Optimal Binary Linear Codes from Maximal Arcs

Near-MDS Codes from Maximal Arcs in PG$(2,q)$

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…

Optimal Binary $(5,3)$ Projective Space Codes from Maximal Partial Spreads

Recently a construction of optimal non-constant dimension subspace codes, also termed projective space codes, has been reported in a paper of Honold-Kiermaier-Kurz. Restricted to binary codes in a 5-dimensional ambient space with minimum subspace distance 3, these optimal codes were interpreted in terms of maximal partial spreads of 2-dimensional subspaces. In a parallel development, an optimal binary (5,3) code was obtained by a minimal change strategy on a nearly optimal example of Etzion and Vardy. In this article, we report several examples of optimal binary (5,3) codes obtained by the app…

关于以交替形式为最大全 isotropic 子空间的二进制码

On Binary Codes That Are Maximal Totally Isotropic Subspaces with Respect to an Alternating Form

本文研究了以交替形式为最大全 isotropic 子空间的二进制码，分类了n≤24的情况，并推导了与该形式下正交补集的重量枚举式相关的MacWilliams型恒等式。

关于具有2、3和4个距离的极大二元码的规模

On the size of maximal binary codes with 2, 3, and 4 distances

本文研究了具有少数距离的二元码和二元等重码的最大规模，确定了所有长度$n\ge 6$下具有两个距离的极大二元码的精确规模，以及多种权重和长度下具有2、3和4个距离的极大二元等重码的精确规模。

具有最优列距离的二进制卷积码

Binary convolutional codes with optimal column distances

本文针对卷积码列距离构造研究较少的问题，提出一种在二进制域上构造具有最优列距离的卷积码的新方法。

最优码和弧的广义汉明重量

Optimal codes and arcs for the generalized Hamming weights

本文研究了广义汉明重量的最优码构造，通过几何方法证明了格里斯默界，并确定了低维下的最优二进制和三进制码参数。