In this letter, we address the problem of decoding nonbinary low-density parity-check (LDPC) codes over finite fields GF(q), with reasonable complexity and good performance. In the first part of the letter, we recall the original belief propagation (BP) decoding algorithm and its Fourier domain implementation. We show that the use of tensor notations for the messages is very convenient for the algorithm description and understanding. In the second part of the letter, we introduce a simplified decoder which is inspired by the min-sum decoder for binary LDPC codes. We called this decoder extended min-sum (EMS). We show that it is possible to greatly reduce the computational complexity of the check-node processing by computing approximate reliability measures with a limited number of values in a message. By choosing appropriate correction factors or offsets, we show that the EMS decoder performance is quite good, and in some cases better than the regular BP decoder. The optimal values of the factor and offset correction are obtained asymptotically with simulated density evolution. Our simulations on ultra-sparse codes over very-high-order fields show that nonbinary LDPC codes are promising for applications which require low frame-error rates for small or moderate codeword lengths. The EMS decoder is a good candidate for practical hardware implementations of such codes

Decoding Algorithms for Nonbinary LDPC Codes Over GF $(q)$

In this paper, a method to design regular (2, dc)- LDPC codes over GF(q) with both good waterfall and error floor properties is presented, based on the algebraic properties of their binary image. First, the algebraic properties of rows of the parity check matrix H associated with a code are characterized and optimized to improve the waterfall. Then the algebraic properties of cycles and stopping sets associated with the underlying Tanner graph are studied and linked to the global binary minimum distance of the code. Finally, simulations are presented to illustrate the excellent performance of the designed codes.

/pdf/design-of-regular-2-d-sub-c-ldpc-codes-over-gf-q-using-their-4hlcc048qj.pdf

Design of regular (2,d/sub c/)-LDPC codes over GF(q) using their binary images

When quantum key distribution is composed with other secure protocols the overall security has to be guaranteed, which adds further security requirements. Here, the authors demonstrate continuous-variable quantum key distribution with composable security and one-sided-device independence.

/pdf/implementation-of-continuous-variable-quantum-key-izqdalj2os.pdf

Implementation of continuous-variable quantum key distribution with composable and one-sided-device-independent security against coherent attacks

In this paper, we propose a new implementation of the Extended Min-Sum (EMS) decoder for non-binary LDPC codes. A particularity of the new algorithm is that it takes into accounts the memory problem of the non-binary LDPC decoders, together with a significant complexity reduction per decoding iteration. The key feature of our decoder is to truncate the vector messages of the decoder to a limited number nm of values in order to reduce the memory requirements. Using the truncated messages, we propose an efficient implementation of the EMS decoder which reduces the order of complexity to ?(nm log2 nm). This complexity starts to be reasonable enough to compete with binary decoders. The performance of the low complexity algorithm with proper compensation is quite good with respect to the important complexity reduction, which is shown both with a simulated density evolution approach and actual simulations.

/pdf/low-complexity-decoding-for-non-binary-ldpc-codes-in-high-dtnw9lhd6f.pdf

Low-complexity decoding for non-binary LDPC codes in high order fields

Low-density parity-check (LDPC) codes constructed over the Galois field GF(q), which are also called nonbinary LDPC codes, are an extension of binary LDPC codes with significantly better performance. Although various kinds of low-complexity quasi-optimal iterative decoding algorithms have been proposed, the VLSI implementation of nonbinary LDPC decoders has rarely been discussed due to their hardware unfriendly properties. In this brief, an efficient selective computation algorithm, which totally avoids the sorting process, is proposed for Min-Max decoding. In addition, an efficient VLSI architecture for a nonbinary Min-Max decoder is presented. The synthesis results are given to demonstrate the efficiency of the proposed techniques.

An Efficient VLSI Architecture for Nonbinary LDPC Decoders

We present a modification of belief propagation that enables us to decode LDPC codes defined on high order Galois fields with a complexity that scales as p log/sub 2/ (p), p being the field order. With this low complexity algorithm, we are able to decode GF(2/sup q/) LDPC codes up to a field order value of 256. We show by simulation that ultra-sparse regular LDPC codes in GF(64) and GF(256) exhibit very good performance.

Fast decoding algorithm for LDPC over GF(2/sup q/)

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Fast decoding algorithm for LDPC over GF(2/sup q/)