论文标题
pg $（2，q）$中最大弧的近MDS代码
Near-MDS Codes from Maximal Arcs in PG$(2,q)$
论文作者
论文摘要
$ [n，k，d] $线性代码$ {\ cal c} $的单例缺陷定义为$ s（{\ cal c}）= n-k+1-d $。 $ s（{\ cal c}）= 0 $的代码称为最大距离可分离（MDS）代码，并且具有$ s（\ cal c）= s（\ cal c ^{\ bot}）= 1 $的代码接近最大距离可分开（NMDS）代码。 MDS代码和NMDS代码在有限的投影几何形状中均具有良好的表示。 MDS代码超过$ f_q $，带有长度$ n $和PG $中的$ n $ -Arcs（K-1，Q）$是等效的对象。当$ k = 3 $时，长度$ n $的NMDS代码相当于$（n，3）$ - Arcs in Pg $（2，Q）$。在本文中，我们使用尺寸3的NMDS代码处理。通过在PG $（2，Q）$的最大弧中添加一些合适的投影点，我们可以获得两类的$（q+5,3）$ - 弧（或等效地[q+5,3，q+2] $ NMDS代码） 我们还确定了确切的重量分布 以及此类NMDS代码及其双重的局部性。事实证明，所得的NMDS代码及其双重既是距离最佳和最佳的局部可回收代码。
The singleton defect of an $[n,k,d]$ linear code ${\cal C}$ is defined as $s({\cal C})=n-k+1-d$. Codes with $S({\cal C})=0$ are called maximum distance separable (MDS) codes, and codes with $S(\cal C)=S(\cal C ^{\bot})=1$ are called near maximum distance separable (NMDS) codes. Both MDS codes and NMDS codes have good representations in finite projective geometry. MDS codes over $F_q$ with length $n$ and $n$-arcs in PG$(k-1,q)$ are equivalent objects. When $k=3$, NMDS codes of length $n$ are equivalent to $(n,3)$-arcs in PG$(2,q)$. In this paper, we deal with the NMDS codes with dimension 3. By adding some suitable projective points in maximal arcs of PG$(2,q)$, we can obtain two classes of $(q+5,3)$-arcs (or equivalently $[q+5,3,q+2]$ NMDS codes) for any prime power $q$. We also determine the exact weight distribution and the locality of such NMDS codes and their duals. It turns out that the resultant NMDS codes and their duals are both distance-optimal and dimension-optimal locally recoverable codes.
