**3. Fuzzy** *pk* **-linear code**

In the previous section, we study define and fuzzy linear codes over the ring *pk* in the previous section. Now define the notion on **fuzzy Gray map**, we are going to use it in the construction of the fuzzy *pk* -linear codes which is different from the fuzzy linear codes over the ring *pk* .

## **3.1 Fuzzy Gray map**

When we order and enumerate a binary sequences of a fixed length we obtain the code of Gray in it original form. For the length two which interest us directly we have the following Gray code:

$$\begin{array}{c} \mathbf{0} \mapsto \mathbf{0} \mathbf{0} \\\\ \mathbf{1} \mapsto \mathbf{0} \mathbf{1} \\\\ \mathbf{2} \mapsto \mathbf{1} \mathbf{1} \\\\ \mathbf{3} \mapsto \mathbf{1} \mathbf{0}. \end{array}$$

Let *<sup>ϕ</sup>* : 2<sup>2</sup> ! <sup>2</sup> <sup>2</sup> the Gray map.

Using the extension principle [16], we will define the fuzzy Gray map between two fuzzy spaces by what is follow.

**Definition 3.1.** Consider the Gray map *<sup>ϕ</sup>* : 2<sup>2</sup> ! <sup>2</sup> 2. Let <sup>F</sup> <sup>2</sup> ð Þ<sup>2</sup> , <sup>F</sup> <sup>2</sup> 2 � � the set of all the fuzzy subset on 2<sup>2</sup> and <sup>2</sup> <sup>2</sup> respectively. The fuzzy Gray map is the map *<sup>ϕ</sup>*^ : <sup>F</sup> <sup>2</sup> ð Þ!F <sup>2</sup> <sup>2</sup> 2 � �, such that for all <sup>F</sup>⊓ ∈ <sup>F</sup> <sup>2</sup> ð Þ<sup>2</sup> , *<sup>ϕ</sup>*^ð Þ <sup>F</sup><sup>⊓</sup> ð Þ¼ *<sup>y</sup> sup A x* f g ð Þj*<sup>y</sup>* <sup>¼</sup> *<sup>ϕ</sup>*ð Þ *<sup>x</sup>* .

The next Theorem is straightforward.

**Theorem 3.2.** *The fuzzy Gray map ψ*^ *is a bijection.*

**Proof**: It is due to the fact that *<sup>ψ</sup>* is one to one function. □

As in crisp case, we have the following Proposition which is very important. **Proposition 3.3.** *If* F⊓ *is a fuzzy linear code over* 2<sup>2</sup> *and ϕ the Gray map, then <sup>ϕ</sup>*^ð Þ <sup>F</sup><sup>⊓</sup> *is no always a fuzzy linear code over the field* 2*.*

The Gray map give a way to construct the nonlinear codes as binary image of the linear codes, we have for example the case of Kerdock, Preparata, and Goethals codes which have very good properties and also useful (We refer reader for it on [17, 18]). Moreover if C is a linear code of length *n* over 4, then *C* ¼ *ψ*ð Þ C is a nonlinear code of length 2*n* over <sup>2</sup> in generally [18]. In that way we construct a fuzzy Kerdock code in the following example.

**Example 3.4.** *Let G* ¼ 1 0002111 01 001213 001 01321 0001 1132 0 BBBBBBB@ 1 CCCCCCCA *be a generating matrix for a*

*linear code* C *of length* 8 *over* 4*. Then his image under the Gray map ϕ give a Kerdock code C.*

Let <sup>F</sup><sup>⊓</sup> : <sup>8</sup> <sup>4</sup> ! ½ � 0, 1 , *x* ↦ 1, *if x*∈C; 0, *otherwise:* ( Thus F⊓ is a fuzzy linear code over 4.

Since *<sup>ϕ</sup>* is a bijection, we construct *<sup>ϕ</sup>*^ð Þ <sup>F</sup><sup>⊓</sup> : <sup>16</sup> <sup>2</sup> ! ½ � 0, 1 , *<sup>y</sup>* <sup>↦</sup> 1, *<sup>y</sup>* <sup>∈</sup> <sup>E</sup>; 0, *otherwise:* � ,

where E¼f*y*∈<sup>16</sup> <sup>2</sup> ∣*y* ¼ *ϕ*ð Þ *x* and *x*∈*C*g.

Noted that as <sup>E</sup> is not a linear code over 2, then *<sup>ϕ</sup>*^ð Þ *Fu* is a fuzzy 2-linear code but not a fuzzy linear code over 2.

*<sup>ϕ</sup>*^ð Þ <sup>F</sup><sup>⊓</sup> is a fuzzy Kerdock code of length 16.

By the Example, we remark that a fuzzy 4-linear code is not in generally a fuzzy linear code over 2.

If we define the fuzzy binary relation *<sup>R</sup><sup>ϕ</sup>* on 2<sup>2</sup> � <sup>2</sup> <sup>2</sup> by *Rϕ*ð Þ¼ *x*, *y*

1, if *y* ¼ *ψ*ð Þ *x* ; 0, otherwise*:* It is easy to see [19] that *<sup>ϕ</sup>*^ð Þ <sup>F</sup><sup>⊓</sup> ð Þ¼ *<sup>y</sup> sup*f g <sup>F</sup>⊓ð Þj *<sup>x</sup> <sup>y</sup>* <sup>¼</sup> *<sup>ϕ</sup>*ð Þ *<sup>x</sup>* can be represented by *<sup>ϕ</sup>*^ð Þ <sup>F</sup><sup>⊓</sup> ð Þ¼ *<sup>y</sup> sup min* <sup>F</sup>⊓ð Þ *<sup>x</sup>* , *<sup>R</sup>ϕ*ð Þ *<sup>x</sup>*, *<sup>y</sup>* <sup>j</sup>*x*∈<sup>2</sup> 2 .

We now define fuzzy generalized gray map. First we consider the generalized Gray map as in [8] <sup>Φ</sup> : *pk* ! *<sup>p</sup>k*�<sup>1</sup> *<sup>p</sup>* .

**Definition 3.5.** The map <sup>Φ</sup>^ : <sup>F</sup> *pk* ! F *pk*�<sup>1</sup> *p* , such that for any <sup>F</sup>⊓ ∈ <sup>F</sup> *pk* ,

$$\hat{\Phi}(\mathcal{F}\sqcap)(\boldsymbol{\jmath}) = \begin{cases} \sup\{\mathcal{F}\sqcap(\boldsymbol{\kappa})|\boldsymbol{\jmath} = \boldsymbol{\Phi}(\boldsymbol{\kappa})\}, & \text{if } \boldsymbol{\alpha} \text{ such } \boldsymbol{\kappa} \text{ exists;}\\ \boldsymbol{0}, & \text{otherwise.} \end{cases}$$

Is called a fuzzy generalized gray map. **Remark 3.6.**

1.The Definition 3.5 can be simply write <sup>Φ</sup>^ ð Þ *<sup>A</sup>* ð Þ¼ *<sup>y</sup>* F⊓ð Þ *x* , if *y* ¼ Φð Þ *x* ; 0, otherwise*:* 

This because <sup>Φ</sup> : *pk* ! *<sup>p</sup>k*�<sup>1</sup> *<sup>p</sup>* cannot give more than one image for one element.

2.Let <sup>F</sup>⊓<sup>1</sup> <sup>∈</sup> <sup>F</sup> *pk*�<sup>1</sup> *p* such that <sup>F</sup>⊓1ð Þ¼ *<sup>y</sup> <sup>t</sup>* 6¼ 0 for any *<sup>y</sup>*∈*pk*�<sup>1</sup> *<sup>p</sup>* . There does not exist a fuzzy subset F⊓ ∈ F *pk* such that <sup>Φ</sup>^ ð Þ¼F <sup>F</sup><sup>⊓</sup> <sup>⊓</sup>1.

Thus Φ^ is not a bijection map.
