Error-correcting codes (Q1327871)

An elementary introduction to the theory of error correcting codes over a finite field is given. The initial example is the four game football pool corresponding to the ternary code of length four. The single error- correcting Hamming codes are introduced and generalized to arbitrary linear codes over a finite field, with notions of Hamming distance, generator and parity check matrices used. Cyclic codes are discussed as ideals in the ring \(F_ q [x]/ (x^ n -1)\). The BCH bound on the minimum distance of a cyclic code is established using the usual Vandermonde determinant approach. Golay and Reed-Solomon codes are introduced and their application to deep space communication briefly discussed.