**2.2 On fuzzy linear codes over** *pk*

Now we bring fuzzy logic in linear codes and introduce the notion of fuzzy linear code over the ring *pk* .

**Definition 2.13.** Let *<sup>M</sup>* <sup>¼</sup> *<sup>n</sup> pk* be a *pk* -module. The fuzzy submodule F⊓ of *M* is called fuzzy linear code of length *n* over *pk* .

Using the transfer principle of Kondo [13], we have what is follow. **Proposition 2.14.** *Let A be a fuzzy set on <sup>n</sup> pk .*

*A* is a fuzzy linear code of length *n* over *pk* if and only if for any *t* ∈½ � 0, 1 , if *At* 6¼ ∅, then *At* is a linear code of length *n* over *pk* .

**Corollary 2.15.** *Let A be a fuzzy set on <sup>n</sup> pk .*

*A* is a fuzzy linear code of length *n* over *pk* if and only if the characteristic function of any upper *t*-level cut *At* 6¼ ∅ for *t*∈½ � 0, 1 is a fuzzy linear code of length *n* over *pk* .

**Example 2.16.** *Consider a fuzzy subset* F⊓ *on* <sup>4</sup> *as follows:*

$$\mathcal{F} \sqcap \colon \mathbb{Z}\_4 \to [0,1], \mathfrak{x} \mapsto \begin{cases} 1 & \text{if } \; x = 0; \\ \frac{1}{3} & \text{if } \; x = 1; \\ \frac{1}{3} & \text{if } \; x = 2; \\ \frac{1}{3} & \text{if } \; x = 3. \end{cases}$$

*Then* F⊓ *is a fuzzy submodule on* 4*-module* 4*, hence* F⊓ *is a fuzzy linear code over* 4*.*

**Remark 2.17.** Let <sup>F</sup><sup>⊓</sup> be a fuzzy linear code of length *<sup>n</sup>* over *pk* , since *<sup>n</sup> pk* is a finite set, then *Im*ð Þ¼ F <sup>F</sup><sup>⊓</sup> <sup>⊓</sup>ð Þj *<sup>x</sup> <sup>x</sup>*<sup>∈</sup> *<sup>n</sup> pk* n o is finite. Let *Im*ð Þ <sup>F</sup><sup>⊓</sup> is set as: *t*<sup>1</sup> > *t*<sup>2</sup> > ⋯ >*tm* (where *ti* ∈ ½ � 0, 1 ) that is *Im*ð Þ F⊓ have *m* elements.

Since F⊓*ti* is a linear code over *pk* , let *Gti* his generator matrix, F⊓ can be determined by *m* matrixes *Gt*<sup>1</sup> , *Gt*<sup>2</sup> , ⋯, *Gtm* as in the below Theorem 2.31.

There are some know notions of the orthogonality in fuzzy space, but no one of them does not hold here because these definitions does not meet the transfer principle in the sense of the orthogonality for the *t*-level cut sets. So we have to introduce an new notion of orthogonality on fuzzy submodules.

**Definition 2.18.** Let <sup>F</sup>⊓<sup>1</sup> and *Fu*<sup>2</sup> be two fuzzy submodules on module *<sup>n</sup> pk* over the ring *pk* . We said that F⊓<sup>1</sup> and F⊓<sup>2</sup> are orthogonal if *Im*ð Þ¼ F⊓<sup>2</sup> f g <sup>1</sup> � *<sup>α</sup>*<sup>j</sup> *<sup>α</sup>*∈*Im*ð Þ <sup>F</sup>⊓<sup>1</sup> and for all *<sup>t</sup>*<sup>∈</sup> ½ � 0, 1 , ð Þ <sup>F</sup>⊓<sup>2</sup> <sup>1</sup>�*<sup>t</sup>* ¼ Fð Þ <sup>⊓</sup><sup>1</sup> *<sup>t</sup>* � �<sup>⊥</sup> <sup>¼</sup> <sup>f</sup>*y*∈*<sup>n</sup> pk* ∣<*x*, *y*> ¼ 0, for all *x*∈ð Þ F⊓<sup>1</sup> *<sup>t</sup>* g. Where < , > is the standard inner prod-

uct on *<sup>n</sup> pk* .

Noted that F⊓<sup>1</sup> ⊥ F⊓<sup>2</sup> means F⊓<sup>1</sup> and F⊓<sup>2</sup> are orthogonal. We what is follow as an example.

**Example 2.19.** *Consider the two fuzzy submodules* F⊓<sup>1</sup> *and* F⊓<sup>2</sup> *on* <sup>4</sup> *defined as follows:*

$$\begin{split} & \mathcal{F}\sqcap\_1: \mathbb{Z}\_4 \to [0,1], \mathbf{x} \mapsto \begin{cases} \frac{1}{2} & \text{if } \mathbf{x} = \mathbf{0}; \\ \frac{1}{4} & \text{if } \mathbf{x} = \mathbf{1}; \\ \frac{1}{3} & \text{if } \mathbf{x} = \mathbf{2}; \\ \frac{1}{3} & \text{if } \mathbf{x} = \mathbf{3}. \end{cases} \text{ and } \mathcal{F}\sqcap\_2: \mathbb{Z}\_4 \to [\mathbf{0},\mathbf{1}], \\ & \qquad \left\{ \begin{array}{ll} \frac{3}{4} & \text{if } \mathbf{x} = \mathbf{0}; \\ \frac{1}{2} & \text{if } \mathbf{x} = \mathbf{1}; \\ \frac{1}{2} & \text{if } \mathbf{x} = \mathbf{1}; \\ \frac{2}{3} & \text{if } \mathbf{x} = \mathbf{2}; \\ \frac{1}{2} & \text{if } \mathbf{x} = \mathbf{3}. \end{cases} \right. \\ & \text{We easily observe that: } \\ & \left(\mathcal{F}\sqcap\_1\right) = \left\{\mathbf{0}\right\} \text{ and } \left(\mathcal{F}\sqcap\_2\right)\_1 = \mathbb{Z}\_4, \\ & \left(\mathcal{F}\sqcap\_1\right)\_1 = \left\{\mathbf{0}, \mathbf{2}\right\} \text{ and } \left(\mathcal{F}\sqcap\_1\right)\_2 = \left\{\mathbf{0}\right\}, \\ & \left(\mathcal{F}\sqcap\_1\right)\_3 = \left\{\mathbf{0}, \mathbf{2}\right\} \text{ and } \left(\mathcal{F}\sqcap\_1\right)\_3 = \left\{\mathbf{0}, \mathbf{2}\right\}. \end{split}$$

3 Thus F⊓1⊥F⊓2.

**Remark 2.20.** Let <sup>F</sup>⊓<sup>1</sup> be a fuzzy submodule on module *<sup>n</sup> pk* such that <sup>∀</sup>*x*∈*<sup>n</sup> pk* , <sup>F</sup>⊓1ð Þ¼ *<sup>x</sup> <sup>γ</sup>* (with *<sup>γ</sup>* <sup>∈</sup> ½ � 0, 1 ), then it does not exists a fuzzy set <sup>F</sup><sup>⊓</sup> on *<sup>n</sup> pk* such that F⊓1⊥F⊓.

The previous Remark 2.20 show that the orthogonal of some fuzzy submodule in our sense does not always exists, so it is important to see under which condition the orthogonal of fuzzy submodule exists. The following theorem show the existence of the orthogonal of some fuzzy submodule.

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

**Theorem 2.21.** *Let* <sup>F</sup>⊓<sup>1</sup> *be a fuzzy submodule on a finite module <sup>n</sup> pk . If Im*ð Þ F⊓<sup>1</sup> *have more that one element and for all ς*∈*Im*ð Þ F⊓<sup>1</sup> *there exist* ϵ∈ *Im*ð Þ F⊓<sup>1</sup> *such that <sup>A</sup><sup>ς</sup>* <sup>¼</sup> ð Þ *<sup>A</sup>*<sup>ϵ</sup> <sup>⊥</sup>*, then there always exists a fuzzy submodule* <sup>F</sup>⊓<sup>2</sup> *on <sup>n</sup> pk such that* F⊓1⊥F⊓2*.*

**Proof**. Let <sup>F</sup>⊓<sup>1</sup> be a fuzzy submodule on *<sup>n</sup> pk* . Assume that ∣*Im*ð Þ F⊓<sup>1</sup> ∣ ¼ *m* > 1 and for any *ς*∈*Im A*ð Þ there exist ϵ∈*Im*ð Þ F⊓<sup>1</sup> such that ð Þ F⊓<sup>1</sup> *<sup>ς</sup>* ¼ Fð Þ ⊓<sup>1</sup> <sup>ϵ</sup> � �<sup>⊥</sup> .

Assume that *Im*ð Þ¼ F⊓<sup>1</sup> f g *t*<sup>1</sup> >*t*<sup>2</sup> > ⋯ >*tm* . Let the sets *Mi* ¼

*x*∈*<sup>n</sup> pk* jF⊓1ð Þ¼ *x ti* n o, *<sup>i</sup>* <sup>¼</sup> 1, <sup>⋯</sup>, *<sup>m</sup>*. These sets form a partition of *<sup>n</sup> pk* .

Let us define a fuzzy set F⊓ as follow:

<sup>F</sup><sup>⊓</sup> : *<sup>n</sup> pk* ! ½ � 0, 1 , *x* ↦ 1 � *tm*�*i*þ1, if *x*∈ *Mi*.

Since *Im*ð Þ¼ F⊓<sup>1</sup> f g *t*<sup>1</sup> >*t*<sup>2</sup> > ⋯ >*tm* , we have ð Þ F⊓<sup>1</sup> *<sup>t</sup>*<sup>1</sup> ⊆ð Þ F1 *<sup>t</sup>*<sup>2</sup> ⊆⋯⊆ ð Þ F⊓<sup>1</sup> *tm* . As for any *<sup>ς</sup>*∈*Im*ð Þ <sup>F</sup>⊓<sup>1</sup> there exist <sup>ϵ</sup>∈*Im A*ð Þ such that *<sup>A</sup><sup>ς</sup>* <sup>¼</sup> ð Þ *<sup>A</sup>*<sup>ϵ</sup> <sup>⊥</sup>, then *Ati* <sup>¼</sup> *Atm*�*i*þ<sup>1</sup> � �<sup>⊥</sup> .

$$\begin{aligned} \text{Thus } \mathcal{F}\sqcap\_{1-t\_{m-i+1}} &= \left\{ \mathbf{x} \in \mathbb{Z}\_{p^k}^n \middle| \mathcal{F}\sqcap(\mathbf{x}) \ge \mathbf{1} - t\_{m-i+1} \right\} = M\_i \cup M\_{i-1} \cup \cdots \cup M\_1 = \left\{ (\mathcal{F}\sqcap\_1)\_{t\_{m-i+1}} \right\}^\perp. \\ (\mathcal{F}\sqcap\_1)\_{t\_i} &= \left( (\mathcal{F}\sqcap\_1)\_{t\_{m-i+1}} \right)^\perp. \end{aligned}$$

Then by taken <sup>F</sup>⊓<sup>2</sup> ¼ F<sup>⊓</sup> we obtain the need fuzzy submodule. □ When the orthogonality exist, there is unique. We have the following theorem to

show it.

**Theorem 2.22.** *Let* <sup>F</sup>⊓1*,* <sup>F</sup>⊓<sup>2</sup> *and* <sup>F</sup>⊓<sup>3</sup> *be three fuzzy submodules on module <sup>n</sup> pk , such that* F⊓1⊥F⊓<sup>2</sup> *and* F⊓1⊥F⊓3*, then* F⊓<sup>2</sup> ¼ F⊓3*.*

**Proof**. Consider that F⊓1⊥F⊓<sup>2</sup> and F⊓1⊥F⊓13.

Let *<sup>t</sup>*∈½ � 0, 1 , and *<sup>b</sup>*<sup>∈</sup> ð Þ <sup>F</sup>⊓<sup>2</sup> <sup>1</sup>�*<sup>t</sup>* , then <*a*, *b*> ¼ 0, for all *a*∈ð Þ F⊓<sup>1</sup> *<sup>t</sup>* . Thus *<sup>b</sup>*∈ð Þ <sup>F</sup>⊓<sup>3</sup> <sup>1</sup>�*<sup>t</sup>* and ð Þ <sup>F</sup>⊓<sup>2</sup> <sup>1</sup>�*<sup>t</sup>* <sup>⊆</sup>ð Þ <sup>F</sup>⊓<sup>3</sup> <sup>1</sup>�*<sup>t</sup>* . Therefore ð Þ F⊓<sup>3</sup> *<sup>t</sup>* ⊆ ð Þ F⊓<sup>3</sup> *<sup>t</sup>* .

In the same way, we show that ð Þ F⊓<sup>2</sup> *<sup>t</sup>* ⊆ð Þ F⊓<sup>3</sup> *<sup>t</sup>* . Therefore <sup>F</sup>⊓<sup>2</sup> ¼ F⊓3. □ **Corollary 2.23.** *The orthogonal of a fuzzy set on <sup>n</sup> pk is a fuzzy submodule on <sup>n</sup> pk .*

The orthogonality is an indempotent operator, in fact if F⊓ be a fuzzy submodule on *<sup>n</sup> pk* then <sup>F</sup>⊓<sup>⊥</sup> ð Þ<sup>⊥</sup> ¼ F⊓1.

The notion of equivalence on fuzzy linear code can be define as follow.

**Definition 2.24.** Let F⊓<sup>1</sup> and F⊓<sup>2</sup> be two fuzzy linear codes over *pk* . F⊓<sup>1</sup> and F⊓<sup>2</sup> are said to be equivalent if for all *t*∈½ � 0, 1 , the linear codes ð Þ F⊓<sup>1</sup> *<sup>t</sup>* and ð Þ F⊓<sup>2</sup> *<sup>t</sup>* are equivalent.

**Example 2.25.** *Let CG*<sup>1</sup> *and CG*<sup>2</sup> *be two equivalent linear codes of length n over pk .* We define the equivalent fuzzy linear codes as follow:

$$\begin{aligned} \mathcal{F}\sqcap\_1: \mathbb{Z}\_{p^k}^n &\to [0,1], \mathfrak{x} \mapsto \begin{cases} \mathbf{1} & \text{if } \,\,\mathfrak{x} \in \,\,\, \mathcal{C}\_{G\_1};\\ \mathbf{0} & \text{otherwise.} \end{cases} \text{ and } \mathfrak{x} \\\ \mathcal{F}\sqcap\_2: \mathbb{Z}\_{p^k}^n &\to [0,1], \mathfrak{x} \mapsto \begin{cases} \mathbf{1} & \text{if } \,\,\, \mathfrak{x} \in \,\,\, \mathcal{C}\_{G\_2};\\ \mathbf{0} & \text{otherwise.} \end{cases} . \end{aligned}$$

Thus the 1 and 0 -level cut of the both fuzzy linear codes give ð Þ F⊓<sup>1</sup> <sup>1</sup> ¼ *CG*<sup>1</sup> and ð Þ F⊓<sup>2</sup> <sup>1</sup> ¼ *CG*<sup>2</sup> ,

ð Þ <sup>F</sup>⊓<sup>1</sup> <sup>0</sup> <sup>¼</sup> *<sup>n</sup> pk* and ð Þ <sup>F</sup>⊓<sup>2</sup> <sup>0</sup> <sup>¼</sup> *<sup>n</sup> pk* .

**Remark 2.26.** Two equivalent fuzzy linear codes over *pk* have the same image.
