Study On Reed-Solomon Codes And Turbo Codes

AbstractBoth the algebraic algorithm for decoding Reed-Solomon codes and theprobabilistic algorithm for decoding Turbo codes are investigated in this paper Both theproblems are important subjects from the view of theory as well as from that ofengineering The main results in this paper are fl. The rational interpolation problem is investigated with the elementarymathematical methods. The validity of the Welch-Beflekamp algorithm isproved. There exists one and only one solution with rank not greater than j tothe rational interpolatiott problem with j interpolating points, which has at leastone coInPlementary solution. The Welch-Berlekamp algorithm is a iterativealgoritbIn. In the kth iteration, the Welch-Berlekamp algorithm can find thesolution and one of its comPlements con-esponding to the rational interpolationproblem with k interPolating points form the solution and one of itscomplements corresPOnding to the rational interpolation probIem t,-ith k-linterpoIating points. One of the tuo solutions u.ith lower rank is just thesolution to the minimal rational interpolation problem t'ith k interpolatingpoints.2. A net" key equation is derived from the classical key equation of syndromedecoding algorithm, which transforms equivalently the problem of decoding RScodes into the minima1 rational interpolation problem. Hence the Welch-Berlekamp algorithm can sol\fe the problem of decoding RS codes;3. The Turbo iterative decoding algorithm is described in probability domain. Theiterati'e algorithm is a process tha utilizes and improves continually theextrinsic information. From this, a distance metric is introduced between theextrinsic information vectors. In the sense of the defined distance, the iteratix.ealgorithm is viewed as a continuous transformation o\'er the hyper cube. Thisimportant obsers'ation suggested us a nox.eI method \'isualizing the convergenceof the iterative aIgorithm.4. A useful stopping criterion for the iterative algorithm is presented.5. The simulating results show that the algorithm is very likely convergent formost of the received vectors, but there do exist some received vectors. tbrwhich, the algorithm does not converge.6. The simulating results also show tha the pefformance of the parallelconcatenated codes is not affected by the initialization of the a priori information, an interesting fact.