Symmetric Rank Codes |
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Authors: | Gabidulin E M Pilipchuk N I |
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Affiliation: | (1) Moscow Institute of Physics and Technology (State University), Russia |
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Abstract: | As is well known, a finite field
n
= GF(q
n
) can be described in terms of n × n matrices A over the field
= 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
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
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 = n – k + 1] MRD code
k
containing the above-mentioned one-dimensional symmetric code as a subcode has the following property: the corresponding transposed code is also
n
-linear. Such codes have an extended capability of correcting symmetric errors and erasures. |
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Keywords: | |
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