**2. Fuzzy linear codes over** *pk*

#### **2.1 Preliminaries**

The theory of fuzzy code as we present here were introduce by Shum and Chen De Gang [5], although they have authors such as Hall Diall and Von Kaenel [6, 7] who also worked on it. In this section, we shall formulate the preliminary definitions and results that are required for a good understanding of the sequel (we can see it in [8–10]).

**Definition 2.1.** Let *X* be a non-empty set, let *I* and *J* be two fuzzy subsets in *X*, then:


$$\bullet \ (I+f)(\mathbf{x}) = \max \ \{ I(\mathbf{y}) \land I(\mathbf{z}) | \mathbf{x} = \mathbf{y} + \mathbf{z} \}, \\ \ (I)(\mathbf{x}) = \max \ \{ I(\mathbf{y}) \land I(\mathbf{z}) | \mathbf{x} = \mathbf{y} \mathbf{z} \}.$$

These for all *x*, *y*, *z*∈*X*.

Let denoted by *M* the *pk* -module *<sup>n</sup> pk* , where *p* is a prime integer and *n*, *k*∈ nf g0 .

The following definitions on the fuzzy linear space derive from [11, 12].

**Definition 2.2.** We called a fuzzy submodule of *M*, a fuzzy subset F⊓ of a *pk* module *M* such that for all *x*, *y*∈ *M* and *r*∈*pk* , we have:

• Fð Þ *x* þ *y* ≥ *min* f g F⊓ð Þ *x* , F⊓ð Þ*y* .

$$\bullet \, \_{\mathcal{F}(\mathfrak{X}) \ge \mathcal{F} \sqcap (\mathfrak{X}) .}$$

**Definition 2.3.** Let F⊓ be a fuzzy subset of a nonempty set *M*. For *t* ∈½ � 0, 1 , we called the the upper *t*-level cut and lower *t*-level cut of F⊓, the sets F⊓*<sup>t</sup>* ¼ f g *x*∈ *M*jF⊓ð Þ *x* ≥*t* and F⊓*<sup>t</sup>* ¼ f g *x*∈ *M*jF⊓ð Þ *x* ≤*t* respectively.

**Proposition 2.4.** F⊓ *is a fuzzy submodule of an pk -module M if and only if for all α*, *β* ∈*pk ; x*, *y*∈ *M, we have* F⊓ð Þ *αx* þ *βy* ≥ *min* f g F⊓ð Þ *x* , F⊓ð Þ*y .*

The following difinition recalled the notion on fuzzy ideal of a ring.

**Definition 2.5.** *We called a fuzzy ideal of pk , a fuzzy subset I of a ring pk such that for each x*, *y*∈*pk ;*


**Definition 2.6.** Let *G* be a group and *R* a ring. We denote by **RG** the set of all formal linear combinations of the form *<sup>α</sup>* <sup>¼</sup> <sup>P</sup> *<sup>g</sup>* <sup>∈</sup> *<sup>G</sup>ag g* (where *ag* ∈*R* and *ag* ¼ 0 almost everywhere, that is only a finite number of coefficients are different from zero in each of these sums).

**Definition 2.7.** Let **RG** a ring group which is the group algebra of <*x*> on the ring *pk* (where *x* is an invertible element of *pk* ). A fuzzy subset *I* of **RG** is called a fuzzy ideal of **RG**, if for all *α*, *β* ∈ **RG**,

$$\bullet \ I(a\emptyset) \ge \max\left\{ I(a), I(\beta) \right\}.$$

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

• *I*ð Þ *α* � *β* ≥ *min I* f g ð Þ *α* ,*I*ð Þ *β* .

When we use the transfer principle in [13], we easily get the next Proposition. **Proposition 2.8.** *A is a fuzzy ideal of RG if and only if for all t* ∈½ � 0, 1 *, if At* 6¼ ∅*, then At is an ideal of RG.*

The following is very important, the give the meaning of the linear code over the ring *pk* .

**Definition 2.9.** A submodule of *<sup>n</sup> pk* , is called a linear code of length *n* over *pk* . (with *n* a positive integer).

Contrary to the vector spaces, the module do not admit in general a basis. However it possesses a generating family and therefore a generating matrix, but the decomposition of the elements on this family is not necessarily unique.

**Definition 2.10.** We called generating matrix of a linear code over *pk* all matrix � �, where the lines are the minimal generating family of code.

of M *pk* The equivalence of two codes is define by the following definition.

**Definition 2.11.** Let *Cpk* and *C*<sup>0</sup> *pk* two linear codes over *pk* of generating matrix *G* and *G*<sup>0</sup> respectively. The codes *Cpk* and *C*<sup>0</sup> *pk* are equivalences if there exists a permutation matrix *P*, such that *G*<sup>0</sup> ¼ *GP*.

To define a dual of a code which is helpful when we fine some properties of the codes, we need to know the inner product.

**Definition 2.12.** Let *Cpk* be a linear code of length *n* over *pk* , the dual of the code *Cpk* that we note *C*<sup>⊥</sup> *pk* is the submodule of *<sup>n</sup> pk* define by; *<sup>C</sup>*<sup>⊥</sup> *pk* ¼ f*a*∣ for all

*<sup>b</sup>*∈*Cpk* , *<sup>a</sup>:<sup>b</sup>* <sup>¼</sup> <sup>0</sup>g. where "�" is the natural inner product on the submodule *<sup>n</sup> pk* .

In a linear code *Cpk* of length *n* over *pk* , if for all ð Þ *a*0, ⋯, *an*�<sup>1</sup> ∈*Cpk* , then *s a* ð Þ ð Þ 0, ⋯, *an*�<sup>1</sup> ∈*Cpk* (where *s* is the shift map), then the code is said to cyclic.
