Generalization of Gabidulin Codes over Fields of Rational Functions

We transpose the theory of rank metric and Gabidulin codes to the case of fields which are not finite fields. The Frobenius automorphism is replaced by any element of the Galois group of a cyclic algebraic extension of a base field. We use our framework to define Gabidulin codes over the field of rational functions using algebraic function fields with a cyclic Galois group. This gives a linear subspace of matrices whose coefficients are rational function, such that the rank of each of this matrix is lower bounded, where the rank is comprised in term of linear combination with rational functions. We provide two examples based on Kummer and Artin-Schreier extensions.The matrices that we obtain may be interpreted as generating matrices of convolutional codes.Comment: 21st International Symposium on Mathematical Theory of Networks and Systems (MTNS 2014), Jul 2014, Groningen, Netherlands. https://fwn06.housing.rug.nl/mtns2014

On the Geometry of Balls in the Grassmannian and List Decoding of Lifted Gabidulin Codes

The finite Grassmannian $\mathcal{G}_{q}(k,n)$ is defined as the set of all $k$-dimensional subspaces of the ambient space $\mathbb{F}_{q}^{n}$. Subsets of the finite Grassmannian are called constant dimension codes and have recently found an application in random network coding. In this setting codewords from $\mathcal{G}_{q}(k,n)$ are sent through a network channel and, since errors may occur during transmission, the received words can possible lie in $\mathcal{G}_{q}(k',n)$, where $k'\neq k$. In this paper, we study the balls in $\mathcal{G}_{q}(k,n)$ with center that is not necessarily in $\mathcal{G}_{q}(k,n)$. We describe the balls with respect to two different metrics, namely the subspace and the injection metric. Moreover, we use two different techniques for describing these balls, one is the Pl\"ucker embedding of $\mathcal{G}_{q}(k,n)$, and the second one is a rational parametrization of the matrix representation of the codewords. With these results, we consider the problem of list decoding a certain family of constant dimension codes, called lifted Gabidulin codes. We describe a way of representing these codes by linear equations in either the matrix representation or a subset of the Pl\"ucker coordinates. The union of these equations and the equations which arise from the description of the ball of a given radius in the Grassmannian describe the list of codewords with distance less than or equal to the given radius from the received word.Comment: To be published in Designs, Codes and Cryptography (Springer

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

Reed-Solomon codes and Gabidulin codes have maximum Hamming distance and maximum rank distance, respectively. A general construction using skew polynomials, called skew Reed-Solomon codes, has already been introduced in the literature. In this work, we introduce a linearized version of such codes, called linearized Reed-Solomon codes. We prove that they have maximum sum-rank distance. Such distance is of interest in multishot network coding or in singleshot multi-network coding. To prove our result, we introduce new metrics defined by skew polynomials, which we call skew metrics, we prove that skew Reed-Solomon codes have maximum skew distance, and then we translate this scenario to linearized Reed-Solomon codes and the sum-rank metric. The theories of Reed-Solomon codes and Gabidulin codes are particular cases of our theory, and the sum-rank metric extends both the Hamming and rank metrics. We develop our theory over any division ring (commutative or non-commutative field). We also consider non-zero derivations, which give new maximum rank distance codes over infinite fields not considered before

MRD codes with maximum idealizers

Left and right idealizers are important invariants of linear rank-distance codes. In the case of maximum rank-distance (MRD for short) codes in $\mathbb{F}_q^{n\times n}$ the idealizers have been proved to be isomorphic to finite fields of size at most $q^n$. Up to now, the only known MRD codes with maximum left and right idealizers are generalized Gabidulin codes, which were first constructed in 1978 by Delsarte and later generalized by Kshevetskiy and Gabidulin in 2005. In this paper we classify MRD codes in $\mathbb{F}_q^{n\times n}$ for $n\leq 9$ with maximum left and right idealizers and connect them to Moore-type matrices. Apart from generalized Gabidulin codes, it turns out that there is a further family of rank-distance codes providing MRD ones with maximum idealizers for $n=7$, $q$ odd and for $n=8$, $q\equiv 1 \pmod 3$. These codes are not equivalent to any previously known MRD code. Moreover, we show that this family of rank-distance codes does not provide any further examples for $n\geq 9$.Comment: Reviewers' comments implemented, we changed the titl

MRD codes with maximum idealisers

Left and right idealizers are important invariants of linear rank-distance codes. In the case of maximum rank-distance (MRD for short) codes in GF(q)^(n×n) the idealizers have been proved to be isomorphic to finite fields of size at most q^n. Up to now, the only known MRD codes with maximum left and right idealizers are generalized Gabidulin codes, which were first constructed in 1978 by Delsarte and later generalized by Kshevetskiy and Gabidulin in 2005. In this paper we classify MRD codes in GF(q)^(n×n) for n9

Exceptional scattered sequences

The concept of scattered polynomials is generalized to those of exceptional scattered sequences which are shown to be the natural algebraic counterpart of $\mathbb{F}_{q^n}$-linear MRD codes. The first infinite family in the first nontrivial case is also provided and equivalence issues are considered. As a byproduct, a new infinite family of MRD codes is obtained.Comment: 32 page

Row Reduction Applied to Decoding of Rank Metric and Subspace Codes

We show that decoding of $\ell$-Interleaved Gabidulin codes, as well as list-$\ell$ decoding of Mahdavifar--Vardy codes can be performed by row reducing skew polynomial matrices. Inspired by row reduction of \F[x] matrices, we develop a general and flexible approach of transforming matrices over skew polynomial rings into a certain reduced form. We apply this to solve generalised shift register problems over skew polynomial rings which occur in decoding $\ell$-Interleaved Gabidulin codes. We obtain an algorithm with complexity $O(\ell \mu^2)$ where $\mu$ measures the size of the input problem and is proportional to the code length $n$ in the case of decoding. Further, we show how to perform the interpolation step of list-$\ell$-decoding Mahdavifar--Vardy codes in complexity $O(\ell n^2)$, where $n$ is the number of interpolation constraints.Comment: Accepted for Designs, Codes and Cryptograph