We compare the two duality theories of rank-metric codes proposed by Delsarteand Gabidulin, proving that the former generalizes the latter. We also give anelementary proof of MacWilliams identities for the general case of Delsarterank-metric codes. The identities which we derive are very easy to handle, andallow us to re-establish in a very concise way the main results of the theoryof rank-metric codes first proved by Delsarte employing the theory ofassociation schemes and regular semilattices. We also show that our identitiesimply as a corollary the original MacWilliams identities established byDelsarte. We describe how the minimum and maximum rank of a rank-metric coderelate to the minimum and maximum rank of the dual code, giving some bounds andcharacterizing the codes attaining them. Then we study optimal anticodes in therank metric, describing them in terms of optimal codes (namely, MRD codes). Inparticular, we prove that the dual of an optimal anticode is an optimalanticode. Finally, as an application of our results to a classical problem inenumerative combinatorics, we derive both a recursive and an explicit formulafor the number of $k \times m$ matrices over a finite field with given rank and$h$-trace.
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机译：我们比较了Delsarteand Gabidulin提出的秩度量码的两种对偶理论，证明了前者可以概括后者。对于Delsarterank度量代码的一般情况，我们还提供MacWilliams身份的初步证明。我们得出的身份很容易处理，并允许我们以一种非常简洁的方式重新建立Delsarte首先使用关联方案和规则半格理论证明的秩度量码理论的主要结果。我们还表明，我们的身份暗示着由德尔萨特建立的最初的MacWilliams身份。我们描述了秩度量代码的最小和最大秩如何与对偶代码的最小和最大秩相关，给出了一些界限并表征了获得它们的代码。然后，我们研究排名度量标准中的最佳反码，并根据最佳码（即MRD码）对其进行描述。特别地，我们证明了最佳反码的对偶是最佳反码。最后，将我们的结果应用到经典问题枚举组合中，我们得出了在给定秩和$ h $ -trace的有限域上$ k \ times m $矩阵的数量的递归和显式公式。
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