Nina I. Pilipchuk

Error and erasure correcting algorithms for rank codes

In this paper, transmitted signals are considered as square matrices of the Maximum rank distance (MRD) (n, k, d)-codes. A new composed decoding algorithm is proposed to correct simultaneously rank errors and rank erasures. If the rank of errors and erasures is not greater than the Singleton bound, then the algorithm gives always the correct decision. If it is not a case, then the algorithm gives still the correct solution in many cases but some times the unique solution may not exist.

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.

Decoding of random network codes

We consider the decoding for Silva-Kschischang-Kötter random network codes based on Gabidulin’s rank-metric codes. The model of a random network coding channel can be reduced to transmitting matrices of a rank code through a channel introducing three types of additive errors. The first type is called random rank errors. To describe other types, the notions of generalized row erasures and generalized column erasures are introduced. An algorithm for simultaneous correction of rank errors and generalized erasures is presented. An example is given.

Symmetric matrices and codes correcting rank errors beyond the ⌊(d - 1)/2⌋ bound

Rank codes can be described either as matrix codes over the base field Fq or as vector codes over the extension field Fqn. For any matrix code, there exists a corresponding vector codes, and vice versa. We investigate matrix codes containing a linear subcode of symmetric matrices. The corresponding vector codes contain a linear subspace of so-called symmetric vectors. It is shown that such vector codes are generated by self-orthogonal bases of the field Fqn. If code distance is equal to d, than such codes can correct not only all the errors of rank up to [(d - 1)/2] but also many symmetric errors of rank beyond this bound.

Symmetric Rank Codes

AbstractAs is well known, a finite field

Bounds on cardinality of multicomponent network codes

\mathbb{K}

Decoding new multicomponent codes

n = GF(qn) can be described in terms of n × n matrices A over the field