论文标题
排名代码的注释
A note on rank-metric codes
论文作者
论文摘要
令$ \ mathbb {f} _q $用$ q = p^λ$元素表示有限字段。最大排名代码（简称MRD）是$ M_ {M \ times n}的子集（\ Mathbb {f} _Q）$，其元素数量达到类似Singleton的界限。 Delsarte（1978）和Gabidulin（1985）发现了第一个MRD代码。 Sheekey（2016）在$ \ mathbb {f} Q $上提出了一种新的MRD代码，称为扭曲的Gabidulin代码，还提出了将扭曲的Gabidulin代码概括到代码$ \ MATHCAL {H} _ {k，s}（k，s}（l_1，l_1，l_2）$。 Lunardoni，Trombetti和Zhou（2018）讨论了扭曲的Gabidulin代码的等效性和二元性。 Trombetti-Zhou（2018）找到了$ m_ {2n \ times 2n} $ m_ {2n \ times 2n}（\ mathbb {f} _q）$的新的MRD代码。在这项工作中，我们表征了Sheekey提出的代码类别的等效性，概括了扭曲的Gabidulin代码和Trombetti-Zhou代码所知的结果。在本文的第二部分中，我们将自己限制在$ l_1（x）= x $的情况下，在那里我们呈现其正确的核，中核，delsarte dual dual dual dual dual dual dual dual和expaint代码。在最后一节中，我们介绍了$ \ Mathcal {h} _ {k，s}（x，x，l（x））$的自动形态组并计算其基数。特别是，我们获得了扭曲Gabidulin代码的自动形态组中的元素数量。
Let $\mathbb{F}_q$ denote the finite field with $q=p^λ$ elements. Maximum Rank-metric codes (MRD for short) are subsets of $M_{m\times n}(\mathbb{F}_q)$ whose number of elements attains the Singleton-like bound. The first MRD codes known was found by Delsarte (1978) and Gabidulin (1985). Sheekey (2016) presented a new class of MRD codes over $\mathbb{F}q$ called twisted Gabidulin codes and also proposed a generalization of the twisted Gabidulin codes to the codes $\mathcal{H}_{k,s}(L_1,L_2)$. The equivalence and duality of twisted Gabidulin codes was discussed by Lunardoni, Trombetti, and Zhou (2018). A new class of MRD codes in $M_{2n\times 2n}(\mathbb{F}_q)$ was found by Trombetti-Zhou (2018). In this work, we characterize the equivalence of the class of codes proposed by Sheekey, generalizing the results known for twisted Gabidulin codes and Trombetti-Zhou codes. In the second part of the paper, we restrict ourselves to the case $L_1(x)=x$, where we present its right nucleus, middle nucleus, Delsarte dual and adjoint codes. In the last section, we present the automorphism group of $\mathcal{H}_{k,s}(x,L(x))$ and compute its cardinality. In particular, we obtain the number of elements in the automorphism group of the twisted Gabidulin codes.
