List-Decoding Gabidulin Codes via Interpolation and the Euclidean Algorithm

We show how Gabidulin codes can be list decoded by using a parametrization approach. For this we consider a certain module in the ring of linearized polynomials and find a minimal basis for this module using the Euclidean algorithm with respect to composition of polynomials. For a given received word, our decoding algorithm computes a list of all codewords that are closest to the received word with respect to the rank metric.Comment: Submitted to ISITA 2014, IEICE copyright upon acceptanc

A concatenated coded modulation scheme for error control

A concatenated coded modulation scheme for error control in data communications is presented. The scheme is achieved by concatenating a Reed-Solomon outer code and a bandwidth efficient block inner code for M-ary PSK modulation. Error performance of the scheme is analyzed for an AWGN channel. It is shown that extremely high reliability can be attained by using a simple M-ary PSK modulation inner code and a relatively powerful Reed-Solomon outer code. Furthermore, if an inner code of high effective rate is used, the bandwidth expansion required by the scheme due to coding will be greatly reduced. The proposed scheme is very effective for high speed satellite communications for large file transfer where high reliability is required. A simple method is also presented for constructing codes for M-ary PSK modulation. Some short M-ary PSK codes with good minimum squared Euclidean distance are constructed. These codes have trellis structure and hence can be decoded with a soft decision Viterbi decoding algorithm. Furthermore, some of these codes are phase invariant under multiples of 45 deg rotation

A concatenated coded modulation scheme for error control

A concatenated coded modulation scheme for error control in data communications is presented. The scheme is achieved by concatenating a Reed-Solomon outer code and a bandwidth efficient block inner code for M-ary PSK modulation. Error performance of the scheme is analyzed for an AWGN channel. It is shown that extremely high reliability can be attained by using a simple M-ary PSK modulation inner code and relatively powerful Reed-Solomon outer code. Furthermore, if an inner code of high effective rate is used, the bandwidth expansion required by the scheme due to coding will be greatly reduced. The proposed scheme is particularly effective for high speed satellite communication for large file transfer where high reliability is required. Also presented is a simple method for constructing block codes for M-ary PSK modulation. Some short M-ary PSK codes with good minimum squared Euclidean distance are constructed. These codes have trellis structure and hence can be decoded with a soft decision Viterbi decoding algorithm

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

Solving Shift Register Problems over Skew Polynomial Rings using Module Minimisation

For many algebraic codes the main part of decoding can be reduced to a shift register synthesis problem. In this paper we present an approach for solving generalised shift register problems over skew polynomial rings which occur in error and erasure decoding of $\ell$-Interleaved Gabidulin codes. The algorithm is based on module minimisation and has time complexity $O(\ell \mu^2)$ where $\mu$ measures the size of the input problem.Comment: 10 pages, submitted to WCC 201

Decoding Reed-Solomon codes up to the Sudan radius with the Euclidean algorithm

International audienceWe modify the Euclidean algorithm of Feng and Tzeng to decode Reed-Solomon (RS) codes up to the Sudan radius. The basic steps are the virtual extension to an Interleaved RS code and the reformulation of the multi-sequence shift-register problem of varying length to a multi-sequence problem of equal length. We prove the reformulation and analyze the complexity of our new decoding approach. Furthermore, the extended key equation, that describes the multi-sequence problem, is derived in an alternative polynomial way