LRC codes over characteristic $2$

Abstract

In this work the construction of LRC codes given in [6] is completed, in the case of even characteristic. A general construction is presented, that enables us to obtain linear LRC codes of large length $n \approx q^4$, dimension and distance of order $q^4$, and locality $r =q-1$. In addition, the cases $q = 4$ and $q=8$ are studied.

Figures (6)

• Figure 1: Diagram of three splitting places $P_j$ of $F_0$ in $F_2/F_0$, for $q=2^{2l} > 5$. Each color, in each function field, represent a set $S_i$, for $1 \leq i \leq 4=q-1$, so each place has exactly two places of its same color above it.
• Figure 2: Diagram of the splitting of the rational places $P$ en $F_0$ that splits completely. From left to right, we may label the rational places of $F_2$ as $P_1, P_2, \ldots, P_8$.
• Figure 3: Graph relating the elements of $S_0 = \mathbb{F}_8 \setminus \mathbb{F}_2$, with an edge from $\alpha$ to $\beta$ for each pair that verifies the recursive equation that defines the tower $\mathcal{T}$ .
• Figure 4: Table with the studied examples
• Figure 5: Comparison of some examples in the table \ref{['tabla']}

Theorems & Definitions (35)

• Definition 1
• Theorem 1.1
• Definition 2
• Definition 3: AG Code
• Definition 4: Evaluation Code
• Definition 5: Tower of function fields
• Proposition 3.1
• Lemma 3.2
• proof
• proof