## Abstract

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* = *n* − *k* + 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.

## Keywords

System Theory Nonzero Element Finite Field Symmetric Matrice Vector Representation## Preview

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