Results on error-correcting codes obtained using linear algebra
Author
Ball, S.
Type of activity
Presentation of work at congresses
Name of edition
Real Sociedad Matematica Española
Date of publication
2013
Presentation's date
2013-01-23
Book of congress proceedings
Sesiones Especiales - RSME 2013
First page
16
Last page
16
Abstract
A linear code C is a k-dimensional subspace of Fnq, with respect to a fixed basis.
Let d 2 N be minimal sich that every non-zero vector of C has at least d non-zero coordinates. With such a code, up to b(d ¿ 1)=2c errors in any codeword can be corrected when sending codewords down a noisy channel.
There are many results on error-correcting codes obtained by combinatorial arguments (and these generally apply to non-linear codes as well) such as the sphere packing bound, the Gilbert-Varshamov bou...
A linear code C is a k-dimensional subspace of Fnq, with respect to a fixed basis.
Let d 2 N be minimal sich that every non-zero vector of C has at least d non-zero coordinates. With such a code, up to b(d ¿ 1)=2c errors in any codeword can be corrected when sending codewords down a noisy channel.
There are many results on error-correcting codes obtained by combinatorial arguments (and these generally apply to non-linear codes as well) such as the sphere packing bound, the Gilbert-Varshamov bound, the Plotkin bound, etc. In this talk I will concentrate on results obtained using linear algebra, for example the MacWilliam’s identities, the MDS conjecture, the maximum weight of a codeword. It is also
possible to obtain results for non-linear codes using linear algebra and I will also mention one of these, the Alderson-Gács extension theorem.