New constructions of cyclic constant-dimension subspace codes based on Sidon spaces and subspace polynomials

A characterization of cyclic subspace codes via subspace polynomials

Subspace codes, especially cyclic subspace codes, have attracted a wide attention in the past few decades due to their applications in error correction for random network coding. In 2016, Ben-Sasson et al. gave a systematic approach to constructing cyclic subspace codes by employing subspace polynomials. Inspired by Ben-Sasson’s idea, Chen et al. also provided some constructions of cyclic subspace codes in 2017. In this paper, two constructions of cyclic subspace codes are given to further improve the results of Chen and Roth et al. respectively. Consequently, we obtain more cyclic subspace codes with larger size of codewords without reducing the minimum distance.

Subspace codes have been intensely studied in the last decade due to their application in random network coding. In particular, cyclic subspace codes are very useful subspace codes with their efficient encoding and decoding algorithms. In a recent paper, Ben-Sasson et al. gave a systematic construction of subspace codes using subspace polynomials. In this paper, we mainly generalize and improve their result so that we can obtain larger codes for fixed parameters and also we can increase the density of some possible parameters. In addition, we give some relative remarks and explicit examples.

Gabidulin codes can be seen as the rank-metric equivalent of Reed–Solomon codes. It was recently proved, using subspace polynomials, that Gabidulin codes cannot be list decoded beyond the so-called Johnson radius. In another result, cyclic subspace codes were constructed by inspecting the connection between subspaces and their subspace polynomials. In this paper, these subspace codes are used to prove two bounds on the list size in decoding certain Gabidulin codes. The first bound is an existential one, showing that exponentially sized lists exist for codes with specific parameters. The second bound presents exponentially sized lists explicitly for a different set of parameters. Both bounds rule out the possibility of efficiently list decoding several families of Gabidulin codes for any radius beyond half the minimum distance. Such a result was known so far only for non-linear rank-metric codes, and not for Gabidulin codes. Using a standard operation called lifting, identical results also follow for an important class of constant dimension subspace codes.

Subspace codes have received an increasing interest recently due to their application in error-correction for random network coding. In particular, cyclic subspace codes are possible candidates for large codes with efficient encoding and decoding algorithms. In this paper we consider such cyclic codes. We provide constructions of optimal cyclic codes for which their codewords do not have full length orbits. We further introduce a new way to represent subspace codes by a class of polynomials called subspace polynomials. We present some constructions of such codes which are cyclic and analyze their parameters.

Subspace codes have received an increasing interest recently due to their application in error correction for random network coding. In particular, cyclic subspace codes are possible candidates for large codes with efficient encoding and decoding algorithms. In this paper, we consider such cyclic codes and provide constructions of optimal codes for which their codewords do not have full orbits. We further introduce a new way to represent subspace codes by a class of polynomials called subspace polynomials. We present some constructions of such codes, which are cyclic and analyze their parameters.

Gabidulin codes can be seen as the rank-metric equivalent of Reed-Solomon codes. It was recently proven, using subspace polynomials, that Gabidulin codes cannot be list decoded beyond the so-called Johnson radius. In another result, cyclic subspace codes were constructed by inspecting the connection between subspaces and their subspace polynomials. In this paper, these subspace codes are used to prove two bounds on the minimum possible list size in decoding certain Gabidulin codes. The first bound is an existential one, showing that exponentially-sized lists exist for codes with specific parameters. The second bound presents exponentially-sized lists explicitly, for a different set of parameters. Both bounds rule out the possibility of efficiently list decoding their respective families of codes for any radius beyond half the minimum distance. Such a result was known so far only for non-linear rank-metric codes, and not for Gabidulin codes.

A subspace of a finite extension field is called a Sidon space if the product of any two of its elements is unique up to a scalar multiplier from the base field. Sidon spaces were recently introduced by Bachoc et al. as a means to characterize multiplicative properties of subspaces, and yet no explicit constructions were given. In this paper, several constructions of Sidon spaces are provided. In particular, in some of the constructions the relation between $k$ , the dimension of the Sidon space, and $n$ , the dimension of the ambient extension field, is optimal. These constructions are shown to provide cyclic subspace codes, which are useful tools in network coding schemes. To the best of our knowledge, this constitutes the first set of constructions of non-trivial cyclic subspace codes in which the relation between $k$ and $n$ is polynomial, and in particular, linear. As a result, a conjecture by Trautmann et al. regarding the existence of non-trivial cyclic subspace codes is resolved for most parameters, and multi-orbit cyclic subspace codes are attained, whose cardinality is within a constant factor (close to 1/2) from the sphere-packing bound for subspace codes.

New constructions of large cyclic subspace codes and Sidon spaces

Multi-orbit cyclic subspace codes and linear sets

Subspace codes and particularly constant dimension codes have attracted much attention in recent years due to their applications in random network coding. As a particular subclass of subspace codes, cyclic subspace codes have additional properties that can be applied efficiently in encoding and decoding algorithms. It is desirable to find cyclic constant dimension codes such that both the code sizes and the minimum distances are as large as possible. In this paper, we explore the ideas of constructing cyclic constant dimension codes proposed in \big([2], IEEE Trans. Inf. Theory, 2016\big) and \big([17], Des. Codes Cryptogr., 2016\big) to obtain further results. Consequently, new code constructions are provided and several previously known results in [2] and [17] are extended.

The interest in subspace codes has increased in recent years due to their application in error correction for random network coding. In order to study their properties and find good constructions, the notion of cyclic subspace codes was introduced by using the extension field structure of the ambient space. However, to this date there exists no general construction with a polynomial relation between k, the dimension of the codewords, and n, the dimension of the entire space. Independently of the study of cyclic subspace codes, sSidon spaces were recently introduced by Bachoc et al. as a tool for the study of certain multiplicative properties of subspaces over finite fields. In this paper it is shown that Sidon spaces are necessary and sufficient for obtaining a full-orbit cyclic subspace code with minimum distance 2 k − 2. By presenting several constructions of Sidon spaces, full-orbit cyclic subspace codes are obtained, in which n is quadratic in k. The constructions are based on a variety of tools; namely, Sidon sets, that are sets of integers in which all pairwise sums are distinct, irreducible polynomials, and linearized polynomials. Further, the existence of a Sidon space in which n is linear in k is shown, alongside the fact that any Sidon space induces a Sidon set.

In this paper we construct, using GAP System for Computational Discrete Algebra and Wolfram Mathematica, some cyclic subspace codes, specially an optimal code over the finite field F210 . Further we introduce the q-analogous of theclassic quasi cyclic block codes over finite fields.

This chapter is a survey of the recent results on the constructions of cyclic subspace codes and maximum rank distance codes. Linearized polynomials are the main tools used to introduce both constructions in this chapter. In the construction of cyclic subspace codes, codewords are considered as the root spaces of some subspace polynomials (which are a particular type of linearized polynomials). In this set up, some algebraic manipulations on the coefficients and degrees of such polynomials are applied to provide a systematic construction of cyclic subspace codes. In constructions of maximum rank distance codes, linearized polynomials are used as codewords again, but in a different way. Codewords of rank metric codes are considered as the linear maps that these polynomials represent. All known constructions of maximum rank distance codes in the literature are summarized using this linearized polynomial representation. Connections among the constructions and further explanations are also provided.

Further constructions of large cyclic subspace codes via Sidon spaces