Ernst M. Gabidulin

Ideals over a non-commutative ring and their application in cryptology

A new modification of the McEliece public-key cryptosystem is proposed that employs the so-called maximum-rank-distance (MRD) codes in place of Goppa codes and that hides the generator matrix of the MRD code by addition of a randomly-chosen matrix. A short review of the mathematical background required for the construction of MRD codes is given. The cryptanalytic work function for the modified McEliece system is shown to be much greater than that of the original system. Extensions of the rank metric are also considered.

Maximum rank distance codes as space-time codes

The critical design criterion for space-time codes in asymptotically good channels is the minimum rank between codeword pairs. Rank codes are a two-dimensional matrix code construction where by the rank is the metric of merit. We look at the application of rank codes to space-time code design. In particular, we provide construction methods of full-rank codes over different complex signal constellations, for arbitrary numbers of antennas, and codeword periods. We also derive a Singleton-type bound on the rate of a code for the rank metric, and we show that rank codes satisfy this bound with equality.

The new construction of rank codes

The only known construction of error-correcting codes in rank metric was proposed in 1985. These were codes with fast decoding algorithm. We present a new construction of rank codes, which defines new codes and includes known codes. This is a generalization of E.M. Gabidulin, 1985. Though the new codes seem to be very similar to subcodes of known rank codes, we argue that these are different codes. A fast decoding algorithm is described

Modeling Toroidal Networks with the Gaussian Integers

In this paper we consider a broad family of toroidal networks, denoted as Gaussian networks, which include many previously proposed and used topologies. We will define such networks by means of the Gaussian integers, the subset of the complex numbers with integer real and imaginary parts. Nodes in Gaussian networks are labeled by Gaussian integers, which confer these topologies an algebraic structure based on quotient rings of the Gaussian integers. In this sense, Gaussian integers reveal themselves as the appropriate tool for analyzing and exploiting any type of toroidal network. Using this algebraic approach, we can characterize the main distance-related properties of Gaussian networks, providing closed expressions for their diameter and average distance. In addition, we solve some important applications, like unicast and broadcast packet routing or the perfect placement of resources over these networks.

Linear codes with covering radius 2 and other new covering codes

Infinite families of linear binary codes with covering radius R=2 and minimum distance d=3 and d=4 are given. Using the constructed codes with d=3, R=2, families of covering codes with R>2 are obtained. The parameters of many constructed codes with R >

A Fast Matrix Decoding Algorithm for Rank-Error-Correcting Codes

The so-called term-rank and rank metrics and appropriate codes were introduced and investigated in [1 –7]. These metrics and codes can be used for correcting array errors in a set of parallel channels, for scrambling in channels with burst errors, as basic codes in McEliece public key cryptosystem [8], etc. For codes with maximal rank distance (MRD codes) there exists a fast decoding algorithm based on Euclids Division Algorithm in some non-commutative ring [6]. In this paper a new construction of MRD codes is given and a new fast matrix decoding algorithm is proposed which generalizes Petersons algorithm [9] for BCH codes.

Error and erasure correcting algorithms for rank codes

In this paper, transmitted signals are considered as square matrices of the Maximum rank distance (MRD) (n, k, d)-codes. A new composed decoding algorithm is proposed to correct simultaneously rank errors and rank erasures. If the rank of errors and erasures is not greater than the Singleton bound, then the algorithm gives always the correct decision. If it is not a case, then the algorithm gives still the correct solution in many cases but some times the unique solution may not exist.

Codes for network coding

In [4] a metric for error correction in network coding is introduced. Also constant-dimension codes were introduced and investigated. Nevertheless little is known on codes in this metric in general. In this paper, several classes of codes are defined and investigated.

Perfect Codes for Metrics Induced by Circulant Graphs

An algebraic methodology for defining new metrics over two-dimensional signal spaces is presented in this work. We have mainly considered quadrature amplitude modulation (QAM) constellations which have previously been modeled by quotient rings of Gaussian integers. The metric over these constellations, based on the distance concept in circulant graphs, is one of the main contributions of this work. A detailed analysis of some degree-four circulant graphs has allowed us to detail the weight distribution for these signal spaces. A new family of perfect codes over Gaussian integers will be defined and characterized by providing a solution to the perfect t-dominating set problem over the circulant graphs presented. Finally, we will show how this new metric can be extended to other signal sets by considering hexagonal constellations and circulant graphs of degree six.

Metrics generated by families of subspaces

A new family of metrics is introduced. Each of these is defined by a spanning set F of linear subspaces of a finite vector space. The norm of a vector is defined as the size of a minimal subset of F whose span contains this vector. Some examples and applications are presented. A-class of Varshamov-Gilbert bound based F-metrics is introduced. Connections with combinatorial metrics are discussed.