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.