Symmetric Rank Codes

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.