Abstract
We compare the two duality theories of rank-metric codes proposed by Delsarte and Gabidulin, proving that the former generalizes the latter. We also give an elementary proof of MacWilliams identities for the general case of Delsarte rank-metric codes. The identities which we derive are very easy to handle, and allow us to re-establish in a very concise way the main results of the theory of rank-metric codes first proved by Delsarte employing association schemes and regular semilattices. We also show that our identities imply as a corollary the original MacWilliams identities established by Delsarte. We describe how the minimum and maximum rank of a rank-metric code relate to the minimum and maximum rank of the dual code, giving some bounds and characterizing the codes attaining them. Then we study optimal anticodes in the rank metric, describing them in terms of optimal codes (namely, MRD codes). In particular, we prove that the dual of an optimal anticode is an optimal anticode. Finally, as an application of our results to a classical problem in enumerative combinatorics, we derive both a recursive and an explicit formula for the number of \(k \times m\) matrices over a finite field with given rank and \(h\)-trace.
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Acknowledgments
The author is grateful to Elisa Gorla and to the Referees for many useful suggestions that improved the presentation of the paper. The author was partially supported by the Swiss National Science Foundation through Grant No. 200021_150207.
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Communicated by C. Ding.
Appendix: Explicit form of Theorems 31 and 62
Appendix: Explicit form of Theorems 31 and 62
Using known properties of binomial coefficients one can show that Theorem 31 implies Theorem 3.3 of [6] as an easy corollary. The following result, first proved by Delsarte using the theory of association schemes, may be regarded as the explicit version of Theorem 31.
Theorem 64
Let \(\mathcal {C} \subseteq \text{ Mat }(k \times m, \mathbb {F}_q)\) be a code. Let \({(A_i)}_{i \in \mathbb {N}}\) and \({(B_j)}_{j \in \mathbb {N}}\) be the rank distributions of \(\mathcal {C}\) and \(\mathcal {C}^\perp \), respectively. We have
for \(j =0,...,k\).
Proof
Throughout this proof the rows and columns of matrices are labeled from \(0\) to \(k\) for convenience (instead of from \(1\) to \(k+1\)). Define the matrix \(P \in \text{ Mat }(k+1 \times k+1,\mathbb {F}_q)\) by
for \(j,i \in \{0,...,k\}\). We can write the statement in matrix form as \((B_0,...,B_k)^t= P \cdot (A_0,...,A_k)^t\). Define matrices \(S,T \in \text{ Mat }(k+1 \times k+1,\mathbb {F}_q)\) by
for \(i,j \in \{0,...,k\}\). We notice that \(S\) is invertible, since it is lower-triangular and \(S_{ii}=1\) for \(i=0,...,k\). Theorem 31 reads \(S \cdot (B_1,...,B_k)^t = T \cdot (A_0,...,A_k)^t\), i.e., \((B_1,...,B_k)^t = S^{-1}T \cdot (A_0,...,A_k)^t\). Hence it suffices to prove \(P=S^{-1}T\), i.e., \(T=SP\). Fix arbitrary integers \(i,j \in \{0,...,k\}\). We have
Clearly,
and using the definition of \(q\)-binomial coefficient one finds
Hence we have
where the last equality follows from the \(q\)-Binomial Theorem [24], p. 74]. It follows
as claimed. \(\square \)
Arguing as in the proof of Theorem 62 and replacing Corollary 33 with Theorem 64 we easily obtain the following explicit version of Theorem 62.
Theorem 65
Let \(1 \le k \le m\), \(1 \le h \le k\) and \(0 \le r \le k\) be integers. The number of \(k \times m\) matrices over \(\mathbb {F}_q\) having rank \(r\) and zero \(h\)-trace is
Remark 66
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Ravagnani, A. Rank-metric codes and their duality theory. Des. Codes Cryptogr. 80, 197–216 (2016). https://doi.org/10.1007/s10623-015-0077-3
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DOI: https://doi.org/10.1007/s10623-015-0077-3