**Acknowledgements**

This research has been partially supported by the COST Action IC1104 and the project ARES (Team for Advanced Research on Information Security and Privacy. Funded by Ministry of Economy and Competitivity).

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

constitutes a basis over the prime field *p*, and the field extensions *pn* ffi

**2.3 Generating functions of conjugacy classes in a group**

The automorphism group of the projective line *<sup>q</sup>*

edges are *k*�element sets f g *x*1, … , *xk* ∈

*Probability, Combinatorics and Control*

of the *k*�categories.

*a basis for <sup>n</sup>*

that is, a *PGq*

**2.4 Conclusion**

**Acknowledgements**

**268**

that is, number of conjugacy classes in *A*, then <sup>1</sup>

Consider the normal rational curve over *q*:

Funded by Ministry of Economy and Competitivity).

<sup>1</sup> ≔ *<sup>q</sup>* 1, *x*, *x*<sup>2</sup>

is a (*q* + 1)-arc in the *n*-dimensional projective space *PG n*ð Þ , *q* .

*q* + 1 points, and every set of *n* þ 1 points are linearly independent.

V*n*

*p*½ � *<sup>x</sup> <sup>=</sup> <sup>x</sup>*ð Þ *<sup>n</sup>*�<sup>1</sup> <sup>þ</sup> … <sup>þ</sup> *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> are isomorphic. □

group *PGL*ð Þ 2, *q* . Any finite subgroup *A* ⊂*PGL*ð Þ 2, *q* defines a *k*�uniform Cayley (sum) hypergraph <sup>Γ</sup>*k*ð Þ *<sup>A</sup>* whose vertices are the generating *<sup>k</sup>*�tuples of *<sup>A</sup>* and the

*G k* 

*x*1, … , *xk*. In particular, if *f z*ð Þ is the ordinary generating function that enumerates *A*,

**Definition 2.13.** *In PG n*ð Þ � 1, *q a k*ð Þ� ;*r arc is a set of k points any r of which form*

, … , *<sup>x</sup><sup>n</sup>* <sup>j</sup> *<sup>x</sup>*∈*<sup>q</sup>* ∪ ∞f g

ð Þ *n*, *q* . So we can enumerate how many NRC's are there in a *PG n*ð Þ , *q* .

We see that if *q*≤*n*, the NRC is a basis of a *q*-dimensional projective subspace,

The problem of considering finite subgroups and conjugacy classes in *PGL*ð Þ 2, *q* the automorphism group of the projective line can be generalised to that of finite subgroups in *PGL n*ð Þ , *q* , the collineation group of the normal rational curve.

This research has been partially supported by the COST Action IC1104 and the project ARES (Team for Advanced Research on Information Security and Privacy.

The answer is *ϕ*ð Þ *q*; *n*, *q* , the number of ways of choosing such a set of points in a particular *q*-space. If *q*≥ *n* þ 2, the NRC is an example of a (*q* + 1)-arc. It contains

*q, or in other words, r* � 1 *of them but not r are collinear.*

function enumerating sequences of *k* elements in *A*. If *G* is an abelian group, then *x*<sup>1</sup> þ ⋯ þ *xk* ∈ *A*. In general, we will consider *k*-arcs in Γð Þ *A* which represent casual connections between the variables. Applications are known in statistics, for example the multinomial experiment consists of *n* identical independent trials, and there are *k* possible outcomes (classes, categories or cells) to each trial and the cell counts *n*1, *n*2, … , *nk* are the random variables, the number of observations that fall into each

is the projective linear

represented by random variables

<sup>1</sup>�*f z*ð Þ is the ordinary generating
