Encoder Lipschitz Integers: The Lipschitz integers that have the ”division with small remainder” property

The residue class set of a Lipschitz integer is constructed by modulo function with primitive Lipschitz integer whose norm is a prime integer, i.e. prime Lipschitz integer. In this study, we consider primitive Lipschitz integer whose norm is both a prime integer and not a prime integer. If the norm of each element of the residue class set of a Lipschitz integer is less than the norm of the primitive Lipschitz integer used to construct the residue class set of the Lipschitz integer, then, the Euclid division algorithm works for this primitive Lipschitz integer. The Euclid division algorithm always works for prime Lipschitz integers. In other words, the prime Lipschitz integers have the ”division with small remainder” property. However, this property is ignored in some studies that have a constructed Lipschitz residue class set that lies on primitive Lipschitz integers whose norm is not a prime integer. In this study, we solve this problem by defining Lipschitz integers that have the ”division with small remainder” property, namely, encoder Lipschitz integers set. Therefore, we can define appropriate metrics for codes over Lipschitz integers. Also, we investigate the performances of Lipschitz signal constellations (the left residue class set) obtained by modulo function with Lipschitz integers, which have the ”division with small remainder” property, over the additive white Gaussian noise (AWGN) channel by agency of the constellation figure of merit (CFM), average energy, and signal-to-noise ratio (SNR).

Encoder Lipschitz Integers: The Lipschitz integers that have the ”division with small remainder” property

The residue class set of a Lipschitz integer is constructed by modulo function with primitive Lipschitz integer whose norm is a prime integer, i.e. prime Lipschitz integer. In this study, we consider primitive Lipschitz integer whose norm is both a prime integer and not a prime integer. If the norm of each element of the residue class set of a Lipschitz integer is less than the norm of the primitive Lipschitz integer used to construct the residue class set of the Lipschitz integer, then, the Euclid division algorithm works for this primitive Lipschitz integer. The Euclid division algorithm always works for prime Lipschitz integers. In other words, the prime Lipschitz integers have the ”division with small remainder” property. However, this property is ignored in some studies that have a constructed Lipschitz residue class set that lies on primitive Lipschitz integers whose norm is not a prime integer. In this study, we solve this problem by defining Lipschitz integers that have the ”division with small remainder” property, namely, encoder Lipschitz integers set. Therefore, we can define appropriate metrics for codes over Lipschitz integers. Also, we investigate the performances of Lipschitz signal constellations (the left residue class set) obtained by modulo function with Lipschitz integers, which have the ”division with small remainder” property, over the additive white Gaussian noise (AWGN) channel by agency of the constellation figure of merit (CFM), average energy, and signal-to-noise ratio (SNR).

Irregular Vector Turbo Codes with Low Complexity

The term Block Turbo Code typically refers to the iterative decoding of a serially concatenated two-dimensional systematic block code. This paper introduces a Vector Turbo Code that is irregular but with code rates comparable to those of a Block Turbo Code (BTC) when the Bahl Cocke Jelinek Raviv (BCJR) algorithm is used. In Block Turbo Codes, the horizontal (or vertical) blocks are encoded first and the vertical (or horizontal) blocks second. The irregular Vector Turbo Code (iVTC) uses information bits that participate in varying numbers of trellis sections, which are organized into blocks that are encoded horizontally (or vertical) without vertical (or horizontal) encoding. The decoding requires only one soft-input soft-output (SISO) decoder. In general, a reduction in complexity, in comparison to a Block Turbo Code was achieved for the same very low probability of bit error (10−5 ). Performance in the AWGN channel shows that iVTC is capable of achieving a significant coding gain of 1.28 dB for a 64QAM modulation scheme, at a bit error rate (BER) of 10−5over its corresponding Block Turbo Code. Simulation results also show that some of these codes perform within 0.49 dB of capacity for binary transmission over an AWGN channel.

Encoder Lipschitz Integers: The Lipschitz integers that have the ”division with small division” property

The residue class set of a Lipschitz integer is constructed by modulo function with primitive Lipschitz integer whose norm is a prime integer, i.e. prime Lipschitz integer. In this study, we consider primitive Lipschitz integer whose norm is both a prime integer and not a prime integer. If the norm of each element of the residue class set of a Lipschitz integer is less than the norm of the primitive Lipschitz integer used to construct the residue class set of the Lipschitz integer, then, the Euclid division algorithm works for this primitive Lipschitz integer. The Euclid division algorithm always works for prime Lipschitz integers. In other words, the prime Lipschitz integers have the ”division with small remainder” property. However, this property is ignored in some studies that have a constructed Lipschitz residue class set that lies on primitive Lipschitz integers whose norm is not a prime integer. In this study, we solve this problem by defining Lipschitz integers that have the ”division with small remainder” property, namely, encoder Lipschitz integers set. Therefore, we can define appropriate metrics for codes over Lipschitz integers. Also, we investigate the performances of Lipschitz signal constellations (the left residue class set) obtained by modulo function with Lipschitz integers, which have the ”division with small remainder” property, over the additive white Gaussian noise (AWGN) channel by agency of the constellation figure of merit (CFM), average energy, and signal-to-noise ratio (SNR).

Design of Nonbinary LDPC Cycle Codes with Large Girth from Circulants and Finite Fields

In this paper, we study a class of nonbinary LDPC (NBLDPC) codes whose parity-check matrices have column weight 2, called NBLDPC cycle codes. We propose a design framework of 2 , ρ -regular binary quasi-cyclic (QC) LDPC codes and then construct NBLDPC cycle codes of large girth based on circulants and finite fields by randomly choosing the nonzero field elements in their parity-check matrices. For enlarging the girth values, our approach is twofold. First, we give an exhaustive search of circulants with column/row weight ρ and design a masking matrix with good cycle distribution based on the edge-node relation in undirected graphs. Second, according to the designed masking matrix, we construct the exponent matrix based on finite fields. The iterative decoding performances of the constructed codes on the additive white Gaussian noise (AWGN) channel are finally provided.

BER Performance Analysis of Non-Coherent Q-Ary Pulse Position Modulation Receivers on AWGN Channel

This paper introduces the structure of a Q-ary pulse position modulation (PPM) signal and presents a noncoherent suboptimal receiver and a noncoherent optimal receiver. Aiming at addressing the lack of an accurate theoretical formula of the bit error rate (BER) of a Q-ary PPM receiver in the additive white Gaussian noise (AWGN) channel in the existing literature, the theoretical formulas of the BER of a noncoherent suboptimal receiver and noncoherent optimal receiver are derived, respectively. The simulation results verify the correctness of the theoretical formulas. The theoretical formulas can be applied to a Q-ary PPM system including binary PPM. In addition, the analysis shows that the larger the Q, the better the error performance of the receiver and that the error performance of the optimal receiver is about 2 dB better than that of the suboptimal receiver. The relationship between the threshold coefficient of the suboptimal receiver and the error performance is also given.