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.

The Number of Rational Points of a Family of Algebraic Varieties over Finite Fields

Let 𝔽q stand for the finite field of odd characteristic p with q elements (q = pn, n ∈ ℕ) and [Formula: see text] denote the set of all the nonzero elements of 𝔽q. Let m and t be positive integers. By using the Smith normal form of the exponent matrix, we obtain a formula for the number of rational points on the variety defined by the following system of equations over [Formula: see text] where the integers t > 0, r0 = 0 < r1 < r2 < ⋯ < rt, 1 ≤ n1 < n2 <, ⋯ < nt and 0 ≤ j ≤ t − 1, bk ∊ 𝔽q, ak,i ∊ [Formula: see text] (k = 1, …, m, i = 1, …, rt), and the exponent of each variable is a positive integer. Further, under some natural conditions, we arrive at an explicit formula for the number of 𝔽q-rational points on the above variety. It extends the results obtained previously by Wolfmann, Sun, Wang, Hong et al. Our result gives a partial answer to an open problem raised in [The number of rational points of a family of hypersurfaces over finite fields, J. Number Theory 156 (2015) 135–153].

On exponent matrices of tiled orders

We describe two methods to determine all generators of the additive semigroup of the non-negative exponent [Formula: see text]-matrices, and illustrate them finding all generating [Formula: see text]-exponent matrices. The generating [Formula: see text]-exponent matrices are found using a computer. We consider the Hasse diagram [Formula: see text] of the partially ordered set of non-negative matrices and prove that for an arbitrary non-negative exponent matrix [Formula: see text] there exists an oriented path in [Formula: see text] starting in some matrix unit and ending in [Formula: see text] which does not pass through any other exponent matrix. We also show that for any non-negative exponent matrix [Formula: see text] there exists a chain of non-negative exponent matrices [Formula: see text] such that [Formula: see text] is a [Formula: see text]-matrix, and each [Formula: see text] is obtained from [Formula: see text] by adding a [Formula: see text]-matrix.

Image Transmission Based on QC-LDPC Codes with Simple Recursive Encoding Form

Low encoding delay and complexity is very important for image transmission. This paper proposes a novel image transmission scheme with low encoding complexity. The proposed scheme is based on quasi-cyclic low density parity check (QC-LDPC) codes with a simple recursive encoding form (SREF QC-LDPC code) which results in low encoding complexity and delay. Constructing the SREF QC-LDPC codes in this scheme composes of two main steps, construction of the base matrix and the exponent matrix. We combine the differential evolution and protograph extrinsic information transfer (PEXIT) method to optimize the base matrix of QC-LDPC code. Consequently, the exponent matrix and the parity check matrix are constructed. Simulation results show that the proposed scheme based on SREF QC-LDPC code can provide a good tradeoff between the performance and complexity.

Numerical Algorithms for Solving Restricted Three Body Problem

A precise integration algorithm for solving the restricted three body problem was put forward based on precise integration method, which divided a large integration time into small intervals and only small value matrix participates in the iterative process during the computation of the exponent matrix. And another symplectic algorithm for solving non-separable Hamiltonian system constructed by flow complex was also introduced, which only had periodic variational energy. The results of both algorithms were compared with fourth Runge-Kutta algorithm and their performances and advantages were analyzed, showing the validities of these two algorithms.

EXPONENT MATRICES AND TILED ORDERS OVER DISCRETE VALUATION RINGS

Exponent matrices appear in the theory of tiled orders over a discrete valuation ring. Many properties of such an order and its quiver are fully determined by its exponent matrix. We prove that an arbitrary strongly connected simply laced quiver with a loop in every vertex is realized as the quiver of a reduced exponent matrix. The relations between exponent matrices and finite posets, Markov chains, and doubly stochastic matrices are discussed.

A matrix exponential form for hitting probabilities and its application to a Markov-modulated fluid queue with downward jumps

We consider a fluid queue with downward jumps, where the fluid flow rate and the downward jumps are controlled by a background Markov chain with a finite state space. We show that the stationary distribution of a buffer content has a matrix exponential form, and identify the exponent matrix. We derive these results using time-reversed arguments and the background state distribution at the hitting time concerning the corresponding fluid flow with upward jumps. This distribution was recently studied for a fluid queue with upward jumps under a stability condition. We give an alternative proof for this result using the rate conservation law. This proof not only simplifies the proof, but also explains an underlying Markov structure and enables us to study more complex cases such that the fluid flow has jumps subject to a nondecreasing Lévy process, a Brownian component, and countably many background states.

A matrix exponential form for hitting probabilities and its application to a Markov-modulated fluid queue with downward jumps

We consider a fluid queue with downward jumps, where the fluid flow rate and the downward jumps are controlled by a background Markov chain with a finite state space. We show that the stationary distribution of a buffer content has a matrix exponential form, and identify the exponent matrix. We derive these results using time-reversed arguments and the background state distribution at the hitting time concerning the corresponding fluid flow with upward jumps. This distribution was recently studied for a fluid queue with upward jumps under a stability condition. We give an alternative proof for this result using the rate conservation law. This proof not only simplifies the proof, but also explains an underlying Markov structure and enables us to study more complex cases such that the fluid flow has jumps subject to a nondecreasing Lévy process, a Brownian component, and countably many background states.