Netanel Raviv

Subspace Polynomials and Cyclic 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 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.

Subspace polynomials and cyclic 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 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.

Equidistant codes in the Grassmannian

Equidistant codes over vector spaces are considered. For k -dimensional subspaces over a large vector space the largest code is always a sunflower. We present several simple constructions for such codes which might produce the largest non-sunflower codes. A novel construction, based on the Plucker embedding, for 1-intersecting codes of k -dimensional subspaces over F q n , n ? ( k + 1 2 ) , where the code size is q k + 1 - 1 q - 1 is presented. Finally, we present a related construction which generates equidistant constant rank codes with matrices of size n i? ( n 2 ) over F q , rank n - 1 , and rank distance n - 1 .

Some Gabidulin Codes Cannot Be List Decoded Efficiently at any Radius

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.

Distributed storage systems based on intersecting subspace codes

Distributed storage systems based on intersecting constant dimension (equidistant) codes are presented. These intersecting codes are constructed using the Plücker embedding, which is essential in the repair and the reconstruction algorithms. These systems possess several useful properties such as high failure resilience, minimum bandwidth, low overall storage, simple algebraic repair and reconstruction algorithms, good locality, and compatibility with small fields.

Constructions of High-Rate Minimum Storage Regenerating Codes Over Small Fields

A novel technique for construction of minimum storage regenerating (MSR) codes is presented. Based on this technique, three explicit constructions of MSR codes are given. The first two constructions provide access-optimal MSR codes, with two and three parities, respectively, which attain the sub-packetization bound for access-optimal codes. The third construction provides longer MSR codes with three parities (i.e., codes with larger number of systematic nodes). This improvement is achieved at the expense of the access-optimality and the field size. In addition to a minimum storage in a node, all three constructions allow the entire data to be recovered from a minimal number of storage nodes. That is, given storage

Some Gabidulin codes cannot be list decoded efficiently at any radius

\ell

Constructions of high-rate minimum storage regenerating codes over small fields

in each node, the entire stored data can be recovered from any

Asymptotically optimal regenerating codes over any field

2\log _{2} \ell

Coding for locality in reconstructing permutations

for two parity nodes, and either