Prof. Dr.-Ing. Werner Henkel
RS codes can be defined using a DFT matrix which is known to diagonolize a Toeplitz matrix. This property is commonly used in multi-carrier modulation, since the channel realises a convolution which can be represented by a Toeplitz matrix. A more general diagonolization for parallel decomposition of a channel is provided by SVD. The question is now, if there is another way to obtain a discrete code construction other than SVD what would be the properties of such a code, especially if it guarantees a certain minimum Hamming distance? In order to check for another option , the so-called Smith Normal Form (Invariant Factor Theorem) is considered. Similar to Singular Value Decomposition, the Smith Normal Form is used to decompose a matrix into two unimodular matrices and a diagonal matrix. However, such type of diagonolization is different from the one provided by the Singular Value Decomposition. Elementary row and column operations are used to diagonolize a matrix. The aim of this project is to study the properties of the unimodular matrices, and the possibility of a code construction using the Smith Normal Form.
[Report] Smith Normal Form — a possible basis for an SVD-like code construction?