关于秩度量码的一点注记
A note on rank-metric codes
摘要：
令$\mathbb{F}_q$表示具有$q=p^λ$元素的有限域。最大秩度量码（Maximum Rank-metric codes，简称MRD）是$M_{m\times n}(\mathbb{F}_q)$的子集，其元素个数达到类Singleton界。已知的第一个MRD代码是由Delsarte（1978）和Gabidulin（1985）发现的。Sheekey（2016）提出了一类新的基于$\mathbb{F}q$的MRD码，称为扭曲Gabidulin码，并提出了扭曲Gabidulin码到码$\mathcal{H}_{k,s}(L_1,L_2)$的推广。扭曲Gabidulin码的等价性和对偶性由Lunardoni，Trombetti和Zhou（2018）讨论。Trombetti-Zhou（2018）在$M_{2n\times 2n}(\mathbb{F}_q)$中发现了一类新的MRD代码。在这项工作中，我们刻画了一类代码提出的Sheekey的等价性，推广了已知的扭曲Gabidulin码和Trombetti-Zhou码的结果。在本文的第二部分中，我们将我们自己限制在$L_1(x)=x$的情况下，在那里我们给出了它的右核，中核，Delsarte对偶和伴随码。最后给出了$\mathcal{H}_{k,s}(x,L(x))$的自同构群，并计算了它的基数。特别地，我们得到了扭曲Gabidulin码的自同构群中的元素个数。
英文摘要：
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.
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