List Decoding of Matrix-Product Codes from nested codes: an application to Quasi-Cyclic codes

A list decoding algorithm for matrix-product codes is provided when $C_1,..., C_s$ are nested linear codes and $A$ is a non-singular by columns matrix. We estimate the probability of getting more than one codeword as output when the constituent codes are Reed-Solomon codes. We extend this list decoding algorithm for matrix-product codes with polynomial units, which are quasi-cyclic codes. Furthermore, it allows us to consider unique decoding for matrix-product codes with polynomial units

Decoding of Matrix-Product Codes

We propose a decoding algorithm for the $(u\mid u+v)$-construction that decodes up to half of the minimum distance of the linear code. We extend this algorithm for a class of matrix-product codes in two different ways. In some cases, one can decode beyond the error correction capability of the code

Non Binary Low Density Parity Check Codes Decoding Over Galois Field

Conventional LDPC codes have a low decoding complexity but may have high encoding complexity. The encoding complexity is typically of the order O(n2)[5]. Also high storage space may be required to explicitly store the generator matrix. For long blocknbsp lengths the storage space required would be huge. The above factors make the implementation of the Conventional LDPC codes less attractive. These codes are usually decoded using the sum-product algorithm, which is anbsp message passing algorithm working on the Tanner graph of the code[5]. The sparseness of the parity check matrix is essential for attaining good performance with sum-product decoding. The time complexity of the sum- product algorithm is linear in code length. This property makes it possible to implement a practical decoder for long lengths.nbs

Iterative Soft Input Soft Output Decoding of Reed-Solomon Codes by Adapting the Parity Check Matrix

An iterative algorithm is presented for soft-input-soft-output (SISO) decoding of Reed-Solomon (RS) codes. The proposed iterative algorithm uses the sum product algorithm (SPA) in conjunction with a binary parity check matrix of the RS code. The novelty is in reducing a submatrix of the binary parity check matrix that corresponds to less reliable bits to a sparse nature before the SPA is applied at each iteration. The proposed algorithm can be geometrically interpreted as a two-stage gradient descent with an adaptive potential function. This adaptive procedure is crucial to the convergence behavior of the gradient descent algorithm and, therefore, significantly improves the performance. Simulation results show that the proposed decoding algorithm and its variations provide significant gain over hard decision decoding (HDD) and compare favorably with other popular soft decision decoding methods.Comment: 10 pages, 10 figures, final version accepted by IEEE Trans. on Information Theor