**2. Algebraic geometric codes**

Let *<sup>q</sup>* be a finite field of *q* elements, where *q* is a power of a prime. We consider as an alphabet a set P ¼ f g *P*1, … , *PN* of *N* � *<sup>q</sup>* rational points lying on a smooth projective curve C of genus *g* and degree *d* defined over the field *q*. If *D* is a divisor on the curve C, Lð Þ *D* is the linear series attached to this divisor with coefficients in the field.

**Definition 2.1.** *Algebraic Geometric Codes (AGC) are constructed by evaluation of the global sections of a line bundle or a vector bundle on the curve* C *over N N*ð Þ > *g distinct rational places P*1, … , *PN. Namely, let F*∣*<sup>q</sup> be the function field of the curve,* D *the divisor P*<sup>1</sup> þ ⋯ þ *PN and G a divisor of F*∣*<sup>q</sup> of degree s* ≤ *N such that* Supp G ∩ Supp D ¼ ø*. Then the geometric Goppa code associated with the divisors D and G is defined by*

$$\mathbf{C}(D, G) = \{ (\mathfrak{x}(P\_1), \dots, \mathfrak{x}(P\_n)) | \mathfrak{x} \in \mathcal{L}(G) \} \subseteq \mathbb{F}\_{q^n}.$$

*A Geometrical Realisation of Quasi-Cyclic Codes DOI: http://dx.doi.org/10.5772/intechopen.88288*

given metric properties as a correcting code. The codes considered here are codes for channels whose errors are consistent with the weighted Hamming metric

Let C be a non-singular, projective, irreducible curve defined over *q*, as the vanishing locus of a polynomial *F* ∈*q*½ � *x*0, *x*1, *x*<sup>2</sup> . We define the number *N q*ð Þ of

It is a polynomial in *q* with integer coefficients, whenever *q* is a prime power.

*r*¼1

One of the main problems in coding theory is to obtain non-trivial lower bounds

Garcia and Stichtenoth analysed the asymptotic behaviour of the number of rational places and the genus in towers of function fields, [2]. From Garcia-Stichtenoth's second tower one obtains codes over any field *<sup>q</sup>* where *q* is an even

#C *qr* � � *t r r*

!

� �j*F x*ð Þ¼ 0, *<sup>x</sup>*1, *<sup>x</sup>*<sup>2</sup> <sup>0</sup> � �∣*:*

� � on <sup>C</sup> over the extensions *qr* of *<sup>q</sup>* is encoded in an

*:*

� �*<sup>n</sup>*

� �<sup>∞</sup>

� �*<sup>n</sup>* denote the

. Such a code is

*<sup>i</sup>*¼<sup>1</sup> such

*N q*ð Þ¼ <sup>∣</sup> ð Þ *<sup>x</sup>*0, *<sup>x</sup>*1, *<sup>x</sup>*<sup>2</sup> <sup>∈</sup><sup>2</sup> *<sup>q</sup>*

exponential generating series, called the zeta function of C:

*<sup>Z</sup>*ð Þ¼ <sup>C</sup>, *<sup>t</sup> exp* <sup>X</sup><sup>∞</sup>

of the number *N F*ð Þ*<sup>i</sup>* of rational places of towers of function fields *Fi=<sup>q</sup>*

**Notation.** Let *<sup>q</sup>* denote the Galois field of *q* elements and let *<sup>q</sup>*

0, they are in fact extended generalised Reed-Solomon codes.

*n* and dimension *k* over *<sup>q</sup>* is a *k*-dimensional subspace of *<sup>q</sup>*

that *Fi* ⊊ *Fi*þ1. Suitable families of function fields, for example good towers of function fields, have been used to construct families of codes that beat the Gilbert-Varshamov bound. This paper aims to explore this link for the study and construction of quasi-cyclic codes. For example good codes are obtained for curves of genus

vector spaces of all ordered *n*-tuples over *q*. The Hamming weight of a vector *x*, denoted by *wt x*ð Þ is then number of non-zero entries in *x*. A linear code *C* of length

called ½ � *n*, *k*, *d <sup>q</sup>* code if its minimum Hamming distance is *d*. For *d* a positive integer, *α* ¼ ð Þ *α*1, … , *α<sup>m</sup>* is a partition of *d* into *m* parts if the *α<sup>i</sup>* are positive and decreasing.

Let *<sup>q</sup>* be a finite field of *q* elements, where *q* is a power of a prime. We consider as an alphabet a set P ¼ f g *P*1, … , *PN* of *N* � *<sup>q</sup>* rational points lying on a smooth projective curve C of genus *g* and degree *d* defined over the field *q*. If *D* is a divisor on the curve C, Lð Þ *D* is the linear series attached to this divisor with coefficients in

**Definition 2.1.** *Algebraic Geometric Codes (AGC) are constructed by evaluation of*

*the global sections of a line bundle or a vector bundle on the curve* C *over N N*ð Þ > *g distinct rational places P*1, … , *PN. Namely, let F*∣*<sup>q</sup> be the function field of the curve,* D

Supp G ∩ Supp D ¼ ø*. Then the geometric Goppa code associated with the divisors D*

**C**ð Þ¼ *D*, *G* f g ð*x P*ð Þ<sup>1</sup> , … , *x P*ð Þ*<sup>n</sup>* Þj*x*∈Lð Þ *G* ⊆*qn :*

*the divisor P*<sup>1</sup> þ ⋯ þ *PN and G a divisor of F*∣*<sup>q</sup> of degree s* ≤ *N such that*

(WHM).

*q*�rational points on the curve to be

*Probability, Combinatorics and Control*

The number of points C *qr*

**2. Algebraic geometric codes**

the field.

**260**

*and G is defined by*

power of a prime [3].

Recall that *qn* ∣*<sup>q</sup>* is a cyclic Galois extension and it is finitely generated by unique element *<sup>α</sup>* <sup>∈</sup>*qn* <sup>n</sup>*q*. *<sup>α</sup>* is a primitive element and 1, *<sup>α</sup>*, *<sup>α</sup>*2, … , *<sup>α</sup>n*�<sup>1</sup> � � is a basis of the field extension *<sup>q</sup>* ↪ *q*ð Þ *α* , that is, *qn* ffi *<sup>q</sup>* � �*<sup>n</sup>* .

In the sequel, an ½ � *n*, *k <sup>q</sup>*-code *C* is a *k*-dimensional subspace of *<sup>q</sup>* � �*<sup>n</sup> :*
