论文标题
在最大添加性赫米尼亚级级代码上
On maximum additive Hermitian rank-metric codes
论文作者
论文摘要
受周的启发，“关于最大添加剂对称排名距离代码的等效性”（2020年），基于施密特的论文“对称双线性形式，在有限的字段上使用了编码理论的应用”（2015年），我们研究了最大$ d $ d $ d $ d $ d $ - hermitian矩阵的等效问题。更准确地说，在$ \ \ \ mathbb {f} _ {q^2} $上的Hermitian矩阵的$ \ mathrm {h} _n（q^2）$中，我们有两个可能的等价：一个经典的等价：来自地图中的一个从$ \ mathbb {f} $ {f} $ {限制那些保留等级和空间$ \ mathrm {h} _n（q^2）$的地图。我们证明，当$ d <n $和所考虑的代码是最大添加剂$ d $ - 代码和$（n-d）$ - 设计时，这两个等价关系的重合。结果，我们发现，与等级度量代码通常发生的情况不同，此类代码的理想信息不是区分。最后，我们处理已知最大遗产代码的组合特性，并通过这项调查，提出了一个新的最大Hermitian $ 2 $代码的家族，扩展了Longobardi等人提出的结构。在“具有限制的最大添加剂等级指标的自动形态组和新结构”（2020）。
Inspired by the work of Zhou "On equivalence of maximum additive symmetric rank-distance codes" (2020) based on the paper of Schmidt "Symmetric bilinear forms over finite fields with applications to coding theory" (2015), we investigate the equivalence issue of maximum $d$-codes of Hermitian matrices. More precisely, in the space $\mathrm{H}_n(q^2)$ of Hermitian matrices over $\mathbb{F}_{q^2}$ we have two possible equivalence: the classical one coming from the maps that preserve the rank in $\mathbb{F}_{q^2}^{n\times n}$, and the one that comes from restricting to those maps preserving both the rank and the space $\mathrm{H}_n(q^2)$. We prove that when $d<n$ and the codes considered are maximum additive $d$-codes and $(n-d)$-designs, these two equivalence relations coincide. As a consequence, we get that the idealisers of such codes are not distinguishers, unlike what usually happens for rank metric codes. Finally, we deal with the combinatorial properties of known maximum Hermitian codes and, by means of this investigation, we present a new family of maximum Hermitian $2$-code, extending the construction presented by Longobardi et al. in "Automorphism groups and new constructions of maximum additive rank metric codes with restrictions" (2020).
