**2.3 Fuzzy linear codes in a practical way**

As we have said in the introduction, how fuzzy linear code can deal with uncertain information in a practical way? This subsection allow us to use directly fuzzyness in the information theory.

Let us draw the communication channel as follows:

$$F^k \xrightarrow{\quad Encoding} F^n \xrightarrow{\quad} \xrightarrow{\quad} \mathbb{R}^n \xrightarrow{\quad} \xrightarrow{\quad} F^k$$

Assume that *<sup>R</sup><sup>k</sup>* <sup>¼</sup> <sup>2</sup> <sup>2</sup> and *<sup>R</sup><sup>n</sup>* <sup>¼</sup> <sup>3</sup> 2, that means that *<sup>k</sup>* <sup>¼</sup> 2 and *<sup>n</sup>* <sup>¼</sup> 3. Let <sup>C</sup> <sup>⊆</sup> *<sup>R</sup>*<sup>3</sup> be a linear code over *R*, in the classical case, when we send a codeword *a* ¼ ð Þ 101 ∈C through a communication channel, the signal receive can be read as *a*<sup>0</sup> ¼ ð Þ 0*:*98, 0*:*03, 0*:*49 and modulate to *a*<sup>00</sup> ¼ ð Þ 100 . Thus to know if *a*<sup>00</sup> belong to the code C, we use syndrome calculation [14]. Since the modulation have gave a wrong word, we can consider that *a*<sup>0</sup> have more information than *a*00, in the sense that we can estimate a level to which a word 0 is modulate to 1, and a word 1 is modulate to 0. Therefore it is possible to use the idea of fuzzy logic to recover the transmit codeword.

Let <sup>C</sup> be a linear code over <sup>3</sup> 2. To each *a*∈C, we find *t*∈ ½ � 0, 1 such that *t* estimate the degree of which the element of <sup>3</sup> , obtain from *a* through the transmission channel belong to the code <sup>C</sup>. Thus in <sup>3</sup> <sup>2</sup> the information that we handle are certain, whereas in <sup>3</sup> there are uncertain. When we associate to all elements of <sup>3</sup> <sup>2</sup> the degree of which its correspond element obtain through the transmission channel belong to <sup>3</sup> 2, then we obtain a fuzzy code. If the fuzzy code are fuzzy linear code, then we can recover the code C just by using the upper *t*-level cut. Thus we deal directly with the uncertain information to obtain the code C.

The following example illustrate this reconstruction of the code by using uncertain information in the case of fuzzy linear code.

**Example 2.27.** *Let* <sup>3</sup> <sup>2</sup> ¼ f g 000, 001, 010, 100, 110, 101, 011, 111 *and C* ¼ f g 000, 001, 110, 111 *be a linear code over* 2*.*

Assume that after the transmission we obtain respectively f g 000; 0*:*01, 01; 1*:*01, 10; 1*:*001, 1, 0*:*<sup>999</sup> . Let <sup>F</sup><sup>⊓</sup> : <sup>3</sup> <sup>2</sup> ! ½ � 0, 1 such that

$$\begin{aligned} \; \_ {x \mapsto} \begin{cases} \{1\} & \text{if} \; \; x = 000; \\ \{0.99\} & \text{if} \; x = 001; \\ \{0.9\} & \text{if} \; x = 010; \\ \{0.9\} & \text{if} \; x = 100; \\ \{0.99\} & \text{if} \; x = 110; \\ \{0.9\} & \text{if} \; x = 101; \\ \{0.9\} & \text{if} \; x = 011; \\ \{0.99\} & \text{if} \; x = 111. \end{cases} \end{aligned} $$

Then by finding a *<sup>t</sup>*∈½ � 0, 1 such that <sup>F</sup>⊓*<sup>t</sup>* <sup>¼</sup> *<sup>x</sup>*∈<sup>3</sup> <sup>2</sup>j F⊓ð Þ *<sup>x</sup>* <sup>≥</sup>*<sup>t</sup>* � � <sup>¼</sup> *<sup>C</sup>*, we obtain *t*>0*:*9. Thus, for *t* ¼ 0*:*99, we are sure that the receive codeword is in C.

It should be better to investigate in deep this approach.

.
