论文标题
将痕量代码概括为高度的地方
Generalization of trace codes to places of higher degree
论文作者
论文摘要
在本说明中，我们使用$ f $（不一定是第一级）的位置对代数函数字段$ f/ \ mathbb {f} _ {q} $进行代码的构造，并从$ \ mathbb {f} _ {q} $的各种扩展功能进行跟踪函数。这是将几何GOPPA代码的痕量代码概括为更高程度的位置。我们在此代码的维度上计算一个界限。此外，我们给出了一个条件，在该条件下，我们获得了代码的确切维度。我们还根据$ b_ {r}（f）$（$ f $中的$ r $的位置$ r $），$ 1 \ leq r <\ r <\ infty $来确定该代码最小距离的限制。在$ \ mathbb {f} _ {p} $上，几个准循环代码也可以作为这些代码的示例获得。
In this note, we give a construction of codes on algebraic function field $F/ \mathbb{F}_{q}$ using places of $F$ (not necessarily of degree one) and trace functions from various extensions of $\mathbb{F}_{q}$. This is a generalization of trace code of geometric Goppa codes to higher degree places. We compute a bound on the dimension of this code. Furthermore, we give a condition under which we get exact dimension of the code. We also determine a bound on the minimum distance of this code in terms of $B_{r}(F)$ ( the number of places of degree $r$ in $F$), $1 \leq r < \infty$. Few quasi-cyclic codes over $\mathbb{F}_{p}$ are also obtained as examples of these codes.
