Abstract-This work introduces the Generalized Low DensityParity Check (GLDPC)-Staircase codes for the erasure channel, that are constructed by extending LDPC-Staircase codes through Reed Solomon (RS) codes based on "quasi" Hankel matrices. This construction has several key benefits: in addition to the LDPC-Staircase repair symbols, it adds extra-repair symbols that can be produced on demand and in large quantities, which provides small rate capabilities. Additionally, with selecting the best internal parameters of GLDPC graph and under hybrid Iterative/Reed-Solomon/Maximum Likelihood decoding, the GLDPC-Staircase codes feature a very small decoding overhead and a low error floor. These excellent erasure capabilities, close to that of ideal, MDS codes, are obtained both with large and very small objects, whereas, as a matter of comparison, LDPC codes are known to be asymptotically good. Therefore, these properties make GLDPC-Staircase codes an excellent AL-FEC solution for many situations that require erasure protection such as media streaming.

We investigate the efficacy of error correcting codes in improving the performance of underwater wireless optical communication systems. For this purpose, the effectiveness of several coding schemes, i.e., the classical Reed-Solomon and a recent family of low-density parity check codes, is studied in the physical (PHY) and the upper layers assuming negligible water turbulence. The presented numerical results testify to the interest of using efficient codes both at the PHY and upper protocol layers, although we are concerned by a non-fading channel. Furthermore, we discuss the choice of coding schemes and the appropriate degree of data protection in the PHY and upper layers.

Reed Solomon RS(n, k) codes are Maximum Distance Separable (MDS) ideal codes that can be put into a systematic form, which makes them well suited to many situations. In this work we consider use-cases that rely on a software RS codec and for which the code is not fixed. This means that the application potentially uses a different RS(n, k) code each time, and this code needs to be built dynamically. A lightweight code creation scheme is therefore highly desirable, otherwise this stage would negatively impact the encoding and decoding times. Constructing such an RS code is equivalent to constructing its systematic generator matrix. Using the classic Vandermonde matrix approach to that purpose is feasible but adds significant complexity. In this paper we propose an alternative solution, based on Hankel matrices as the base matrix. We prove theoretically and experimentally that the code construction time and the number of operations performed to build the target RS code are largely in favor of the Hankel approach, which can be between 3.5 to 157 times faster than the Vandermonde approach, depending on the (n, k) parameters.

In a previous work we introduced the Generalized LDPCStaircase codes for the Binary Erasure Channel, based on LDPC-Staircase codes and Reed Solomon as component codes. In this paper we perform an asymptotic analysis, in terms of EXtrinsic Information Transfer functions and we derive an upper bound of the ML decoding threshold based on the area theorem. We use this analysis to study the impact of the internal LDPC-Staircase code rate on the performance, and show that the proposed Generalized LDPC-Staircase codes closely approach the channel capacity, with only a small number (E = 2, 3) of extra-repair symbols per check node.

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