Problems of Information Transmission

, Volume 40, Issue 2, pp 103–117 | Cite as

Symmetric Rank Codes

  • E. M. Gabidulin
  • N. I. Pilipchuk


As is well known, a finite field \(\mathbb{K}\) n = GF(q n ) can be described in terms of n × n matrices A over the field \(\mathbb{K}\) = GF(q) such that their powers A i , i = 1, 2, ..., q n − 1, correspond to all nonzero elements of the field. It is proved that, for fields \(\mathbb{K}\) n of characteristic 2, such a matrix A can be chosen to be symmetric. Several constructions of field-representing symmetric matrices are given. These matrices A i together with the all-zero matrix can be considered as a \(\mathbb{K}\) n -linear matrix code in the rank metric with maximum rank distance d = n and maximum possible cardinality q n . These codes are called symmetric rank codes. In the vector representation, such codes are maximum rank distance (MRD) linear [n, 1, n] codes, which allows one to use known rank-error-correcting algorithms. For symmetric codes, an algorithm of erasure symmetrization is proposed, which considerably reduces the decoding complexity as compared with standard algorithms. It is also shown that a linear [n, k, d = nk + 1] MRD code \(\mathcal{V}\) k containing the above-mentioned one-dimensional symmetric code as a subcode has the following property: the corresponding transposed code is also \(\mathbb{K}\) n -linear. Such codes have an extended capability of correcting symmetric errors and erasures.


System Theory Nonzero Element Finite Field Symmetric Matrice Vector Representation 
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Copyright information

© MAIK “Nauka/Interperiodica” 2004

Authors and Affiliations

  • E. M. Gabidulin
    • 1
  • N. I. Pilipchuk
    • 1
  1. 1.Moscow Institute of Physics and Technology (State University)Russia

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