Symmetric Rank Codes

Symmetric Rank Codes 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Problems of Information Transmission Springer Journals

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Publisher
Springer Journals
Copyright
Copyright © 2004 by MAIK “Nauka/Interperiodica”
Subject
Engineering; Communications Engineering, Networks; Electrical Engineering; Information Storage and Retrieval; Systems Theory, Control
ISSN
0032-9460
eISSN
1608-3253
D.O.I.
10.1023/B:PRIT.0000043925.67309.c6
Publisher site
See Article on Publisher Site

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.

Journal

Problems of Information TransmissionSpringer Journals

Published: Dec 16, 2004

References

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