Giuliano Garrammone

On Decoding Complexity of Reed-Solomon Codes on the Packet Erasure Channel

We study the finite-length complexity of the Berlekamp-Massey algorithm (BM) and of the Gaussian elimination algorithm (GE) for decoding a Reed-Solomon (RS) code of length n on the packet erasure channel (PEC). In particular, we are interested in the dependency of the complexity on the packet size g. We show that, for large packet sizes, the complexity of both algorithms is O(g) and the constants hidden by the O-notation are comparable. As an example, we consider a RS code from the Digital Video Broadcasting - Handhelds (DVB-H) standard and we show that, although the asymptotic complexity of the GE is O(n3) and the one of the BM is O(n2), the finite-length complexity of both algorithms is comparable already for small/moderate packet sizes, in particular for all packet sizes considered within the standard, making the GE not inferior to the BM, in this context.

Analysis of LT Codes over Finite Fields under Optimal Erasure Decoding

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.

Non-binary low-density parity-check codes for the q-ary erasure channel

The finite-length design of non-binary low-density parity-check (LDPC) codes for the q-ary erasure channel under maximum a posteriori (MAP) decoding is addressed. A low-complexity MAP decoding algorithm is reviewed for which a code design strategy is proposed. In particular, it is illustrated how a judicious code design permits to find a trade-off between performance in terms of codeword error rate (CER) and decoding complexity. As an example, the performance curve of a short (400, 200) code on the memoryless 4-ary erasure channel tightly approaches the Singleton bound at least down to a CER of 10-8.

A Novel Protocol to Transmit Alert Messages during Crises over GNSS

Alert messages are usually encoded according to the CAP (Common Alerting Protocol) de-facto standard, which is usually translated into an appropriate XML schema for transport over HTTP. The large overhead introduced by XML penalizes the delivery of alert messages to final recipients because of the latency introduced in GNSS channels. To compensate this effect, this paper introduces a novel protocol actually translating the CAP format and providing a more efficient transport of alert messages.

Non-Binary LDPC Erasure Codes With Separated Low-Degree Variable Nodes

The code design of non-binary low-density paritycheck codes for the erasure channel, under maximum a posteriori decoding, is addressed. In particular, a partially structured ensemble of codes, characterized by a careful control of the amount and of the connectivity of the variable nodes of small degrees, is proposed. The identified ensemble of codes is analyzed in terms of asymptotic thresholds and weight distribution and it is shown that codes from the ensemble provide a remarkable trade-off between waterfall performance, error floor, and decoding complexity. As an example, the performance curve of a short (256,128) code on the memoryless 16-ary erasure channel tightly approaches the Singleton bound at least down to a codeword error rate of 10-9, at low decoding complexity.

Short Erasure Correcting LDPC IRA Codes over GF(q)

This paper investigates non-binary low-density parity-check (LDPC) erasure correcting codes suitable to guarantee reliable transmission in wireless communications systems. In particular, irregular repeat-accumulate (IRA) codes are considered, characterized by linear-time encoding complexity. The performance of non-binary IRA codes is compared with their binary counterparts on the packet erasure channel (PEC), with considerable advantages for the non-binary construction. Particularly, it is illustrated that the performance of short-block-length erasure correcting IRA codes over Galois fields (GFs) of order q > 2 approaches, under maximum-likelihood (ML) decoding, the performance of ideal maximum distance separable (MDS) codes. This is especially appealing in the context of satellite communications, where efficient codes are required to cope with small link margins.

Weight distributions of non-binary multi-edge type LDPC code ensembles

Non-binary multi-edge type ensembles of low-density parity-check codes are analyzed in terms of non-binary codeword weight distribution and its growth rate. In particular, an exact expression of the growth rate for small weights is developed. As a side result, the stopping set distributions of these ensembles are developed. Examples of weight distributions are provided, showing that the derived closed-form expressions can be easily evaluated. The obtained results can thus be exploited to analyze and design non-binary low-density parity-check codes that fall within the multi-edge type framework such as, but not limited to, protograph-based codes.

Recent Findings on Erasure Codes for Space Communications

Implementation of automatic repeat request strategies is the main reliability option available in the Consultative Committee for Space Data Systems protocols to support data communications in deep-space missions. The large latency often experienced in such scenarios, however, penalizes the efficiency of retransmissions, making the overall applicability of automatic repeat request problematic. On the contrary, the use of erasure codes is a more appealing solution to mitigate packet losses. In this regard, preliminary studies on the use of binary low-density parity-check codes under maximum likelihood/iterative decoding have already shown the performance benefit they can bring with respect to traditional schemes based on retransmissions. This paper extends the analysis conducted in previous studies toward non-binary low-density parity-check codes. Performance assessment is carried out with respect to reliability metrics (codeword error rate) and encoding/decoding complexity. Finally, the integration of erasure codes into the Consultative Committee for Space Data Systems protocol stack is discussed, by presenting the implementation in the Licklider Transmission Protocol.

Quasi-Cyclic Doubly Generalized LDPC Codes

We introduce a class of structured (protograph-based) doubly generalized low-density parity-check (DGLDPC) codes. The proposed class leads to quasi-cyclic codes for which efficient encoder and decoder implementations are possible. In particular, we illustrate that on the additive white Gaussian noise (AWGN) channel, the introduced structured DGLDPC codes do not lose in performance with respect to unstructured DGLDPC codes. Furthermore, a sufficient condition on the code graph is derived, which allows unveiling the quasi-cyclic nature of the designed codes.

Weight Distributions of Non-Binary Multi-Edge Type LDPC Code Ensembles: Analysis and Efficient Evaluation

The non-binary codeword weight distribution and its growth rate are developed for non-binary multi-edge type ensembles of low-density parity-check codes. Moreover, an analysis of the growth rate for small weights is provided. The derived expressions can serve as powerful and flexible tools to analyze and design the non-binary low-density parity-check codes that fall within the multi-edge type framework.