Gerd Richter

Fast decoding of rank-codes with rank errors and column erasures

This paper describes the decoding of Rank-Codes with different decoding algorithms. A new modified Berlekamp-Massey algorithm for correcting rank errors and column erasures is described. These algorithms consist of two decoding steps. The first step is the puncturing of the code and the decoding in the punctured code. The second step is the column erasure decoding in the original code. Thus decoding step is about half as complex as the known algorithms

On a Construction Method of Irregular LDPC Codes Without Small Stopping Sets

In this paper, we present a construction method based on the progressive edge-growth (PEG) algorithm to design irregular low-density parity-check (LDPC) codes without small stopping sets. We show how to choose the connections in the PEG algorithm when having multiple choices to connect a variable node with a check node. Since preventing small stopping sets also prevents a low minimum distance, our construction method also leads to LDPC codes with a higher minimum distance. Furthermore, we show by simulation that our construction method improves the performance over the binary erasure channel and over the additive white Gaussian noise channel for a low erasure probability and a high signal-to-noise ratio, respectively.

Irregular Low-Density Parity-Check Convolutional Codes Based on Protographs

Irregular low-density parity-check (LDPC) codes constructed from small protographs are one of the most powerful LDPC block codes. In this paper, we introduce the convolutional version of these codes. LDPC convolutional codes constructed from protographs have some advantages in comparison to LDPC block codes constructed from protographs, e.g., an effective pipeline decoding. We show that LDPC convolutional codes constructed from protographs have much lower error rates than analogous LDPC block codes with the same complexity and that they operate with low error rates near the Shannon limits even for a relatively small memory size

Optimization of a reduced-complexity decoding algorithm for LDPC codes by density evolution

In this paper, an algorithm for low-density parity-check (LDPC) codes with reduced complexity is presented. The complexity reduction is achieved by calculating a linear function for updating the check nodes in each iteration, instead of an exponential and logarithmic function. The parameters of the linear function are optimized by density evolution. Simulation results show that there is nearly no loss in the performance by using this approximation compared to the exact calculation of the belief propagation decoding algorithm, even for very large block lengths.

Linearized Shift-Register Synthesis

An efficient algorithm synthesizing all shortest <i>q</i> -linearized-feedback shift-registers generating a given sequence of length <i>N</i> over a finite field \BBF<i>qm</i> is derived and its correctness is proved. This algorithm, which is a generalization of the Berlekamp-Massey algorithm, has time complexity <i>O</i>(<i>lN</i>) <;~<i>O</i>(<i>N</i>²) operations in \BBF<i>qm</i>, where <i>l</i> is the linearized complexity of the sequence. The algorithm can be applied for efficiently solving the key equation when decoding Gabidulin codes.

On the Mapping of Low-Density Parity-Check Codes for Bit-Interleaved Coded Modulation

Low-density parity-check (LDPC) codes are very powerful error correction codes for bit-interleaved coded modulation (BICM) schemes. In BICM schemes the bits of one symbol have unequal error protection. In this paper, we show how to improve the performance of given LDPC codes by mapping the variable nodes of certain degree in a special way to the different bit levels. Since the equivalent binary-input component channels for each individual bit level are not symmetric, we use the tool of i.i.d. channel adapters to force symmetry. After that, we optimize the mapping of the LDPC code to the different bit levels by a downhill algorithm that uses a generalized discretized density evolution to calculate the thresholds. We show by thresholds and by simulations that a special mapping of LDPC codes lowers the bit error rates compared to a random mapping of LDPC codes.

Optimized Asymptotic Puncturing Distributions for Different LDPC Code Constructions

In this paper, we describe a method, how to optimize the asymptotic puncturing distributions for low-density parity-check codes constructed with different algorithms. Therefore, we generalize the discretized density evolution such that we can take care of the structure of the code. We show by density evolution and by simulations that even for the same degree distributions the optimized asymptotic puncturing distributions vary considerably for different construction algorithms. Furthermore, we demonstrate the performance gain by using the designed puncturing distributions compared to known puncturing distributions

Coding Schemes for Crisscross Error Patterns

This paper addresses two coding schemes which can handle emerging errors with crisscross patterns. First, a code with maximum rank distance, so-called Rank-Codes, is described and a modified Berlekamp–Massey algorithm is provided. Secondly, a Permutation Code based coding scheme for crisscross error patterns is presented. The influence of different types of noise are also discussed.

Channel Correlation Properties in OFDM by using Time-Varying Cyclic Delay Diversity

This paper analyzes the influence of time-varying cyclic delay diversity (TV-CDD) on the channel fading correlation properties in orthogonal frequency division mul- tiplexing (OFDM) based systems. The underlying trans- mit diversity technique CDD only increases the frequency diversity at the receiver. In contrast, TV-CDD introduces additionally time diversity which can be exploited without the need of additional complexity at the receiver. This paper gives investigations regarding the resulting channel characteristics from TV-CDD and the impact on the system performance. Due to the increased frequency and time selectivity, an unintended higher channel estimation effort is possible. Therefore, we analyze the impact of choice of the maximum cyclic delay. We show that the resulting channel for TV-CDD can be seen as an uncorrelated Rayleigh fading channel (except for the first sub-carrier) for a large maximum cyclic delay. Furthermore, analysis and simulation results demonstrate a feasible choice of small time-varying cyclic delays for guaranteeing the standard conformability of the TV-CDD technique at the receiver without significant performance degradations.

On the design of woven convolutional encoders with outer warp row permutors

In this paper, permutor design aspects for woven convolutional encoders (WCEs) with outer warp (OW) are discussed. We show how the minimum distance of the overall code can be increased by considering all warp permutors together. A bound on the minimum distance of a code generated by a WCE with OW and designed permutors is presented that considerably exceeds those with known designs.