the set of all non-empty subsets of <sup>H</sup>. Then, a map <sup>⊛</sup> : H�H!P<sup>∗</sup> ð Þ <sup>H</sup> , where ð Þ *<sup>h</sup>*1, *<sup>h</sup>*<sup>2</sup> <sup>↦</sup> *<sup>h</sup>*<sup>1</sup> <sup>⊛</sup> *<sup>h</sup>*<sup>2</sup> <sup>⊆</sup> <sup>H</sup> is called a hyperoperation and the couple ð Þ H, , ⊛, is called a hypergroupoid.

For all non-empty subsets *A* and *B* of H and *h*∈ H, we define *A* ⊛ *B* ¼ ⋃*<sup>a</sup>* <sup>∈</sup> *<sup>A</sup>*,*b*∈*<sup>B</sup>a* ⊛ *b*, *A* ⊛ *h* ¼ *A* ⊛ f g*h* and *h* ⊛ *B* ¼ f g*h* ⊛ *B*.

**Definition 4.1.** A canonical hypergroup ð Þ R, ⊕ is an algebraic structure in which the following axioms hold:


**Remark 4.2.** Note that, in the classical group ð Þ *R*, þ , the concept of opposite of *x*∈*R* is the same as inverse.

A canonical hypergroup with a multiplicative operation which satisfies the following conditions is called a Krasner hyperring.

**Definition 4.3.** An algebraic hyperstructure ð Þ R, ⊕ , � , where "�" is usual multiplication on R, is called a Krasner hyperring when the following axioms hold:

1.ð Þ R, ⊕ is a canonical hypergroup with 0 as additive identity,


A Krasner hyperring is called commutative (with unit element) if ð Þ R, � is a commutative semigroup (with unit element) and such is denoted ð Þ *R*, ⊕ , �, 0, 1 .

**Definition 4.4.** Let ð Þ *R*, ⊕ , �, 0, 1 be a commutative Krasner hyperring with unit such that ð Þ *R*nf g0 , � , 1 is a group. Then, ð Þ *R*, ⊕ , �, 0, 1 is called a Krasner hyperfield.

This Example is from Krasner.

**Example 4.5.** *[?] Consider a field F*ð Þ , þ , � *and a subgroup G of F*ð Þ nf g0 , � *. Take H* ¼ *F=G* ¼ f g *aG*j *a*∈*F with the hyperoperation and the multiplication given by:*

> *aG* ⊕ *bG* ¼ f g *c* ¼ *cG*j *c*∈ *aG* þ *bG aG* � *bG* <sup>¼</sup> *abG* �

Then ð Þ *H*, ⊕ , � is a Krasner hyperfield.

We now give an example of a finite hyperfield with two elements 0 and 1, that we name F<sup>2</sup> and which will be used it in the sequel.

**Example 4.6.** *Let* F<sup>2</sup> ¼ f g 0, 1 *be the finite set with two elements. Then* F<sup>2</sup> *becomes a Krasner hyperfield with the following hyperoperation "* ⊕ *" and binary operation "*�*".*


and


Let ð Þ R, ⊕ , � be a hyperring, *A* and *B* be a non-empty subset of R. Then, *A* is said to be a subhyperring of R if (*A*, ⊕ , �) is itself a hyperring. A subhyperring *A* of a hyperring R is a left (right) hyperideal of R if *r* � *a*∈ *A* (*a* � *r*∈ *A*) for all *r*∈ R, *a*∈ *A*. Also, *A* is called a hyperideal if *A* is both a left and a right hyperideal. We define *A* ⊕ *B* by *A* ⊕ *B* ¼ f*x*∣ *x*∈*a* ⊕ *b* for some *a*∈ *A*, *b*∈*B*g and the product *A* � *B* is defined by *<sup>A</sup>* � *<sup>B</sup>* ¼ f*x*<sup>∣</sup> *<sup>x</sup>*∈P*<sup>n</sup> <sup>i</sup>*¼<sup>1</sup>*ai* � *bi*, with *ai* <sup>∈</sup> *<sup>A</sup>*, *bi* <sup>∈</sup>*B*, *<sup>n</sup>* <sup>∈</sup> <sup>∗</sup> <sup>g</sup>. If *<sup>A</sup>* and *<sup>B</sup>* are hyperideals of R, then *A* ⊕ *B* and *A* � *B* are also hyperideals of R.

**Definition 4.7.** An algebraic structure ð Þ R, ⊕ , � (where ⊕ and � are both hyperoperations) is called additive-multiplicative hyperring if the satisfies the following axioms:


An additive-multiplicative hyperring ð Þ R, ⊕ , � is said to be commutative if ð Þ R, � is a commutative semihypergroup. and ð Þ R, ⊕ , � is called a hyperring with multiplicative identity if there exists *e*∈ R such that *x* � *e* ¼ *x* ¼ *e* � *x* for every *x*∈ R.

We close this section with the following definition of the ideal in a additivemultiplicative hyperring.

**Definition 4.8** A non-empty subset *A* of an additive-multiplicative hyperring R is a left (right) hyperideal if,

1. for all *a*, *b*∈ *A*, then *a* ⊕ *b*<sup>0</sup> ⊆ *A*,

2. for all *a*∈ *A*, *r*∈ R, then *r* � *a*⊆ *A* (*a* � *r*⊆ *A*).
