**2.4 Fuzzy cyclic code over** *pk*

Let the module *<sup>n</sup> pk* , in this subsection we will consider the case where the integers *n* and *p* are coprime.

**Definition 2.28.** A fuzzy module <sup>F</sup><sup>⊓</sup> on the module *<sup>n</sup> pk* is called a fuzzy cyclic code of length *<sup>n</sup>* over *pk* if for all ð Þ *<sup>a</sup>*0, *<sup>a</sup>*1, <sup>⋯</sup>, *an*�<sup>1</sup> <sup>∈</sup>*<sup>n</sup> pk* , then

F⊓ð Þ ð Þ *an*�1, *a*0, ⋯, *an*�<sup>2</sup> ≥ F⊓ð Þ ð Þ *a*0, *a*1, ⋯, *an*�<sup>1</sup> .

The following proposition give a caracterization of the fuzzy cyclic codes. **Proposition 2.29.** *[15] A fuzzy submodule* <sup>F</sup><sup>⊓</sup> *on on <sup>n</sup> pk is a fuzzy cyclic code if and only if for all.*

ð Þ *<sup>a</sup>*0, *<sup>a</sup>*1, <sup>⋯</sup>, *an*�<sup>1</sup> <sup>∈</sup>*<sup>n</sup> pk* , we have F⊓ðð Þ *a*0, *a*1, ⋯, *an*�<sup>1</sup> Þ ¼ F⊓ðð Þ *an*�1, *a*0, ⋯, *an*�<sup>2</sup> Þ ¼ ⋯ ¼

$$\mathcal{F} \sqcap ((a\_1, a\_2, \dots, a\_{n-1}, a\_0)).$$

**Proposition 2.30.** F⊓ *is a fuzzy cyclic code of length n over pk if and only if for all <sup>t</sup>*∈½ � 0, 1 *, if* ð Þ <sup>F</sup><sup>⊓</sup> *<sup>t</sup>* 6¼ <sup>∅</sup>*, then* ð Þ <sup>F</sup><sup>⊓</sup> *<sup>t</sup> is a ideal of the factor ring pk* ½ � *<sup>X</sup> <sup>X</sup><sup>n</sup>* ð Þ �<sup>1</sup> *.*

**Proof**. Assume that F⊓ is a fuzzy cyclic code over *pk* and *t* ∈½ � 0, 1 such that ð Þ F⊓ *<sup>t</sup>* 6¼ ∅. Then ð Þ F⊓ *<sup>t</sup>* is a cyclic code over *pk* .

Let *ψ* : *<sup>n</sup> pk* ! *pk* ½ � *<sup>X</sup> <sup>X</sup><sup>n</sup>* ð Þ �<sup>1</sup> , *<sup>c</sup>* <sup>¼</sup> ð Þ *<sup>c</sup>*0, <sup>⋯</sup>,*cn*�<sup>1</sup> <sup>↦</sup> *<sup>ψ</sup>*ð Þ¼ *<sup>c</sup>* <sup>P</sup>*n*�<sup>1</sup> *<sup>i</sup>*¼<sup>0</sup> *ciX<sup>i</sup>* .

It is prove by easy way that *ψ* is a isomorphism of *pk* -module, which send a cyclic codes over *pk* onto the ideals of the factor ring *pk* ½ � *<sup>X</sup> <sup>X</sup><sup>n</sup>* ð Þ �<sup>1</sup> . Therefore, <sup>∀</sup>*t*∈½ � 0, 1 , <sup>F</sup>⊓*<sup>t</sup>* is a ideal of *pk* ½ � *<sup>X</sup> <sup>X</sup><sup>n</sup>* ð Þ �<sup>1</sup> .

Conversely, assume that, ∀*t* ∈½ � 0, 1 such that F⊓*<sup>t</sup>* 6¼ ∅, F⊓*<sup>t</sup>* is a ideal of factor ring *pk* ½ � *X <sup>X</sup><sup>n</sup>* ð Þ �<sup>1</sup> . Since <sup>F</sup>⊓*<sup>t</sup>* is a ideal of factor ring *pk* ½ � *<sup>X</sup> <sup>X</sup><sup>n</sup>* ð Þ �<sup>1</sup> , then <sup>F</sup>⊓*<sup>t</sup>* is a submodule of *pk* module *<sup>n</sup> pk* . Hence F⊓*<sup>t</sup>* 6¼ ∅, is a linear code over *pk* , then F⊓ is a fuzzy linear code. Due to *ψ*, ∀*t*∈½ � 0, 1 , F⊓*<sup>t</sup>* is a cyclic code over *pk* , then F⊓ is a fuzzy cyclic code over *pk* . □

Since *pk* is a finite ring, then *Im*ð Þ¼ F <sup>F</sup><sup>⊓</sup> <sup>⊓</sup>ð Þ *<sup>x</sup>* <sup>∈</sup>½ �j 0, 1 *<sup>x</sup>*∈*<sup>n</sup> pk* n o is also finite. Assume that *Im*ð Þ¼ F⊓ f g *t*<sup>1</sup> >*t*<sup>2</sup> > ⋯ >*tm* , then F⊓*<sup>t</sup>*<sup>1</sup> ⊆ F⊓*<sup>t</sup>*<sup>2</sup> ⊆⋯⊆ F⊓*tm*�<sup>1</sup> ⊆ *Atm* ¼ *n pk* .

Let *g* ð Þ*k <sup>i</sup>* ð Þ *X* ∈ *pk* ½ � *X* the generator polynomial of F⊓*ti* , note that *g* ð Þ*k <sup>i</sup>* ð Þ *X* is the Hensel lifting of order *k* of some polynomial *gi* ð Þ *<sup>X</sup>* <sup>∈</sup> *p*½ � *<sup>X</sup>* which divide *<sup>X</sup><sup>n</sup>* � 1, the cyclic code <*g* ð Þ*k <sup>i</sup>* ð Þ *X* > ⊂ *pk* ½ � *X <sup>X</sup><sup>n</sup>* ð Þ �<sup>1</sup> is called the **lifted code** of the cyclic code < *gi* ð Þ *X* > ⊂ *p*½ � *X <sup>X</sup><sup>n</sup>* ð Þ �<sup>1</sup> [8].

Since <sup>F</sup>⊓*<sup>t</sup>*<sup>1</sup> <sup>⊆</sup> <sup>F</sup>⊓*<sup>t</sup>*<sup>2</sup> ⊆⋯⊆ <sup>F</sup>⊓*tm*�<sup>1</sup> <sup>⊆</sup> <sup>F</sup>⊓*tm* <sup>¼</sup> *<sup>n</sup> pk* , then *g* ð Þ*k <sup>i</sup>*þ<sup>1</sup>ð Þ *<sup>X</sup>* <sup>∣</sup>*<sup>g</sup>* ð Þ*k <sup>i</sup>* ð Þ *X* , *i* ¼ 1, <sup>⋯</sup>, *<sup>m</sup>* � 1. So we define the polynomial *<sup>h</sup>*ð Þ*<sup>k</sup> <sup>i</sup>* ð Þ¼ *<sup>X</sup> <sup>X</sup><sup>n</sup>* ð Þ � <sup>1</sup> *<sup>=</sup><sup>g</sup>* ð Þ*k <sup>i</sup>* ð Þ *X* which is called the check polynomial of the cyclic code F⊓*ti* ¼ <*g* ð Þ*k <sup>i</sup>* ð Þ *X* > , *i* ¼ 1, ⋯, *m*.

**Theorem 2.31.** *Let* G ¼ *g* ð Þ*k* <sup>1</sup> ð Þ *X* , *g* ð Þ*k* <sup>2</sup> ð Þ *<sup>X</sup>* , <sup>⋯</sup>, *<sup>g</sup>*ð Þ*<sup>k</sup> <sup>m</sup>* ð Þ *X* n o *be a set of polynomial in pk* ½ � *X , such that gi* ð Þ *<sup>X</sup> divide X<sup>n</sup>* � <sup>1</sup>*, i* <sup>¼</sup> 1, <sup>⋯</sup>, *m. If g*ð Þ*<sup>k</sup> <sup>i</sup>*þ<sup>1</sup>ð Þ *<sup>X</sup>* <sup>∣</sup>*<sup>g</sup>* ð Þ*k <sup>i</sup>* ð Þ *X for i* ¼ 1, 2, <sup>⋯</sup>, *<sup>m</sup>* � <sup>1</sup> *and* <sup>&</sup>lt; *<sup>g</sup>*ð Þ*<sup>k</sup> <sup>m</sup>* ð Þ *<sup>X</sup>* <sup>&</sup>gt; <sup>¼</sup> *<sup>n</sup> pk , then the set* G *can determined a fuzzy cyclic code* F⊓ *and* <*g* ð Þ*k <sup>i</sup>* ð Þ *X* >j*i* ¼ 1, ⋯, *m* n o *is the family of upper level cut cyclic subcodes of* <sup>F</sup>⊓*.* The proof is leave for the reader but he can check it in [15].
