Rank-metric codes and their duality theory

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.

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

$$\begin{aligned} B_j=\frac{1}{|\mathcal {C}|} \sum \limits _{i=0}^k A_i \sum \limits _{s=0}^k {(-1)}^{j-s} q^{ms+ \left( {\begin{array}{c}j-s\\ 2\end{array}}\right) } \begin{bmatrix} k-s \\ k-j \end{bmatrix} \begin{bmatrix} k-i \\ s \end{bmatrix} \end{aligned}$$

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

$$\begin{aligned} P_{ji}{:=}\frac{1}{|\mathcal {C}|} \sum _{s=0}^k (-1)^{j-s} q^{ms+ \left( {\begin{array}{c}j-s\\ 2\end{array}}\right) } \begin{bmatrix} k-s \\ k-j \end{bmatrix} \begin{bmatrix} k-i \\ s \end{bmatrix} \end{aligned}$$

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

$$\begin{aligned} S_{ij}{:=} \begin{bmatrix} k-j \\ i-j \end{bmatrix}, \ \ \ \ \ T_{ij}{:=} \frac{q^{mi}}{|\mathcal {C}|} \begin{bmatrix} k-j \\ i \end{bmatrix} \end{aligned}$$

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

$$\begin{aligned} (SP)_{ij}= & {} \frac{1}{|\mathcal {C}|}\sum _{r=0}^k \begin{bmatrix} k-r \\ i-r \end{bmatrix} \sum _{s=0}^k (-1)^{r-s} q^{ms+ \left( {\begin{array}{c}r-s\\ 2\end{array}}\right) } \begin{bmatrix} k-s \\ k-r \end{bmatrix} \begin{bmatrix} k-j \\ s \end{bmatrix} \\= & {} \frac{1}{|\mathcal {C}|} \sum _{s=0}^k q^{ms} \begin{bmatrix} k-j \\ s \end{bmatrix} \sum _{r=0}^k \begin{bmatrix} k-r \\ i-r \end{bmatrix} (-1)^{r-s} q^{\left( {\begin{array}{c}r-s\\ 2\end{array}}\right) } \begin{bmatrix} k-s \\ k-r \end{bmatrix}. \end{aligned}$$

Clearly,

$$\begin{aligned} \begin{bmatrix} k-r \\ i-r \end{bmatrix}= \begin{bmatrix} k-r \\ k-i \end{bmatrix}, \end{aligned}$$

and using the definition of $q$-binomial coefficient one finds

$$\begin{aligned} \begin{bmatrix} k-s \\ k-r \end{bmatrix} \begin{bmatrix} k-r \\ k-i \end{bmatrix} = \begin{bmatrix} k-s \\ k-i \end{bmatrix} \begin{bmatrix} i-s \\ r-s \end{bmatrix}. \end{aligned}$$

Hence we have

$$\begin{aligned} \sum _{r=0}^k \begin{bmatrix} k-r \\ i-r \end{bmatrix} (-1)^{r-s} q^{\left( {\begin{array}{c}r-s\\ 2\end{array}}\right) } \begin{bmatrix} k-s \\ k-r \end{bmatrix}= & {} \sum _{r=0}^k \begin{bmatrix} k-s \\ k-i \end{bmatrix} \begin{bmatrix} i-s \\ r-s \end{bmatrix} (-1)^{r-s} q^{\left( {\begin{array}{c}r-s\\ 2\end{array}}\right) } \\= & {} \begin{bmatrix} k-s \\ k-i \end{bmatrix} \sum _{r=0}^k \begin{bmatrix} i-s \\ r-s \end{bmatrix} (-1)^{r-s} q^{\left( {\begin{array}{c}r-s\\ 2\end{array}}\right) } \\= & {} \begin{bmatrix} k-s \\ k-i \end{bmatrix} \sum _{r=-s}^{k-s} \begin{bmatrix} i-s \\ r \end{bmatrix} (-1)^{r} q^{\left( {\begin{array}{c}r\\ 2\end{array}}\right) } \\= & {} \begin{bmatrix} k-s \\ k-i \end{bmatrix} \sum _{r=0}^{i-s} \begin{bmatrix} i-s \\ r \end{bmatrix} (-1)^{r} q^{\left( {\begin{array}{c}r\\ 2\end{array}}\right) } \\= & {} \left\{ \begin{array}{ll} 1 &{}\;\, {\hbox {if}}\, s=i, \\ 0 &{} \text{ otherwise, } \end{array} \right. \ \end{aligned}$$

where the last equality follows from the $q$-Binomial Theorem [24], p. 74]. It follows

$$\begin{aligned} (SP)_{ij}= \frac{1}{|\mathcal {C}|} q^{mi} \begin{bmatrix} k-j \\ i \end{bmatrix}=T_{ij}, \end{aligned}$$

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

$$\begin{aligned} n_q(k\times m, r,h)=\frac{1}{q} \sum \limits _{s=0}^k {(-1)}^{r-s} q^{ms+ \left( {\begin{array}{c}r-s\\ 2\end{array}}\right) } \begin{bmatrix} k-s \\ k-r \end{bmatrix} \left( \begin{bmatrix} k \\ s \end{bmatrix} + (q-1) \begin{bmatrix} k-h \\ s \end{bmatrix}\right) . \end{aligned}$$

Remark 66

Theorem 65 generalizes the works cited in Remark 61.