Symmetric Rank Codes

Abstract

As is well known, a finite field \mathbb{K}n e GF(qn) can be described in terms of n × n matrices A over the field \mathbb{K} e GF(q) such that their powers Ai, i e 1, 2, …, qn − 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 Ai 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 e n and maximum possible cardinality qn. These codes are called symmetric rank codes. In the vector representation, such codes are maximum rank distance (MRD) linear ln, 1, nr 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 ln, k, d e n − k + 1r 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.