Maximum Rank Distance Codes Research Articles

Zeta functions for tensor codes

In this work, we introduce a new class of optimal tensor codes related to the Ravagnani-type anticodes defined in 2023. We show that it extends the family of [Formula: see text]-maximum rank distance codes and contains the [Formula: see text]-binomial moment determined codes (with respect to the Ravagnani-type anticodes) as a proper subclass. We define and study the zeta function for tensor codes. We establish connections between this object and the weight enumerator of a tensor code with respect to the Ravagnani-type anticodes. We introduce a new refinement of the invariants of tensor codes exploiting the structure of product lattices of some classes of anticodes classified in 2023 and we derive the corresponding MacWilliams identities. In this framework, we also define a multivariate version of the tensor weight enumerator and we establish relations with the corresponding zeta function. As an application, we derive connections on the tensor weights.

On the equivalence issue of a class of 2-dimensional linear maximum rank distance codes

In Neri et al. (2022) [20], the authors extended the family of 2-dimensional Fq2t-linear MRD codes recently found in Longobardi et al. (2023) [13]. Also, for t≥5 they determined equivalence classes of the elements in this new family and provided the exact number of inequivalent codes in it. In this article, we complete the study of the equivalence issue removing the restriction t≥5. Moreover, we prove that in the case when t=4, the linear sets of the projective line PG(1,q8) ensuing from codes in the relevant family, are not equivalent to any one known so far.

Linear Maximum Rank Distance Codes of Exceptional Type

Scattered polynomials of a given index over finite fields are intriguing rare objects with many connections within mathematics. Of particular interest are the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">exceptional</i> ones, as defined in 2018 by the first author and Zhou, for which partial classification results are known. In this paper we propose a unified algebraic description of F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>qⁿ</i></sub> -linear maximum rank distance codes, introducing the notion of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">exceptional</i> linear maximum rank distance codes of a given index. Such a connection naturally extends the notion of exceptionality for a scattered polynomial in the rank metric framework and provides a generalization of Moore sets in the monomial MRD context. We move towards the classification of exceptional linear MRD codes, by showing that the ones of index zero are generalized Gabidulin codes and proving that in the positive index case the code contains an exceptional scattered polynomial of the same index.

On a family of linear MRD codes with parameters $$[8\times 8,16,7]_q$$

In this paper we consider a family \(\mathcal {F}\) of 2n-dimensional \({\mathbb {F}}_q\)-linear rank metric codes in \({\mathbb {F}}_q^{n\times n}\) arising from polynomials of the form \(x^{q^s}+\delta x^{q^{\frac{n}{2}+s}}\in {\mathbb {F}}_{q^n}[x]\). The family \(\mathcal {F}\) was introduced by Csajbók et al. (JAMA 548:203–220) as a potential source for maximum rank distance (MRD) codes. Indeed, they showed that \(\mathcal {F}\) contains MRD codes for \(n=8\), and other subsequent partial results have been provided in the literature towards the classification of MRD codes in \(\mathcal {F}\) for any n. In particular, the classification has been reached when n is smaller than 8, and also for n greater than 8 provided that s is small enough with respect to n. In this paper we deal with the open case \(n=8\), providing a classification for any large enough odd prime power q. The techniques are from algebraic geometry over finite fields, since our strategy requires the analysis of certain 3-dimensional \({\mathbb {F}}_q\)-rational algebraic varieties in a 7-dimensional projective space. We also show that the MRD codes in \(\mathcal {F}\) are not equivalent to any other MRD codes known so far.

Constructions of optimal rank-metric codes from automorphisms of rational function fields

&lt;p style='text-indent:20px;'&gt;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.&lt;/p&gt;

Extending two families of maximum rank distance codes

In this paper, we properly extend the family of rank-metric codes recently found by Longobardi and Zanella (2021) and by Longobardi, Marino, Trombetti and Zhou (2021). These codes are Fq2t-linear of dimension 2 in the space of linearized polynomials over Fq2t, where t is any integer greater than 2, and we prove that they are maximum rank distance codes. For t≥5, we determine their equivalence classes and these codes turn out to be inequivalent to any other construction known so far, and hence they are really new.

Scattered subspaces and related codes

After a seminal paper by Shekeey (Adv Math Commun 10(3):475-488, 2016), a connection between maximum h-scattered $${{\mathbb {F}}}_{q}$$ -subspaces of $$V(r,q^n)$$ and maximum rank distance (MRD) codes has been established in the extremal cases $$h=1$$ and $$h=r-1$$ . In this paper, we propose a connection for any $$h\in \{1,\ldots ,r-1\}$$ , extending and unifying all the previously known ones. As a consequence, we obtain examples of non-square MRD codes which are not equivalent to generalized Gabidulin or twisted Gabidulin codes. We show that, up to equivalence, MRD codes having the same parameters as the ones in our connection come from an h-scattered subspace. Also, we determine the weight distribution of codes related to the geometric counterpart of maximum h-scattered subspaces.

Generalized LMRD Code Bounds for Constant Dimension Codes

In random network coding so-called constant dimension codes (CDCs) are used for error correction and detection. Most of the largest known codes contain a lifted maximum rank distance (LMRD) code as a subset. For some special cases, Etzion and Silberstein have demonstrated that one can obtain tighter upper bounds on the maximum possible cardinality of CDCs if we assume that an LMRD code is contained. The range of applicable parameters was partially extended by Heinlein. Here we fully generalize those bounds, which also sheds some light on recent constructions.

New Constructions of Subspace Codes Using Subsets of MRD Codes in Several Blocks

A basic problem for the constant dimension subspace coding is to determine the maximal possible size A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> (n, d, k) of a set of k-dimensional subspaces in F <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> such that the subspace distance satisfies d(U, V ) = 2k - 2 dim (U ∩ V ) ≥ d for any two different subspaces U and V in this set. We present two new constructions of constant dimension subspace codes using subsets of maximum rank-distance (MRD) codes in several blocks. This method is firstly applied to the linkage construction and secondly to arbitrary number of blocks of lifting MRD codes. In these two constructions, subsets of MRD codes with bounded ranks play an essential role. The Delsarte theorem about the rank distribution of MRD codes is an important ingredient to count codewords in our constructed constant dimension subspace codes. We give many new lower bounds for A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> (n, d, k). More than 110 new constant dimension subspace codes better than previously best known codes are constructed.

On decoding additive generalized twisted Gabidulin codes

In this paper, we consider an interpolation-based decoding algorithm for a large family of maximum rank distance codes, known as the additive generalized twisted Gabidulin codes, over the finite field mathbb {F}_{q^{n}} for any prime power q. This paper extends the work of the conference paper Li and Kadir (2019) presented at the International Workshop on Coding and Cryptography 2019, which decoded these codes over finite fields in characteristic two.

Matrix graphs and MRD codes over finite principal ideal rings

Let R be a finite principal ideal ring and m,n,d positive integers. In this paper, we study the matrix graph over R which is the graph whose vertices are m×n matrices over R and two matrices A and B are adjacent if and only if 0<rank(A−B)<d. We show that this graph is a connected vertex transitive graph. The distance, diameter, independence number, clique number and chromatic number of this graph are also determined. This graph can be applied to study MRD codes over R. We obtain that a maximal independent set of the matrix graph is a maximum rank distance (MRD) code and vice versa. Moreover, we show the existence of linear MRD codes over R.

Asymptotics of Moore exponent sets

Let n be a positive integer and I a k-subset of integers in [0,n−1]. Given a k-tuple A=(α0,⋯,αk−1)∈Fqnk, let MA,I denote the matrix (αiqj) with 0≤i≤k−1 and j∈I. When I={0,1,⋯,k−1}, MA,I is called a Moore matrix which was introduced by E. H. Moore in 1896. It is well known that the determinant of a Moore matrix equals 0 if and only if α0,⋯,αk−1 are Fq-linearly dependent. We call I that satisfies this property a Moore exponent set. In fact, Moore exponent sets are equivalent to maximum rank-distance (MRD) code with maximum left and right idealizers over finite fields. It is already known that I={0,⋯,k−1} is not the unique Moore exponent set, for instance, (generalized) Delsarte-Gabidulin codes and the MRD codes recently discovered in [5] both give rise to new Moore exponent sets. By using algebraic geometry approach, we obtain an asymptotic classification result: for q>5, if I is not an arithmetic progression, then there exists an integer N depending on I such that I is not a Moore exponent set provided that n>N.

Systematic maximum sum rank codes

In the last decade there has been a great interest in extending results for codes equipped with the Hamming metric to analogous results for codes endowed with the rank metric. This work follows this thread of research and studies the characterization of systematic generator matrices (encoders) of codes with maximum rank distance. In the context of Hamming distance these codes are the so-called Maximum Distance Separable (MDS) codes and systematic encoders have been fully investigated. In this paper we investigate the algebraic properties and representation of encoders in systematic form of Maximum Rank Distance (MRD) codes and Maximum Sum Rank Distance (MSRD) codes. We address both block codes and convolutional codes separately and present necessary and sufficient conditions for an encoder in systematic form to generate a code with maximum (sum) rank distance. These characterizations are given in terms of certain matrices that must be superregular in a extension field and that preserve superregularity after some transformations performed over the base field. We conclude the work presenting some examples of Maximum Sum Rank convolutional codes over small fields. For the given parameters the examples obtained are over smaller fields than the examples obtained by other authors.

Dickson Matrix Based Construction of Linear Maximum Rank Distance Codes

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.

Random construction of partial MDS codes

This work deals with partial MDS (PMDS) codes, a special class of locally repairable codes, used for distributed storage systems. We first show that a known construction of these codes, using Gabidulin codes, can be extended to use any maximum rank distance code. Then we define a standard form for the generator matrices of PMDS codes and use this form to give an algebraic description of PMDS generator matrices. This implies that over a sufficiently large finite field a randomly chosen generator matrix in PMDS standard form generates a PMDS code with high probability. This also provides sufficient conditions on the field size for the existence of PMDS codes.

Analysis of space time block codes and cosets for 5G and its applications

The key parameters of 5G communication technology includes high data rate and latency to support Internet of Things (IoTs) in future. There are many ways to boost up system capacity of 5G systems. In this work, design of an efficient space time block code (STBC) is presented based on maximum rank distance (MRD) codes that give enhanced system capacity in terms of diversity gain. The work also includes a comparative study of orthogonal STBC and all of its equivalent group codes including MRD for 2 transmit antennas and 1 receive antenna (2 × 1) STBCs. It is shown through simulation that MRD code and all other related coset codes can achieve full diversity gain for 2 × 1 scenario similar to what orthogonal codes can do. In addition, it is shown that orthogonal code generally outperforms all equivalent codes. A unique case is also discussed in which MRD code outperforms orthogonal STBC. Due to full diversity gain, these codes find their direct applications in 5G, Unmanned Aerial Vehicles (UAVs) used in sensor networks.

An Explicit Construction of Optimal Streaming Codes for Channels With Burst and Arbitrary Erasures

This paper presents a new construction of error correcting codes which achieves optimal recovery of a streaming source over a packet erasure channel. The channel model considered is the sliding-window erasure model, with burst and arbitrary losses, introduced by Badr et al. We present a simple construction, when the rate of the code is at least 1/2, which achieves optimal error correction in this setup. Our proposed construction is explicit and systematic. It uses off-the-shelf maximum distance separable (MDS) codes and maximum rank distance (MRD) Gabidulin block codes as constituent codes and combines them in a simple manner. This is in contrast to other recent works, where the construction involves a careful design of the generator or parity check matrix from first principles. The field size requirement which depends on the constituent MDS and MRD codes is also analyzed.

Private Information Retrieval Over Random Linear Networks

In this paper, the problem of providing privacy to users requesting data over a network from a distributed storage system (DSS) is considered. The DSS, which is considered as the multi-terminal destination of the network from the user's perspective, is encoded by a maximum rank distance (MRD) code to store the data on these multiple servers. A private information retrieval (PIR) scheme ensures that a user can request a file without revealing any information on which file is being requested to any of the servers. In this paper, a novel PIR scheme is proposed, allowing the user to recover a file from a storage system with low communication cost, while allowing some servers in the system to collude in the quest of revealing the identity of the requested file. The network is modeled as a random linear network, i.e., all nodes of the network forward random (unknown) linear combinations of incoming packets. Both error-free and erroneous random linear networks are considered.

New LMRD Code Bounds for Constant Dimension Codes and Improved Constructions

We prove upper bounds for the cardinality of constant dimension codes (CDC) which contain a lifted maximum rank distance (LMRD) code as a subset. Thereby we cover all parameters fulfilling k <; 3d/2, where k is the codeword dimension and d is the minimum subspace distance. The proofs of the bounds additionally show that an LMRD code L can be unioned with a CDC C (of fitting parameters) without violating the subspace distance condition iff each codeword of C intersects the special subspace of L in at least dimension d/2. This connection is used to find the new largest and sometimes bound achieving CDCs for small parameters.

Several classes of optimal Ferrers diagram rank-metric codes

Four constructions for Ferrers diagram rank-metric (FDRM) codes are presented. The first one makes use of a characterization on generator matrices of a class of systematic maximum rank distance codes. By introducing restricted Gabidulin codes, the second construction is presented, which unifies many known constructions for FDRM codes. The third and fourth constructions are based on two different ways to represent elements of a finite field Fqm (vector representation and matrix representation). Each of these constructions produces optimal codes with different diagrams and parameters.