Subspace codes and network coding

Maximum rank distance (MRD for short) codes lately attract more attention due to their various applications in storage systems, network coding, cryptography and space time coding. Similar to Reed-Solomon codes in classical coding theory, Gabidulin codes are the most prominent family of MRD codes. Due to their poor performance in list decoding or in constructing McEliece-type cryptosystems, the focus moves from Gabidulin codes to other non-Gabidulin codes. A natural following challenge is then to see if we can construct an infinite family of MRD codes that are not equivalent to Gabidulin codes. In this paper, we utilize Dickson matrices to construct an infinite family of Fq-linear MRD codes. Our codes are characterized by each of their codewords corresponding to a linearized polynomial with leading coefficient determined by one of any other coefficients. The family of codes corresponding to the set of linearized polynomials with leading coefficients dependent on the linear terms provides an extension to both Twisted Gabidulin codes and generalized Twisted Gabidulin codes for dimensions 1 and n-1. Lastly, we also provide some analysis on the equivalence between our proposed codes with some known families of MRD codes.

Linear complementary-dual (LCD for short) codes are linear codes that intersect with their duals trivially. LCD codes have been used in certain communication systems. It is recently found that LCD codes can be applied in cryptography. This application of LCD codes renewed the interest in the construction of LCD codes having a large minimum distance. Maximum distance separable (MDS) codes are optimal in the sense that the minimum distance cannot be improved for given length and code size. Constructing LCD MDS codes is thus of significance in theory and practice. Recently, Jin constructed several classes of LCD MDS codes through generalized Reed-Solomon codes. In this paper, a different approach is proposed to obtain new LCD MDS codes from generalized Reed-Solomon codes. Consequently, new code constructions are provided and certain previously known results by Jin are extended.

To consider construction of strongly secure network coding scheme without universality, this paper focuses on properties of MDS(maximum distance separable) codes, especially, Reed-Solomon codes. Our scheme applies Reed-Solomon codes in coset coding scheme to achieve the security based on the classical underlying network coding. Comparing with the existing scheme, MRD(maximum rank distance) code and a necessary condition based on MRD are not required in the scheme. Furthermore, considering the conditions between the code for security and the underlying network code, the scheme could be applied for more situations on fields.

New quantum MDS codes with large minimum distance and short length from generalized Reed–Solomon codes

Quantum maximum-distance-separable (MDS) codes are a significant class of quantum codes. In this paper, we mainly utilize classical Hermitian self-orthogonal generalized Reed-Solomon codes to construct five new classes of quantum MDS codes with large minimum distance.

Skew and linearized Reed–Solomon codes and maximum sum rank distance codes over any division ring

In wireless multihop networks, multihop packet transmissions over error-prone wireless links cause significant performance degradation. In this paper, we study a multicast system in wireless multihop networks with Forward Error Correction (FEC) for packet erasures. Although FEC is a powerful tool to recover packet erasures, it has an inherent problem that burdens the network with overhead due to redundant packets. In order to solve the problem, we propose a new multicast system with Reed Solomon/network joint coding . In the proposed system, information packets from a source node are encoded by Reed Solomon erasure (RSE) coding and transmitted into the network. At intermediate nodes on multicast paths, packets arriving from different links are encoded by linear network coding (LNC). The joint coding provides highly robust and efficient multicast communication because RSE coding provides a recovery mechanism from packet erasures and LNC reduces the number of relayed packets in the network. From the fact that both RSE coding and LNC are linear coding, we propose a new decoder for the joint coding. In the proposed decoder, a decoding matrix is constructed by combining the parity matrix of RSE coding and a coding matrix of LNC. Destination nodes retrieve information packets by solving a system of simultaneous equations constructed by the decoding matrix. Simulation experiments show that the joint coding provides highly robust and efficient multicast communications.

Obtaining quantum maximum distance separable (MDS) codes from dual containing classical constacyclic codes using Hermitian construction have paved a path to undertake the challenges related to such constructions. Using the same technique, some new parameters of quantum MDS codes have been constructed here. One set of parameters obtained in this paper has achieved much larger distance than work done earlier. The remaining constructed parameters of quantum MDS codes have large minimum distance and were not explored yet.

Determining the insdel distance of linear codes is a very challenging problem. Recently, Ji et al. [‘Strict half-Singleton bound, strict direct upper bound for linear insertion-deletion codes and optimal codes’, IEEE Trans. Inform. Theory69(5) (2023), 2900–2910] proposed their strict half-Singleton bound for linear codes that do not include the all-one vector. A natural question asks for linear codes with both large Hamming distance and insdel distance. Almost maximum distance separable (MDS) codes are a special kind of linear code with good Hamming error-correcting capability. We present a sufficient condition for the insdel distance of almost MDS codes to be bounded by the strict half-Singleton bound. In addition, we construct a class of two-dimensional near MDS codes that achieve the strict half-Singleton bound using twisted Reed–Solomon codes. Finally, we present a construction of optimal almost MDS insdel codes from Reed–Solomon codes for large dimensions.

<p style='text-indent:20px;'>Quantum maximum-distance-separable (MDS) codes are very important in coding theory. In this paper, we construct three classes of new quantum MDS codes with large minimum distance by classical Hermitian self-orthogonal generalized Reed-Solomon codes. Furthermore, these quantum MDS codes have more flexible lengths and larger minimum distance than those of some known quantum MDS codes.</p>

Many recent block ciphers use Maximum Distance Separable (MDS) matrices in their diffusion layer. The main objective of this operation is to spread as much as possible the differences between the outputs of nonlinear Sboxes. So they generally act at nibble or at byte level. The MDS matrices are associated to MDS codes of ratio 1/2. The most famous example is the MixColumns operation of the AES block cipher.In this example, the MDS matrix was carefully chosen to obtain compact and efficient implementations in software and hardware. However, this MDS matrix is dedicated to 8-bit words, and is not always adapted to lightweight applications. Recently, several studies have been devoted to the construction of recursive diffusion layers. Such a method allows to apply an MDS matrix using an iterative process which looks like a Feistel network with linear functions instead of nonlinear.In this paper, we present a generic construction of MDS recursive diffusion layers as proposed in [1], [7], [10], [12], [15] but bridging this construction with the theory of Gabidulin codes. This construction uses Gabidulin codes which have the property to be not only MDS but also MRD (Maximum Rank Distance). This fact gives an additional property to diffusion layers which seems interesting for cryptographic applications.

Systematic maximum sum rank codes

As far as we know, there are many different types of coding bounds in classical coding theory, which measure the efficiency of a code. Similarly, some important and useful coding bounds in classical coding theory are generalized to linear network error correction coding, including the Hamming bound, the Gilbert-Varshamov bound and the Singleton bound. While the former two bounds are both interesting theoretical results, the Singleton bound plays a very important role in the theory of network error correction. In this chapter, we first discuss the Hamming bound and the Singleton bound in detail. In particular, the LNEC maximum distance separable (MDS) codes, defined as the codes meeting the Singleton bound with equality, are studied in detail due to their optimality. Furthermore, we present several constructive algorithms of LNEC codes, particularly, for LNEC MDS codes, and analyze their performance.KeywordsNetwork Error CorrectionCode BoundariesHamming BoundSingleton BoundClassical Coding TheoryThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Recently, network error correction coding (NEC) has been studied extensively. Several bounds in classical coding theory have been extended to NEC, especially the Singleton bound. In this paper, following the research line using the extended global encoding kernels proposed by Zhang in 2008, the refined Singleton bound of NEC can be proved more explicitly. Moreover, we give a constructive proof of the attainability of this bound and indicate that the required field size for the existence of network maximum distance separable (MDS) codes can become smaller further. By this proof, an algorithm is proposed to construct general linear network error correction codes including the linear network error correction MDS codes. Finally, we study the error correction capability of random linear NEC. Motivated partly by the performance analysis of random linear network coding, we evaluate the different failure probabilities defined in this paper in order to analyze the performance of random linear NEC. Several upper bounds on these probabilities are obtained and they show that these probabilities will approach to zero as the size of the base field goes to infinity. Using these upper bounds, we slightly improve on the probability mass function of the minimum distance of random linear network error correction codes in a paper by Balli and colleagues, as well as the upper bound on the field size required for the existence of linear network error correction codes with degradation at most <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> .

The Lifted MRD (Maximum Rank Distance) code was introduced by Silva et al. to correct link errors and erasures for network coding systems. It achieves the generalized asymptotic Singleton bound for network error correction codes, and was shown to be equivalent to the Koetter-Kschischang code. However, in wireless network coding, undetected link errors that occur early in the network may accumulate and propagate beyond the decoding capability of the Lifted MRD code at the destination node. In this paper, we address this problem by a few strategies: detect and track possible error packets as erasure packets as they propagate through the network, reverse the error propagation process at the destination node by exploiting knowledge of the erasure packet positions, and double the MRD decoding capability by using erasure-and-error decoding. Algorithms to implement the above strategies are designed for low network overhead and low computational complexity. Simulation results show that the proposed scheme significantly improves the reliability of network coding over unreliable wireless links.