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We study the row-space partition and the pivot partition on the matrix space $mathbb{F}_q^{n times m}$. We show that both these partitions are reflexive and that the row-space partition is self-dual. Moreover, using various combinatorial methods, we explicitly compute the Krawtchouk coefficients associated with these partitions. This establishes MacWilliams-type identities for the row-space and pivot enumerators of linear rank-metric codes. We then generalize the Singleton-like bound for rank-metric codes, and introduce two new concepts of code extremality. Both of them generalize the notion of MRD codes and are preserved by trace-duality. Moreover, codes that are extremal according to either notion satisfy strong rigidity properties analogous to those of MRD codes. As an application of our results to combinatorics, we give closed formulas for the $q$-rook polynomials associated with Ferrers diagram boards. Moreover, we exploit connections between matrices over finite fields and rook placements to prove that the number of matrices of rank $r$ over $mathbb{F}_q$ supported on a Ferrers diagram is a polynomial in $q$, whose degree is strictly increasing in $r$. Finally, we investigate the natural analogues of the MacWilliams Extension Theorem for the rank, the row-space, and the pivot partitions.
Integer partitions express the different ways that a positive integer may be written as a sum of other positive integers. Here we explore the analytic properties of a polynomial $f_lambda(x)$ that we call the partition polynomial for the partition $l
It is well known that linear rank-metric codes give rise to $q$-polymatroids. Analogously to classical matroid theory one may ask whether a given $q$-polymatroid is representable by a rank-metric code. We provide a partial answer by presenting exampl
Lattice-based Cryptography is considered to have the characteristics of classical computers and quantum attack resistance. We will design various graphic lattices and matrix lattices based on knowledge of graph theory and topological coding, since ma
In this note, we provide a description of the elements of minimum rank of a generalized Gabidulin code in terms of Grassmann coordinates. As a consequence, a characterization of linearized polynomials of rank at most $n-k$ is obtained, as well as par
We propose a novel application of coded computing to the problem of the nearest neighbor estimation using MatDot Codes [Fahim. et.al. 2017], that are known to be optimal for matrix multiplication in terms of recovery threshold under storage constrain