**3.2 Fuzzy** *pk* **-linear code**

In the following, we will note <sup>Φ</sup>^ the map on <sup>F</sup> *<sup>n</sup> pk* onto <sup>F</sup> *<sup>n</sup>:pk*�<sup>1</sup> *p* which spreads the fuzzy generalized Gray map.

Let define fuzzy *pk* -linear code.

**Definition 3.7.** A fuzzy code *Fu* over *<sup>p</sup>* is a fuzzy *pk* -linear code if it is an image under the fuzzy generalized Gray map of a fuzzy linear code over the ring *pk* .

For a fuzzy *pk* -linear code, if it is the image under the generalized Gray map of a cyclic code over the ring *pk* . Then the fuzzy code *Fu* is called a fuzzy *pk* -cyclic code.

**Remark 3.8.** A fuzzy *pk* -linear code is a fuzzy code over the fields *p*. **Example 3.9.**

$$\text{Let } \mathcal{F}\sqcap : \mathbb{Z}\_2^6 \to [0, 1], w = (a, b, c, d, e, f) \mapsto \begin{cases} 1, & \text{if } e = f = \mathbf{0}; \\ 0, & \text{otherwise}. \end{cases}$$

F⊓ is a fuzzy linear code of length 6 over 2.

$$\text{Let } \mathcal{F}\sqcap': \mathbb{Z}\_4^3 \to [0,1], \\ \nu = (\varkappa, y, z) \mapsto \begin{cases} 1, & \text{if } z = 0; \\ 0, & \text{otherwise}. \end{cases}$$

F⊓<sup>0</sup> is a fuzzy linear code of length 3 over 4.

Since <sup>F</sup><sup>⊓</sup> <sup>¼</sup> *<sup>ϕ</sup>*^ <sup>F</sup>⊓<sup>0</sup> ð Þ. Then <sup>F</sup>⊓<sup>0</sup> is a fuzzy 4-linear code. Using crisp case technic we prove he following Proposition.

**Proposition 3.10.** *Let Fu be a fuzzy pk -linear code, then Fu is no always a fuzzy linear code over the field p.*

**Proof**. The need here is to construct an counter-example, which is done in the Example 3.9. □

The following diagram give a construct the fuzzy *pk* -linear code. This holds because the fuzzy generalized Gray map image of fuzzy linear code can be a fuzzy linear code over the field *p*:

We construct some codes using the both methods. **Example 3.11.**

1. *Let* <sup>F</sup><sup>⊓</sup> : *<sup>n</sup> pk* ! ½ � 0, 1 *be a linear code such that* F⊓ *have three upper t-level cut* F⊓*<sup>t</sup>*<sup>3</sup> ⊆ F⊓*<sup>t</sup>*<sup>2</sup> ⊆ F⊓*<sup>t</sup>*<sup>1</sup> *. Let* F⊓<sup>0</sup> *<sup>t</sup>*<sup>3</sup> ¼ Φ F⊓*<sup>t</sup>*<sup>3</sup> ð Þ*,* F⊓<sup>0</sup> *<sup>t</sup>*<sup>2</sup> ¼ Φ F⊓*<sup>t</sup>*<sup>2</sup> ð Þ *and* F⊓<sup>0</sup> *<sup>t</sup>*<sup>1</sup> ¼ Φ F⊓*<sup>t</sup>*<sup>1</sup> ð Þ*, we have* F⊓<sup>0</sup> *<sup>t</sup>*<sup>3</sup> ¼ Φ F⊓*<sup>t</sup>*<sup>3</sup> ð Þ⊆ F⊓<sup>0</sup> *<sup>t</sup>*<sup>2</sup> ¼ Φð Þ F⊓*t*<sup>2</sup> ⊆ F⊓<sup>0</sup> *<sup>t</sup>*<sup>1</sup> ¼ Φ F⊓*<sup>t</sup>*<sup>1</sup> ð Þ*. We construct* <sup>F</sup>⊓<sup>0</sup> <sup>¼</sup> <sup>Φ</sup>^ ð Þ <sup>F</sup><sup>⊓</sup> *as follow:* <sup>F</sup>⊓<sup>0</sup> : *<sup>n</sup>:pk*�<sup>1</sup> *<sup>p</sup>* ! ½ � 0, 1 , *y* ↦ *t*3, *if y*∈ F⊓<sup>0</sup> *t*3 ; *t*2, *if y*∈ F⊓<sup>0</sup> *t*2 ; *t*1, *if y*∈ F⊓<sup>0</sup> *t*1 ; 0, *otherwise:* 8 >>>< >>>: 2. Let F⊓ : <sup>4</sup> ! ½ � 0, 1 , *x* ↦ 1 <sup>2</sup> *if x* <sup>¼</sup> 0; 1 <sup>3</sup> *if x* <sup>¼</sup> 2; 1 <sup>4</sup> *if x* <sup>¼</sup> 1, 3*:* 8 >>>>>< >>>>>:

be a fuzzy linear code over 4. Then <sup>F</sup>⊓<sup>1</sup> <sup>2</sup> <sup>¼</sup> f g<sup>0</sup> , <sup>F</sup>⊓<sup>1</sup> <sup>3</sup> <sup>¼</sup> f g 0, 2 and <sup>F</sup>⊓<sup>1</sup> <sup>4</sup> ¼ 4. We construct <sup>F</sup>⊓0<sup>1</sup> <sup>2</sup> <sup>¼</sup> f g <sup>00</sup> , <sup>F</sup>⊓0<sup>1</sup> <sup>3</sup> <sup>¼</sup> f g 00, 11 and <sup>F</sup>⊓<sup>0</sup> <sup>1</sup> <sup>4</sup> <sup>¼</sup> <sup>2</sup> 2, the Gray map image of F⊓<sup>1</sup> 2 , <sup>F</sup>⊓<sup>1</sup> <sup>3</sup> and <sup>F</sup>⊓<sup>1</sup> <sup>4</sup> respectively, we define

$$\begin{aligned} \mathcal{F}\sqcap' &: \mathbb{Z}\_2^2 \to [0,1], \mathfrak{y} \mapsto \begin{cases} \frac{1}{2} & \text{if } \mathfrak{x} \in \mathcal{F}\sqcap\_{\mathbb{Z}}^1, \ \mathfrak{y} = \mathfrak{ϕ}(\mathfrak{x}) \end{cases}; \\\frac{1}{3} & \text{if } \mathfrak{x} \in \mathcal{F}\sqcap\_{\mathbb{Z}}^1 \backslash \mathcal{F}\sqcap\_{\mathbb{Z}}^1, \ \mathfrak{y} = \mathfrak{ϕ}(\mathfrak{x}); \\\frac{1}{4} & \text{if } \mathfrak{x} \in \mathcal{F}\sqcap\_{\mathbb{Z}}^1 \backslash \mathcal{F}\sqcap\_{\mathbb{Z}}^1, \ \mathfrak{y} = \mathfrak{ϕ}(\mathfrak{x}). \end{aligned}$$

We obtain the same code <sup>F</sup>⊓<sup>0</sup> and *<sup>ϕ</sup>*^ð Þ <sup>F</sup><sup>⊓</sup> .

*Non Classical Structures and Linear Codes DOI: http://dx.doi.org/10.5772/intechopen.97471*

**Proposition 3.12.** *[15] If for all t*∈½ � 0, 1 *,* F⊓<sup>0</sup> *<sup>t</sup>* ¼ Φð Þ F⊓*<sup>t</sup> (when* F⊓*<sup>t</sup>* 6¼ ∅*) is a linear code over p, then this two constructions of the fuzzy p-linear code above are give the same fuzzy code.*

**Proof**. This follows directly from the definition of the fuzzy generalized Gray map and the fact that the image under the generalized Gray map of a linear code is not a linear code in general. □
