There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

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.

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

国際会議参加報告:2014 IEEE International Symposium on Information Theory

A method and system for storage and retrieval of blockchains with Galois Fields. One or more blocks for a blockchain are securely stored and retrieved with a modified Galois Fields on a cloud or peer-to-peer (P2P) communications network. The modified Galois Field provides at least additional layers for security and privacy for blockchains. The blocks and blockchains are securely stored and retrieved for cryptocurrency transactions including, but not limited to, BITCOIN transactions and other cryptocurrency transactions.

Method and system for storage and retrieval of blockchain blocks using galois fields

Unmanned aerial vehicles (UAVs) have stroke great interested both by the academic community and the industrial community due to their diverse military applications and civilian applications. Furthermore, UAVs are also envisioned to be part of future airspace traffic. The application functions delivery relies on information exchange among UAVs as well as between UAVs and ground stations (GSs), which further closely depends on aeronautical channels. However, there is a paucity of comprehensive surveys on aeronautical channel modeling in line with the specific aeronautical characteristics and scenarios. To fill this gap, this paper focuses on reviewing the air-to-ground (A2G), ground-to-ground (G2G), and air-to-air (A2A) channel measurements and modeling for UAV communications and aeronautical communications under various scenarios. We also provide the design guideline for managing the link budget of UAV communications taking account of link losses and channel fading effects. Moreover, we also analyze the receive/transmit diversity gain and spatial multiplexing gain achieved by multiple-antenna-aided UAV communications. Finally, we discuss the remaining challenge and open issues for the future development of UAV communication channel modeling.

/pdf/a-comprehensive-survey-on-uav-communication-channel-modeling-41snqetr75.pdf

A Comprehensive Survey on UAV Communication Channel Modeling

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. Index Terms—Channel coding, error correction coding, non- binary LDPC codes, iterative decoding, binary image.

Design of Regular (2,d c )-LDPC Codes over GF(q )U sing Their Binary Images

The progressive edge-growth (PEG) construction is a well known algorithm for constructing bipartite graphs with good girth properties. In this letter, we propose some improvements in the PEG algorithm which greatly improve the girth properties of the resulting graphs: given a graph size, they increase the girth g achievable by the algorithm, and when the girth cannot be increased, our modified algorithm minimizes the number of cycles of length g. As a main illustration, we focus on regular column-weight two graphs (dv = 2), although our algorithm can be applied to any graph connectivity. The class of dv = 2 graphs is often used for non-binary low density parity check codes that can be seen as monopartite graphs: for a given target girth gt, this new instance of the PEG algorithm allows to construct cages, i.e. graphs with the minimal size such that a graph of girth gt exists, which is the best result one might hope for.

/pdf/design-of-cages-with-a-randomized-progressive-edge-growth-18teqnk6eg.pdf

Design of cages with a randomized progressive edge-growth algorithm

We propose nonbinary LDPC codes concatenated with multiplicative repetition codes. By multiplicatively repeating the (2,3)-regular nonbinary LDPC mother code of rate 1/3, we construct rate-compatible codes of lower rates 1/6, 1/9, 1/12,.... Surprisingly, such simple low-rate nonbinary LDPC codes outperform the best low-rate binary LDPC codes so far. Moreover, we propose the decoding algorithm for the proposed codes, which can be decoded with almost the same computational complexity as that of the mother code.

/pdf/multiplicatively-repeated-nonbinary-ldpc-codes-1vj5e92xn1.pdf

Multiplicatively Repeated Nonbinary LDPC Codes

We are interested in the analysis and optimization of Raptor codes under a joint decoding framework, that is, when the precode and the fountain code exchange soft information iteratively. We develop an analytical asymptotic convergence analysis of the joint decoder, derive an optimization method for the design of efficient output degree distributions, and show that the new optimized distributions outperform the existing ones, both at long and moderate lengths. We also show that jointly decoded Raptor codes are robust to channel variation: they perform reasonably well over a wide range of channel capacities. This robustness property was already known for the erasure channel but not for the Gaussian channel. Finally, we discuss some finite length code design issues. Contrary to what is commonly believed, we show by simulations that using a relatively low rate for the precode (Rp ≃ 0.9), we can improve greatly the error floor performance of the Raptor code.

/pdf/jointly-decoded-raptor-codes-analysis-and-design-for-the-4ln1pfej0y.pdf

Charly Poulliat

Papers

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

Design of Regular (2,d c )-LDPC Codes over GF(q )U sing Their Binary Images

Design of cages with a randomized progressive edge-growth algorithm

Multiplicatively Repeated Nonbinary LDPC Codes

Jointly decoded raptor codes: analysis and design for the BIAWGN channel