Rank-metric codes and their duality theory

We review the main results of the theory of error-correcting codes with the rank metric, introducing combinatorial techniques for their analysis. We study their duality theory and MacWilliams identities, comparing in particular rank-metric codes in vector and matrix representation. We then investigate the structure of MRD codes and cardinality-optimal anticodes in the rank metric, describing how they relate to each other.

Evasive subspaces, generalized rank weights and near MRD codes

In the 1960's Florence MacWilliams proved two important results in coding theory. The first result is that all linear codes over finite fields satisfy the MacWilliams identities. The second result is the MacWilliams Extension Theorem. This theorem proved the equivalence of two notions of code equivalence over finite fields with respect to Hamming weight. This paper studies the evolution of the extension theorem from the classical case of linear codes defined over finite fields, to the case of linear codes defined over finite rings and finally to linear codes defined over finite modules. Keywords: Codes over modules, MacWilliams, extension theorem, Frobenius.

This paper establishes information-theoretic limits in estimating a finite field low-rank matrix given random linear measurements of it. These linear measurements are obtained by taking inner products of the low-rank matrix with random sensing matrices. Necessary and sufficient conditions on the number of measurements required are provided. It is shown that these conditions are sharp and the minimum-rank decoder is asymptotically optimal. The reliability function of this decoder is also derived by appealing to de Caen's lower bound on the probability of a union. The sufficient condition also holds when the sensing matrices are sparse - a scenario that may be amenable to efficient decoding. More precisely, it is shown that if the n\times n-sensing matrices contain, on average, \Omega(nlog n) entries, the number of measurements required is the same as that when the sensing matrices are dense and contain entries drawn uniformly at random from the field. Analogies are drawn between the above results and rank-metric codes in the coding theory literature. In fact, we are also strongly motivated by understanding when minimum rank distance decoding of random rank-metric codes succeeds. To this end, we derive distance properties of equiprobable and sparse rank-metric codes. These distance properties provide a precise geometric interpretation of the fact that the sparse ensemble requires as few measurements as the dense one. Finally, we provide a non-exhaustive procedure to search for the unknown low-rank matrix.

A rank-metric code is a subset of F q n × m where F q is a finite field. Gabidulin codes are a well-known class of algebraic rank-metric codes that meet the Singleton bound on the minimum rank distance of a code. The construction, encoding, and decoding of Gabidulin codes use the extension field F q m, where the code is regarded as a linear block code of length n over F q m. However, the parameter m can be large in certain applications and therefore, performing field operations over F q m can become very complex. In this paper, we investigate methods for constructing and decoding rank-metric codes by looking into linear codes of length nm over the base field Fq. Random coding bounds are derived on the minimum distance of such codes and an explicit structure is demonstrated to construct maximum rank distance codes. It is shown how to construct sparse parity-check matrices for these structures which enables low complexity parallelized decoders with complexity that scales linearly with m.

<p style='text-indent:20px;'>We define a class of automorphisms of rational function fields of finite characteristic and employ these to construct different types of optimal linear rank-metric codes. The first construction is of generalized Gabidulin codes over rational function fields. Reducing these codes over finite fields, we obtain maximum rank distance (MRD) codes which are not equivalent to generalized twisted Gabidulin codes. We also construct optimal Ferrers diagram rank-metric codes which settles further a conjecture by Etzion and Silberstein.</p>

Rank-metric codes recently attract a lot of attention due to their possible application to network coding, cryptography, space-time coding and distributed storage. An optimal-cardinality algebraic code construction in rank metric was introduced some decades ago by Delsarte, Gabidulin and Roth. This Reed–Solomon-like code class is based on the evaluation of linearized polynomials and is nowadays called Gabidulin codes. This dissertation considers block and convolutional codes in rank metric with the objective of designing and investigating efficient decoding algorithms for both code classes. After giving a brief introduction to codes in rank metric and their properties, we first derive sub-quadratic-time algorithms for operations with linearized polynomials and state a new bounded minimum distance decoding algorithm for Gabidulin codes. This algorithm directly outputs the linearized evaluation polynomial of the estimated codeword by means of the (fast) linearized Euclidean algorithm. Second, we present a new interpolation-based algorithm for unique and (not necessarily polynomial-time) list decoding of interleaved Gabidulin codes. This algorithm decodes most error patterns of rank greater than half the minimum rank distance by efficiently solving two linear systems of equations. As a third topic, we investigate the possibilities of polynomial-time list decoding of rank-metric codes in general and Gabidulin codes in particular. For this purpose, we derive three bounds on the list size. These bounds show that the behavior of the list size for both, Gabidulin and rank-metric block codes in general, is significantly different from the behavior of Reed–Solomon codes and block codes in Hamming metric, respectively. The bounds imply, amongst others, that there exists no polynomial upper bound on the list size in rank metric as the Johnson bound in Hamming metric, which depends only on the length and the minimum rank distance of the code. Finally, we introduce a special class of convolutional codes in rank metric and propose an efficient decoding algorithm for these codes. These convolutional codes are (partial) unit memory codes, built upon rank-metric block codes. This structure is crucial in the decoding process since we exploit the efficient decoders of the underlying block codes in order to decode the convolutional code.

We investigate two fundamental questions intersecting coding theory and combinatorial geometry, with emphasis on their connections. These are the problem of computing the asymptotic density of MRD codes in the rank metric, and the Critical Problem for combinatorial geometries by Crapo and Rota. In the first part of the paper, we use methods from semifield theory to derive two lower bounds for the density function of full-rank, square MRD codes. The first bound is sharp when the matrix size is a prime number and the underlying field is sufficiently large, while the second bound applies to the binary field. We then take a new look at the Critical Problem for combinatorial geometries, approaching it from a qualitative, often asymptotic, viewpoint. We illustrate the connection between this very classical problem and that of computing the asymptotic density of MRD codes. Finally, in the third part of the paper we study the asymptotic density of some special families of codes in the rank metric, including the symmetric, alternating, and Hermitian ones. In particular, we show that the optimal codes in these three contexts are sparse.

We introduce a general class of regular weight functions on finite abelian groups, and study the combinatorics, the duality theory, and the metric properties of codes endowed with such functions. The weights are obtained by composing a suitable support map with the rank function of a graded lattice satisfying certain regularity properties. A regular weight on a group canonically induces a regular weight on the character group, and invertible MacWilliams identities always hold for such a pair of weights. Moreover, the Krawtchouk coefficients of the corresponding MacWilliams transformation have a precise combinatorial significance, and can be expressed in terms of the invariants of the underlying lattice. In particular, they are easy to compute in many examples. Several weight functions traditionally studied in Coding Theory belong to the class of weights introduced in this paper. Our lattice-theory approach also offers a control on metric structures that a regular weight induces on the underlying group. In particular, it allows to show that every finite abelian group admits weight functions that, simultaneously, give rise to MacWilliams identities, and endow the underlying group with a metric space structure. We propose a general notion of extremality for (not necessarily additive) codes in groups endowed with semi-regular supports, and establish a Singleton-type bound. We then investigate the combinatorics and duality theory of extremal codes, extending classical results on the weight and distance distribution of error-correcting codes. Finally, we apply the theory of MacWilliams identities to enumerative combinatorics problems, obtaining closed formulae for the number of rectangular matrices over a finite having prescribed rank and satisfying some linear conditions.

We further the study of the duality theory of linear space-time codes over finite fields by showing that the only finite linear ''omnibus'' codes (defined herein) with a duality theory are the column distance codes and the rank codes. We introduce weight enumerators for both these codes and show that they have MacWilliams-type functional equations relating them to the weight enumerators of their duals. We also show that the complete weight enumerator for finite linear sum-of-ranks codes satisfies such a functional equation. We produce an analogue of Gleason's Theorem for formally self-dual linear finite rank codes. Finally, we relate the duality matrices of $n\times n$ linear finite rank codes and length $n$ vector codes under the Hamming metric.

We give a short survey of the results obtained in the last several decades that develop the theory of linear codes and polylinear recurrences over finite rings and modules following the well-known results on codes and polylinear recurrences over finite fields. The first direction contains the general results of theory of linear codes, including: the concepts of a reciprocal code and the MacWilliams identity; comparison of linear code properties over fields and over modules; study of weight functions on finite modules, that generalize in some natural way the Hamming weight on a finite field; the ways of representation of codes over fields by linear codes over modules. The second one develops the general theory of polylinear recurrences; describes the algebraic relations between the families of linear recurrent sequences and their periodic properties; studies the ways of obtaining “good” pseudorandom sequences from them. The interaction of these two directions leads to the results on the representation of linear codes by polylinear recurrences and to the constructions of recursive MDS-codes. The common algebraic foundation for the effective development of both directions is the Morita duality theory based on the concept of a quasi-Frobenius module.

In this paper, it is shown that some results in the theory of rank-metric codes over finite fields can be extended to finite commutative principal ideal rings. More precisely, the rank metric is generalized and the rank-metric Singleton bound is established. The definition of Gabidulin codes is extended and it is shown that its properties are preserved. The theory of Gröbner bases is used to give the unique decoding, minimal list decoding, and error-erasure decoding algorithms of interleaved Gabidulin codes. These results are then applied in space-time codes and in random linear network coding as in the case of finite fields. Specifically, two existing encoding schemes of random linear network coding are combined to improve the error correction.

Maximum rank‐distance (MRD) codes are (not necessarily linear) maximum codes in the rank‐distance metric space on ‐by‐ matrices over a finite field . They are diameter perfect and have the cardinality if . We define switching in MRD codes as the replacement of special MRD subcodes by other subcodes with the same parameters. We consider constructions of MRD codes admitting switching, such as punctured twisted Gabidulin codes and direct‐product codes. Using switching, we construct a huge class of MRD codes whose cardinality grows doubly exponentially in if the other parameters (, the code distance) are fixed. Moreover, we construct MRD codes with different affine ranks and aperiodic MRD codes.

In analogy with the Singleton defect for classical codes, we propose a definition of rank defect for rank-metric codes. We characterize codes whose rank defect and dual rank defect are both zero, and prove that the rank distribution of such codes is determined by their parameters. This extends a result by Delsarte on the rank distribution of MRD codes. In the general case of codes of positive defect, we show that the rank distribution is determined by the parameters of the code, together with the number of codewords of small rank. Moreover, we prove that if the rank defect of a code and its dual are both one, and the dimension satisfies a divisibility condition, then the number of minimum-rank codewords and dual minimum-rank codewords is the same. Finally, we discuss how our results specialize to \(\mathbb {F}_{q^m}\)-linear rank-metric codes in vector representation.

Partitions of matrix spaces with an application to [formula omitted]-rook polynomials