Maximum Rank Distance Codes Research Articles

Non-linear maximum rank distance codes

By exploring some geometry of Segre varieties and Veronese varieties, new families of non-linear maximum rank distance codes and optimal constant rank codes are provided.

Convolutional Codes in Rank Metric With Application to Random Network Coding

Random network coding recently attracts attention as a technique to disseminate information in a network. This paper considers a non-coherent multi-shot network, where the unknown and time-variant network is used several times. In order to create dependencies between the different shots, particular convolutional codes in rank metric are used. These codes are so-called (partial) unit memory ((P)UM) codes, i.e., convolutional codes with memory one. First, distance measures for convolutional codes in rank metric are shown and two constructions of (P)UM codes in rank metric based on the generator matrices of maximum rank distance codes are presented. Second, an efficient error-erasure decoding algorithm for these codes is presented. Its guaranteed decoding radius is derived and its complexity is bounded. Finally, it is shown how to apply these codes for error correction in random linear and affine network coding.

Gaussian integers and interleaved rank codes for space–time block codes

SummaryThis paper presents the design of space–time block codes (STBCs) over maximum rank distance (MRD) codes, energy‐efficient STBCs, STBCs using interleaved‐MRD codes, the use of Gaussian integers for STBCs modulation, and Gabidulin's decoding algorithm for decoding STBCs. The design fundamentals of STBCs using MRD codes are firstly put forward for different number of transmit antennas. Extension finite fields (Galois fields) are used to design these linear block codes. Afterward, a comparative study of MRD‐based STBCs with corresponding orthogonal and quasi‐orthogonal codes is also included in the paper. The simulation results show that rank codes, for any number of transmit antennas, exhibit diversity gain at full rate contrary to orthogonal codes, which give diversity gain at full rate only for two transmit antennas case. Secondly, an energy‐efficient MRD‐STBC is proposed, which outperforms orthogonal STBC at least for 2 × 1 antenna system. Thirdly, interleaved‐MRD codes are used to construct higher‐order transmit antenna systems. Using interleaved‐MRD codes further reduces the complexity (compared with normal MRD codes) of the decoding algorithm. Fourthly, the use of Gaussian integers is utilized in mapping MRD‐based STBCs to complex constellations. Furthermore, it is described how an efficient and computationally less complex Gabidulin's decoding algorithm can be exploited for decoding complex MRD‐STBCs. The decoding results have been compared against hard‐decision maximum likelihood decoding. Under this decoding scheme, MRD‐STBCs have been shown to be potential candidate for higher transmit antenna systems as the decoding complexity of Gabidulin's algorithm is far less, and its performance for decoding MRD‐STBCs is somewhat reasonable. Copyright © 2015 John Wiley & Sons, Ltd.

Hexi McEliece Public Key Cryptosystem

This paper introduces a new class of hexi codes namely, hexi polynomial codes, hexi Rank Distance codes, hexi Maximum Rank Distance codes, hexi Goppa codes and hexi wild Goppa codes. These codes are useful to create variants of the McEliece public key cryptosystem known as the hexi McEliece public key cryptosystem and its variants; these cryptosystems are secure against attacks carried out on the existing variants of the McEliece public key cryptosystem. This newly introduced cryptosystem has better error correcting capacity and lesser time complexity making it more feasible to use. The security and possible attacks on these variants of the hexi McEliece public key cryptosystem are analysed. The McEliece public key cryptosystem introduced by McEliece in the year 1978 (19), still remains unbroken. The public key cryptosystem is based on binary Goppa codes. Hexi codes were developed in 2013 for error correction in AES (14), further development of these codes is carried out in this paper. These codes are useful to create variants of the McEliece public key cryptosystem called the hexi McEliece public key cryptosystem and its variants. These public key cryptosystems are secure, have better error correcting capacity and lesser time complexity making it more advantageous to use. The organization of the rest of this paper is as follows. The history of the McEliece public key cryptosystem and its several variants are dealt in section two. Section three recalls hexi codes and introduces hexi polynomial codes, hexi Rank Distance (hexi RD) codes, hexi Maximum Rank Distance (hexi MRD) codes, hexi Goppa codes and hexi wild Goppa codes. The decoding, error detecting and error correcting capacity of these codes is discussed in section four. Section five introduces a few variants of the McEliece public key cryptosystems which are based on these new hexi codes; they are called the hexi McEliece public key cryptosystem and its variants. Section six deals with the possible attacks on the hexi McEliece public key cryptosystem and the resistance against these attacks. It also discusses the security of the cryptosystem. Section seven provides a comparison of the hexi McEliece public key cryptosystem with original McEliece public key cryptosystem, in terms of time complexity and error correcting capacity. Conclusions, suggestions and future direction of research are given in section eight.

Codes and Designs Related to Lifted MRD Codes

Lifted maximum rank distance (MRD) codes, which are constant dimension codes, are considered. It is shown that a lifted MRD code can be represented in such a way that it forms a block design known as a transversal design. A slightly different representation of this design makes it similar to a <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$q$</tex> </formula> -analog of a transversal design. The structure of these designs is used to obtain upper bounds on the sizes of constant dimension codes which contain a lifted MRD code. Codes that attain these bounds are constructed. These codes are the largest known constant dimension codes for the given parameters. These transversal designs can also be used to derive a new family of linear codes in the Hamming space. Bounds on the minimum distance and the dimension of such codes are given.

Covering of subspaces by subspaces

Lower and upper bounds on the size of a covering of subspaces in the Grassmann graph $$\mathcal{G }_q(n,r)$$ by subspaces from the Grassmann graph $$\mathcal{G }_q(n,k)$$ , $$k \ge r$$ , are discussed. The problem is of interest from four points of view: coding theory, combinatorial designs, $$q$$ -analogs, and projective geometry. In particular we examine coverings based on lifted maximum rank distance codes, combined with spreads and a recursive construction. New constructions are given for $$q=2$$ with $$r=2$$ or $$r=3$$ . We discuss the density for some of these coverings. Tables for the best known coverings, for $$q=2$$ and $$5 \le n \le 10$$ , are presented. We present some questions concerning possible constructions of new coverings of smaller size.

Linear codes using skew polynomials with automorphisms and derivations

In this work the definition of codes as modules over skew polynomial rings of automorphism type is generalized to skew polynomial rings, whose multiplication is defined using an automorphism and a derivation. This produces a more general class of codes which, in some cases, produce better distance bounds than module skew codes constructed only with an automorphism. Extending the approach of Gabidulin codes, we introduce new notions of evaluation of skew polynomials with derivations and the corresponding evaluation codes. We propose several approaches to generalize Reed-Solomon and BCH codes to module skew codes and for two classes we show that the dual of such a Reed-Solomon type skew code is an evaluation skew code. We generalize a decoding algorithm due to Gabidulin for the rank metric and derive families of Maximum Distance Separable and Maximum Rank Distance codes.

Matrix-based memory efficient symmetric key generation and pre-distribution scheme for wireless sensor networks

As wireless sensor networks (WSN) have gained popularity due to their broader applications areas, so does the need for effective security mechanisms. Encryption can be used to protect data communication, but severe resource constraints in WSN make necessary key distribution very difficult. This study addresses the problem of key establishment in WSN using a key pre-distribution scheme. In this study, the authors have proposed a symmetric key generation and pre-distribution scheme, using a symmetric matrix and generator matrix of maximum rank distance (MRD) codes. Sensor nodes are divided into groups and some information is stored at each node to enable it to generate link keys. This node division substantially improves the memory required at each node, which is approximately equal to the group size. The proposed scheme also reduces the communication overhead to setup a link key. It requires only two messages to setup a link key between any two nodes. Furthermore this scheme also provides the highest level of network connectivity and scalability. Any two nodes in the network can establish a link key and new nodes can be added at any time without changing any information on previously deployed nodes.

A Rank-Metric Approach to Error Control in Random Network Coding

The problem of error control in random linear network coding is addressed from a matrix perspective that is closely related to the subspace perspective of K\"otter and Kschischang. A large class of constant-dimension subspace codes is investigated. It is shown that codes in this class can be easily constructed from rank-metric codes, while preserving their distance properties. Moreover, it is shown that minimum distance decoding of such subspace codes can be reformulated as a generalized decoding problem for rank-metric codes where partial information about the error is available. This partial information may be in the form of erasures (knowledge of an error location but not its value) and deviations (knowledge of an error value but not its location). Taking erasures and deviations into account (when they occur) strictly increases the error correction capability of a code: if $\mu$ erasures and $\delta$ deviations occur, then errors of rank $t$ can always be corrected provided that $2t \leq d - 1 + \mu + \delta$, where $d$ is the minimum rank distance of the code. For Gabidulin codes, an important family of maximum rank distance codes, an efficient decoding algorithm is proposed that can properly exploit erasures and deviations. In a network coding application where $n$ packets of length $M$ over $F_q$ are transmitted, the complexity of the decoding algorithm is given by $O(dM)$ operations in an extension field $F_{q^n}$.

MRD Hashing

We propose two new classes of hash functions which are motivated by Maximum Rank Distance (MRD) codes. We analise the security of these schemes. The system setup phase is computationally expensive for general field extensions. To overcome this limitation we derive an algebraic solution which avoids computations in special extension fields in the intended operational range of the hash functions.

Symmetric Rank Codes

As is well known, a finite field \mathbb{K}n e GF(qn) can be described in terms of n × n matrices A over the field \mathbb{K} e GF(q) such that their powers Ai, i e 1, 2, …, qn − 1, correspond to all nonzero elements of the field. It is proved that, for fields \mathbb{K}n of characteristic 2, such a matrix A can be chosen to be symmetric. Several constructions of field-representing symmetric matrices are given. These matrices Ai together with the all-zero matrix can be considered as a \mathbb{K}n-linear matrix code in the rank metric with maximum rank distance d e n and maximum possible cardinality qn. These codes are called symmetric rank codes. In the vector representation, such codes are maximum rank distance (MRD) linear ln, 1, nr codes, which allows one to use known rank-error-correcting algorithms. For symmetric codes, an algorithm of erasure symmetrization is proposed, which considerably reduces the decoding complexity as compared with standard algorithms. It is also shown that a linear ln, k, d e n − k + 1r MRD code \mathcal{V}k containing the above-mentioned one-dimensional symmetric code as a subcode has the following property: the corresponding transposed code is also \mathbb{K}n-linear. Such codes have an extended capability of correcting symmetric errors and erasures.