# Number theory days

Jun 23 – 27, 2014
Université Lille 1
Europe/Paris timezone

## Skew Generalized Quasi-Cyclic Codes

Jun 25, 2014, 11:45 AM
1h
Salle de réunions (Université Lille 1)

### Salle de réunions

#### Université Lille 1

U.M.R. CNRS 8524 U.F.R. de Mathématiques 59 655 Villeneuve d'Ascq Cédex
Noncommutative algebra

### Speaker

Patrick Solé (Telecom ParisTech)

### Description

In this article we introduce skew generalized quasi-cyclic codes over finite field $F$ with Galois automorphism $\theta$. This is a generalization of quasi-cyclic codes and skew polynomial codes. These codes have an added advantage over quasi-cyclic codes, since the length of the code $C$ need not be a multiple of the index of $C$. After a brief description of the skew polynomial ring $F[x;\theta]$, it is shown that a skew generalized quasi-cyclic code $C$ is a left submodule of $R_1\times R_2\times \cdots R_l$, where $R_i = F[x;\theta]/(x^{m_i}-1),\,\left\vert \left\langle \theta\right\rangle \right\vert = m$ and $m|m_{i}$ for all $i=1,\ldots ,l$. This method provides a direct construction of many codes with best known parameters over $GF(4)$. Joint work with T. Abualrub, P. Seneviratne

### Presentation materials

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