A construction of Abelian non-cyclic orbit codes

Vortrag von Prof. Dr. Joan-Josep Climent

Sprecher eingeladen von: Prof. Dr. Joachim Rosenthal

Datum: 20.12.17  Zeit: 11.00 - 12.00  Raum:

A constant dimension code consists of a set of $k$-dimensional subspaces of $\mathbb{F}_{q}^{n}$, where $\mathbb{F}_{q}$ is a finite field of $q$ elements. Orbit codes are constant dimension codes which are defined as orbits under the action of a subgroup of the general linear group on the set of all $k$-dimensional subspaces of $\mathbb{F}_{q}^{n}$. If the acting group is Abelian, we call the corresponding orbit code Abelian orbit code.
In this paper we present a construction of an Abelian non-cyclic orbit code for which we compute its cardinality and its minimum subspace distance. Moreover, the minimum subspace distance of our code is maximal.
More specifically, we will assume that $q=p^{r}$, with $r \geq 1$, $p$ a prime number and, $n$ and $k$ such that $p^{r-1} \leq n-2k < k < n-k \leq p^{r} = q$.
In this talk we present an Abelian non-cyclic subgroup of $GL_{n}$ which is a direct product of two cyclic subgroups of orders $p^{r}-1$ and $p^{r} (p^{r}-1)$; then, the corresponding orbit code defined by the orbit of the standard $k$-dimensional vector subspace under the action of that subgroup has size $p^{r} (p^{r}-1)$, length $n$, and minimum subspace distance $2k$.
Moreover, the minimum subspace distance of the orbit code will be maximal for $k = p^{r-1}(p-1)$ and $n = p^{r-1}(2p-1)$.