MRD Codes Research Articles

Identifiers for MRD-codes

For any admissible value of the parameters n and k there exist [n,k]-Maximum Rank Distance Fq-linear codes. Indeed, it can be shown that if field extensions large enough are considered, almost all rank metric codes are MRD. On the other hand, very few families up to equivalence of such codes are currently known. In the present paper we study some invariants of MRD codes and evaluate their value for the known families, providing a new characterization of generalized twisted Gabidulin codes.

Rank-metric complementary dual codes

In this paper, we investigate the rank-metric codes which are proposed by Delsarte and Gabidulin to be complementary dual codes. We first point out the relationship between Delsarte complementary dual codes (Delsarte LCD codes) and Gabidulin complementary dual codes (Gabidulin LCD codes). We then construct two classes of Gabidulin LCD MRD codes by self-dual basis (or almost self-dual basis) of the finite field \(\mathbb {F}_{q^{m}}\) over base field \(\mathbb {F}_{q}\). Finally, we give an interesting application of rank-metric LCD codes in decoding algorithm.

A New Family of MRD Codes in <inline-formula> <tex-math notation="LaTeX">$\mathbb{F_q}^{2n\times2n}$ </tex-math> </inline-formula> With Right and Middle Nuclei <inline-formula> <tex-math notation="LaTeX">$\mathbb F_{q^n}$ </tex-math> </inline-formula>

In this paper, we present a new family of maximum rank distance (MRD for short) codes in $\mathbb F_{q}^{2n\times 2n}$ of minimum distance $2\leq d\leq 2n$. In particular, when $d=2n$, we can show that the corresponding semifield is exactly a Hughes-Kleinfeld semifield. The middle and right nuclei of these MRD codes are both equal to $\mathbb F_{q^n}$. We also prove that the MRD codes of minimum distance $2<d<2n$ in this family are inequivalent to all known ones. The equivalence between any two members of this new family is also determined.

Classification of large partial plane spreads in $${{\,\mathrm{PG}\,}}(6,2)$$ PG ( 6 , 2 ) and related combinatorial objects

In this article, the partial plane spreads in $PG(6,2)$ of maximum possible size $17$ and of size $16$ are classified. Based on this result, we obtain the classification of the following closely related combinatorial objects: Vector space partitions of $PG(6,2)$ of type $(3^{16} 4^1)$, binary $3\times 4$ MRD codes of minimum rank distance $3$, and subspace codes with parameters $(7,17,6)_2$ and $(7,34,5)_2$.

On dually almost MRD codes

In this paper we define and study a family of codes which are coming close to MRD codes. Thus we call them AMRD codes (almost MRD). An AMRD code is a code with rank defect equal to 1. These codes can be viewed as q-analogs of classical AMDS codes as considered in [3,7,9]. AMRD codes whose duals are AMRD as well are called dually AMRD. These codes have important symmetry properties. For instance, the number of codewords of minimum rank in C and C⊥ are equal if the size of the matrices divides the dimension of C[4]. Necessary and sufficient conditions are given for codes to be dually AMRD and we give a construction of such codes of minimal dimension. We also construct self-dual AMRD codes. Furthermore, we show that dually AMRD codes coincide with codes of rank defect one and maximum 2-generalized weight if the size of the matrices divides the dimension. The results may be seen as q-analogs of results established in classical coding theory [7,9].

On [formula omitted]-Steiner systems from rank metric codes

In this paper we prove that rank metric codes with special properties imply the existence of q-analogs of suitable designs. More precisely, we show that the minimum weight vectors of a [2d,d,d] dually almost MRD code C≤Fqm2d(2d≤m) which has no code words of rank weight d+1 form a q-Steiner system S(d−1,d,2d)q. This is the q-analog of a result in classical coding theory and it may be seen as a first step to prove a q-analog of the famous Assmus–Mattson Theorem.

Generalized twisted Gabidulin codes

Let C be a set of m by n matrices over Fq such that the rank of A−B is at least d for all distinct A,B∈C. Suppose that m⩽n. If #C=qn(m−d+1), then C is a maximum rank distance (MRD for short) code. Until 2016, there were only two known constructions of MRD codes for arbitrary 1<d<m−1. One was found by Delsarte (1978) [8] and Gabidulin (1985) [10] independently, and it was later generalized by Kshevetskiy and Gabidulin (2005) [16]. We often call them (generalized) Gabidulin codes. Another family was recently obtained by Sheekey (2016) [22], and its elements are called twisted Gabidulin codes. In the same paper, Sheekey also proposed a generalization of the twisted Gabidulin codes. However the equivalence problem for it is not considered, whence it is not clear whether there exist new MRD codes in this generalization. We call the members of this putative larger family generalized twisted Gabidulin codes. In this paper, we first compute the Delsarte duals and adjoint codes of them, then we completely determine the equivalence between different generalized twisted Gabidulin codes. In particular, it can be proven that, up to equivalence, generalized Gabidulin codes and twisted Gabidulin codes are both proper subsets of this family.

Generalized bilinear forms graphs and MRD codes over a residue class ring

We investigate the generalized bilinear forms graph Γd over a residue class ring Zps. The graph Γd is a connected vertex-transitive graph, and we completely determine its independence number, clique number, chromatic number and maximum cliques. We also prove that cores of both Γd and its complement are maximum cliques. The graph Γd is useful for error-correcting codes. We show that there is a largest independent set of Γd which is a linear MRD code over Zps.

A generalization of the normal rational curve in $$\mathop {\mathrm{PG}}(d,q^n)$$ PG ( d , q n ) and its associated non-linear MRD codes

Let A and B be two points of $$\mathop {\mathrm{PG}}(d,q^n)$$ and let $$\Phi $$ be a collineation between the stars of lines with vertices A and B, that does not map the line AB into itself. In this paper we prove that if $$d=2$$ or $$d\ge 3$$ and the lines $$\Phi ^{-1}(AB), AB, \Phi (AB) $$ are not in a common plane, then the set $$\mathcal{C}$$ of points of intersection of corresponding lines under $$\Phi $$ is the union of $$q-1$$ scattered $${\mathbb {F}}_{q}$$ -linear sets of rank n together with $$\{A,B\}$$ . As an application we will construct, starting from the set $$\mathcal{C}$$ , infinite families of non-linear $$(d+1, n, q;d-1)$$ -MRD codes, $$d\le n-1$$ , generalizing those recently constructed in Cossidente et al. (Des Codes Cryptogr 79:597–609, 2016) and Durante and Siciliano (Electron J Comb, 2017).

Weight distribution of rank-metric 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.

On self-dual MRD codes

We investigate self-dual MRD codes. In particular we prove that a Gabidulin code in $(\mathbb{F}_q)^{n\times n}$ is equivalent to a self-dual code if and only if its dimension is $n^2/2$, $n \equiv 2 \pmod 4$, and $q \equiv 3 \pmod 4$. On the way we determine the full automorphism group of Gabidulin codes in $(\mathbb{F}_q)^{n\times n}$.

Algebraic structures of MRD codes

Based on results in finite geometry we prove the existence of MRD codes in $(\mathbb{F}_q)_{n,n}$ with minimum distance $n$ which are essentially different fromGabidulin codes. The construction results from algebraic structures which are closely related to those of finite fields. Some of the results may be known to experts, but to our knowledge have never been pointed out explicitly in the literature.

Rank-metric codes and their duality theory

We compare the two duality theories of rank-metric codes proposed by Delsarte and Gabidulin, proving that the former generalizes the latter. We also give an elementary proof of MacWilliams identities for the general case of Delsarte rank-metric codes. The identities which we derive are very easy to handle, and allow us to re-establish in a very concise way the main results of the theory of rank-metric codes first proved by Delsarte employing association schemes and regular semilattices. We also show that our identities imply as a corollary the original MacWilliams identities established by Delsarte. We describe how the minimum and maximum rank of a rank-metric code relate to the minimum and maximum rank of the dual code, giving some bounds and characterizing the codes attaining them. Then we study optimal anticodes in the rank metric, describing them in terms of optimal codes (namely, MRD codes). In particular, we prove that the dual of an optimal anticode is an optimal anticode. Finally, as an application of our results to a classical problem in enumerative combinatorics, we derive both a recursive and an explicit formula for the number of $$k \times m$$ matrices over a finite field with given rank and $$h$$ -trace.

Tree based Key Generation and Distribution Scheme in WSN

Wireless sensor network is become more popular because of its various applications in day to day life. Communication security is one of the most important challenges in wireless sensor network. Key generation and distribution are also important in wireless sensor network, so we have effective security mechanism for that. Tree based key generation techniques uses shortest path matrix and MRD code matrix for key generation and distribution. It raises the security as well as scalability of the network than the other techniques. Communication established between the two nodes and If a node get added after that, then the node information is updated without changing the information of those two nodes. General Terms Matrix, Tree,

Errata to “Codes and Designs Related to Lifted MRD Codes” [Feb 13 1004-1017

In the above titled paper (ibid., vol. 59, no. 2, pp. 1004-1017, Feb. 2013), the matrix in Example 7 (p. 1014) is incorrect. The correct matrix is presented here.

Rank subcodes in multicomponent network coding

A new class of subcodes in rank metric is proposed; based on it, multicomponent network codes are constructed. Basic properties of subspace subcodes are considered for the family of rank codes with maximum rank distance (MRD codes). It is shown that nonuniformly restricted rank subcodes reach the Singleton bound in a number of cases. For the construction of multicomponent codes, balanced incomplete block designs and matrices in row-reduced echelon form are used. A decoding algorithm for these network codes is proposed. Examples of codes with seven and thirteen components are given.

Vulnerability of MRD-Code-based Universal Secure Network Coding against Stronger Eavesdroppers

Silva et al. proposed a universal secure network coding scheme based on MRD codes, which can be applied to any underlying network code. This paper considers a stronger eavesdropping model where the eavesdroppers possess the ability to re-select the tapping links during the transmission. We give a proof for the impossibility of attaining universal security against such adversaries using Silva et al.'s code for all choices of code parameters, even with restricted number of tapped links. We also consider the cases with restricted tapping duration and derive some conditions for this code to be secure.