Independent Spaces of q-Polymatroids

This paper is devoted to the study of independent spaces of q-polymatroids. With the aid of an auxiliary q-matroid it is shown that the collection of independent spaces satisfies the same properties as for q-matroids. However, in contrast to q-matroids, the rank value of an independent space does not agree with its dimension. Nonetheless, the rank values of the independent spaces fully determine the q-polymatroid, and this fact can be exploited to derive a cryptomorphism of q-polymatroids. Finally, the notions of minimal spanning spaces, maximally strongly independent spaces, and bases will be elaborated on.

$q$-Polymatroids and Their Relation to Rank-Metric Codes

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 examples of $q$-matroids that are not representable by ${mathbb F}_{q^m}$-linear rank-metric codes. We then go on and introduce deletion and contraction for $q$-polymatroids and show that they are mutually dual and that they correspond to puncturing and shortening of rank-metric codes. Finally, we introduce a closure operator along with the notion of flats and show that the generalized rank weights of a rank-metric code are fully determined by the flats of the associated $q$-polymatroid.

Automorphism Groups and Isometries for Cyclic Orbit Codes

We study orbit codes in the field extension ${mathbb F}_{q^n}$. First we show that the automorphism group of a cyclic orbit code is contained in the normalizer of the Singer subgroup if the orbit is generated by a subspace that is not contained in a proper subfield of ${mathbb F}_{q^n}$. We then generalize to orbits under the normalizer of the Singer subgroup. In that situation some exceptional cases arise and some open cases remain. Finally we characterize linear isometries between such codes.

Anticodes in the Sum-Rank Metric

We study the structure of anticodes in the sum-rank metric for arbitrary fields and matrix blocks of arbitrary sizes. Our main result is a complete classification of optimal linear anticodes. We also compare the cardinality of the ball in the sum-rank metric with that of an optimal linear anticode, showing that the latter is strictly larger over sufficiently large finite fields. Finally, we give examples of parameters for which the largest anticode is neither a ball nor a linear anticode

Fundamental Properties of Sum-Rank Metric Codes

This paper investigates the theory of sum-rank metric codes for which the individual matrix blocks may have different sizes. Various bounds on the cardinality of a code are derived, along with their asymptotic extensions. The duality theory of sum-rank metric codes is also explored, showing that MSRD codes (the sum-rank analogue of MDS codes) dualize to MSRD codes only if all matrix blocks have the same number of columns. In the latter case, duality considerations lead to an upper bound on the number of blocks for MSRD codes. The paper also contains various constructions of sum-rank metric codes for variable block sizes, illustrating the possible behaviours of these objects with respect to bounds, existence, and duality properties.

Distance Distributions of Cyclic Orbit Codes

The distance distribution of a code is the vector whose $i^text{th}$ entry is the number of pairs of codewords with distance $i$. We investigate the structure of the distance distribution for cyclic orbit codes, which are subspace codes generated by the action of $mathbb{F}_{q^n}^*$ on an $mathbb{F}_q$-subspace $U$ of $mathbb{F}_{q^n}$. We show that for optimal full-length orbit codes the distance distribution depends only on $q,,n$, and the dimension of $U$. For full-length orbit codes with lower minimum distance, we provide partial results towards a characterization of the distance distribution, especially in the case that any two codewords intersect in a space of dimension at most 2. Finally, we briefly address the distance distribution of a union of optimal full-length orbit codes.

On the Sparseness of Certain MRD Codes

We determine the proportion of $[3times 3;3]$-MRD codes over ${mathbb F}_q$ within the space of all $3$-dimensional $3times3$-rank-metric codes over the same field. This shows that for these parameters MRD codes are sparse in the sense that the proportion tends to $0$ as $qrightarrowinfty$. This is so far the only parameter case for which MRD codes are known to be sparse. The computation is accomplished by reducing the space of all such rank-metric codes to a space of specific bases and subsequently making use of a result by Menichetti (1973) on 3-dimensional semifields.

Skew-Polynomial Rings and Skew-Cyclic Codes

This is a survey on the theory of skew-cyclic codes based on skew-polynomial rings of automorphism type. Skew-polynomial rings have been introduced and discussed by Ore (1933). Evaluation of skew polynomials and sets of (right) roots were first considered by Lam (1986) and studied in great detail by Lam and Leroy thereafter. After a detailed presentation of the most relevant properties of skew polynomials, we survey the algebraic theory of skew-cyclic codes as introduced by Boucher and Ulmer (2007) and studied by many authors thereafter. A crucial role will be played by skew-circulant matrices. Finally, skew-cyclic codes with designed minimum distance are discussed, and we report on two different kinds of skew-BCH codes, which were designed in 2014 and later.

Partitions of Matrix Spaces With an Application to $q$-Rook Polynomials

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.

Maximal Ferrers Diagram Codes: Constructions and Genericity Considerations

This paper investigates the construction of rank-metric codes with specified Ferrers diagram shapes. These codes play a role in the multilevel construction for subspace codes. A conjecture from 2009 provides an upper bound for the dimension of a rank-metric code with given specified Ferrers diagram shape and rank distance. While the conjecture in its generality is wide open, several cases have been established in the literature. This paper contributes further cases of Ferrers diagrams and ranks for which the conjecture holds true. In addition, the proportion of maximal Ferrers diagram codes within the space of all rank-metric codes with the same shape and dimension is investigated. Special attention is being paid to MRD codes. It is shown that for growing field size the limiting proportion depends highly on the Ferrers diagram. For instance, for $[mtimes 2]$-MRD codes with rank~$2$ this limiting proportion is close to $1/e$.