Birgit Schotsch

The performance of low-density random linear fountain codes over higher order galois fields under maximum likelihood decoding

Digital fountain codes over higher order Galois fields exhibit a better performance than their binary counterparts under maximum likelyhood (ML) decoding when transmitted over a symbol erasure channel (SEC). Especially random linear fountain (RLF) codes exhibit an excellent performance, though at the expense of a high computational complexity for decoding due to their high density generator matrix. For practical applications, we propose RLF codes with a reduced density over higher order Galois fields. Although the reduction of the density results in an error floor at higher reception overheads, the level of this error floor can be well controlled by two parameters. For error floor levels that are tolerable in practical applications, a significant density reduction and thus a reduction of the computational complexity can be achieved. Furthermore, we derive a general upper bound on the symbol erasure rate for Luby Transform (LT) codes over Galois fields Fq of order q. Finally, we propose a method to enhance decoding of Fq-codes in the presence of bit erasures by using the binary images of the Fq-elements, such that not complete Fq-elements have to be discarded if their binary counterparts are impaired by bit erasures.

The Performance of Short Random Linear Fountain Codes under Maximum Likelihood Decoding

In this paper, two particular instances of LT codes with short message blocklength

Analysis of LT Codes over Finite Fields under Optimal Erasure Decoding

k

Finite length LT codes over F q for unequal error protection with biased sampling of input nodes

and maximum likelihood (ML) decoding are investigated, i.e., random linear fountain (RLF) codes and (nearly) check-concentrated LT codes. Both show an almost equally good performance. The focus of this paper will be on RLF codes, a type of LT codes whose generator matrices are constructed from independent Bernoulli trials and have a binomial check node degree distribution. A new simple expression for an upper bound on the bit erasure probability under ML decoding is derived for RLF codes with density 0.5, i.e., with check node degree distribution

Properties and performance bounds of linear analog block codes

\Omega(x) = 2^{-k}(1+x)^k

Graph-Based Turbo DeCodulation with LDPC Codes

. It is shown that RLF codes with a minimum density far less than 0.5 are equally well suited to achieve a certain bit erasure probability for a given reception overhead. Furthermore, a characteristic term from a general upper bound on the bit erasure probability under ML decoding is identified that can be used to optimise check node degree distributions. Its implications on the performance of LT codes are qualitatively analysed.

Design of unequally error protecting low-density random linear fountain codes

The erasure correction performance of Luby transform (LT) code ensembles over higher order Galois fields is analysed under optimal, \ie maximum likelihood (ML) erasure decoding. We provide the complete set of four bounds on the erasure probability after decoding on word as well as on symbol level. Especially the upper bounds are extremely close to the simulated residual erasure rates after decoding and can thus be used for code design instead of time-consuming simulations.

Adaptive Exploitation of Residual Redundancy in Iterative Source-Channel Decoding

Finite length LT codes over higher order Galois fields Fq for unequal error protection (UEP) are analysed under maximum likelihood (ML) decoding. We consider a biased sampling method to create the LT code graph. In contrast to a previous approach by Rahnavard et al., where a predetermined number of edges is created per importance class given a check node of degree d, our procedure allows to precisely adjust the desired class weights. Moreover, we provide upper and lower bounds on the symbol erasure probability for each importance class.

Separation of Recursive Convolutional Codes into Sub-Codes using Galois Field Arithmetic

Linear analog block codes have been considered for transmission of discrete-time and continuous-amplitude data. In this paper, the fidelity measure parameter SNR (pSNR) at the receiver is derived for an arbitrary generator matrix P using an additive white Gaussian noise (AWGN) channel. In contrast, it is shown that the performance of linear analog block codes is dependent on the eigenvalues of the matrix PTP and not only on the dimensions of the matrix P. Surprisingly, the quality of the received values is independent of the code rate r, and e.g. a simple identity matrix has the optimal eigenvalues. Furthermore, the theoretical fidelity bound OPTA (Optimum Performance Theoretically Attainable) is used to assess the performance of a transmission system of continuous-amplitude data.

Efficient Iterative Source-Channel Decoding Using Irregular Index Assignments

Turbo DeCodulation is the combination of iterative demodulation and iterative source-channel decoding in a multiple Turbo process. The receiver structures of bit-interleaved coded modulation with iterative decoding (BICM-ID) and iterative source-channel decoding (ISCD) are merged to one joint Turbo system, which we further enhance in this paper by using a low- density parity check (LDPC) code for channel coding, resulting in a third iterative loop. We propose to use a special LDPC code structure with short sub-codes, which can be implemented very effectively in parallel. The quadripartite Tanner graph of the Turbo DeCodulation is presented, showing that the processing of all receiver nodes could be parallelized. Simulation results including an EXIT chart analysis demonstrate the excellent capabilities of Turbo DeCodulation with its performance gain exceeding the combined gain of BICM-ID and ISCD.