A new class of permutation trinomials constructed from Niho exponents

Permutation polynomials over finite fields are an interesting subject due to their important applications in the areas of mathematics and engineering. In this paper we investigate the trinomial $f(x)=x^{(p-1)q+1}+x^{pq}-x^{q+(p-1)}$ over the finite field $\mathbb{F}_{q^2}$, where $p$ is an odd prime and $q=p^k$ with $k$ being a positive integer. It is shown that when $p=3$ or $5$, $f(x)$ is a permutation trinomial of $\mathbb{F}_{q^2}$ if and only if $k$ is even. This property is also true for more general class of polynomials $g(x)=x^{(q+1)l+(p-1)q+1}+x^{(q+1)l+pq}-x^{(q+1)l+q+(p-1)}$, where $l$ is a nonnegative integer and $\gcd(2l+p,q-1)=1$. Moreover, we also show that for $p=5$ the permutation trinomials $f(x)$ proposed here are new in the sense that they are not multiplicative equivalent to previously known ones of similar form.Comment: 17 pages, three table

Deterministic Construction of Binary Measurement Matrices with Various Sizes

We introduce a general framework to deterministically construct binary measurement matrices for compressed sensing. The proposed matrices are composed of (circulant) permutation submatrix blocks and zero submatrix blocks, thus making their hardware realization convenient and easy. Firstly, using the famous Johnson bound for binary constant weight codes, we derive a new lower bound for the coherence of binary matrices with uniform column weights. Afterwards, a large class of binary base matrices with coherence asymptotically achieving this new bound are presented. Finally, by choosing proper rows and columns from these base matrices, we construct the desired measurement matrices with various sizes and they show empirically comparable performance to that of the corresponding Gaussian matricesComment: 5 pages, 3 figure

Alternating direction algorithms for ℓ0\ell_0 regularization in compressed sensing

In this paper we propose three iterative greedy algorithms for compressed sensing, called \emph{iterative alternating direction} (IAD), \emph{normalized iterative alternating direction} (NIAD) and \emph{alternating direction pursuit} (ADP), which stem from the iteration steps of alternating direction method of multiplier (ADMM) for $\ell_0$-regularized least squares ($\ell_0$-LS) and can be considered as the alternating direction versions of the well-known iterative hard thresholding (IHT), normalized iterative hard thresholding (NIHT) and hard thresholding pursuit (HTP) respectively. Firstly, relative to the general iteration steps of ADMM, the proposed algorithms have no splitting or dual variables in iterations and thus the dependence of the current approximation on past iterations is direct. Secondly, provable theoretical guarantees are provided in terms of restricted isometry property, which is the first theoretical guarantee of ADMM for $\ell_0$-LS to the best of our knowledge. Finally, they outperform the corresponding IHT, NIHT and HTP greatly when reconstructing both constant amplitude signals with random signs (CARS signals) and Gaussian signals.Comment: 16 pages, 1 figur

Nonextensive information theoretical machine

In this paper, we propose a new discriminative model named \emph{nonextensive information theoretical machine (NITM)} based on nonextensive generalization of Shannon information theory. In NITM, weight parameters are treated as random variables. Tsallis divergence is used to regularize the distribution of weight parameters and maximum unnormalized Tsallis entropy distribution is used to evaluate fitting effect. On the one hand, it is showed that some well-known margin-based loss functions such as $\ell_{0/1}$ loss, hinge loss, squared hinge loss and exponential loss can be unified by unnormalized Tsallis entropy. On the other hand, Gaussian prior regularization is generalized to Student-t prior regularization with similar computational complexity. The model can be solved efficiently by gradient-based convex optimization and its performance is illustrated on standard datasets

Bayesian linear regression with Student-t assumptions

As an automatic method of determining model complexity using the training data alone, Bayesian linear regression provides us a principled way to select hyperparameters. But one often needs approximation inference if distribution assumption is beyond Gaussian distribution. In this paper, we propose a Bayesian linear regression model with Student-t assumptions (BLRS), which can be inferred exactly. In this framework, both conjugate prior and expectation maximization (EM) algorithm are generalized. Meanwhile, we prove that the maximum likelihood solution is equivalent to the standard Bayesian linear regression with Gaussian assumptions (BLRG). The $q$-EM algorithm for BLRS is nearly identical to the EM algorithm for BLRG. It is showed that $q$-EM for BLRS can converge faster than EM for BLRG for the task of predicting online news popularity

Johnson Type Bounds on Constant Dimension Codes

Very recently, an operator channel was defined by Koetter and Kschischang when they studied random network coding. They also introduced constant dimension codes and demonstrated that these codes can be employed to correct errors and/or erasures over the operator channel. Constant dimension codes are equivalent to the so-called linear authentication codes introduced by Wang, Xing and Safavi-Naini when constructing distributed authentication systems in 2003. In this paper, we study constant dimension codes. It is shown that Steiner structures are optimal constant dimension codes achieving the Wang-Xing-Safavi-Naini bound. Furthermore, we show that constant dimension codes achieve the Wang-Xing-Safavi-Naini bound if and only if they are certain Steiner structures. Then, we derive two Johnson type upper bounds, say I and II, on constant dimension codes. The Johnson type bound II slightly improves on the Wang-Xing-Safavi-Naini bound. Finally, we point out that a family of known Steiner structures is actually a family of optimal constant dimension codes achieving both the Johnson type bounds I and II.Comment: 12 pages, submitted to Designs, Codes and Cryptograph

Minimum Pseudo-Weight and Minimum Pseudo-Codewords of LDPC Codes

In this correspondence, we study the minimum pseudo-weight and minimum pseudo-codewords of low-density parity-check (LDPC) codes under linear programming (LP) decoding. First, we show that the lower bound of Kelly, Sridhara, Xu and Rosenthal on the pseudo-weight of a pseudo-codeword of an LDPC code with girth greater than 4 is tight if and only if this pseudo-codeword is a real multiple of a codeword. Then, we show that the lower bound of Kashyap and Vardy on the stopping distance of an LDPC code is also a lower bound on the pseudo-weight of a pseudo-codeword of this LDPC code with girth 4, and this lower bound is tight if and only if this pseudo-codeword is a real multiple of a codeword. Using these results we further show that for some LDPC codes, there are no other minimum pseudo-codewords except the real multiples of minimum codewords. This means that the LP decoding for these LDPC codes is asymptotically optimal in the sense that the ratio of the probabilities of decoding errors of LP decoding and maximum-likelihood decoding approaches to 1 as the signal-to-noise ratio leads to infinity. Finally, some LDPC codes are listed to illustrate these results.Comment: 17 pages, 1 figur

Three-flavor Nambu--Jona-Lasinio model at finite isospin chemical potential

QCD at finite isospin chemical potential $\mu_{\text I}$ possesses a positively definite fermion determinant and the lattice simulation can be successfully performed. While the two-flavor effective models may be sufficient to describe the phenomenon of pion condensation, it is interesting to study the roles of the strangeness degree of freedom and the U$_{\rm A}(1)$ anomaly. In this paper, we present a systematic study of the three-flavor Nambu--Jona-Lasinio model with a Kobayashi-Maskawa-'t Hooft (KMT) term that mimics the U$_{\rm A}(1)$ anomaly at finite isospin chemical potential. In the mean-field approximation, the model predicts a phase transition from the vacuum to the pion superfluid phase, which takes place at $\mu_{\rm I}$ equal to the pion mass $m_\pi$. Due to the U$_{\rm A}(1)$ anomaly, the strangeness degree of freedom couples to the light quark degrees of freedom and the strange quark effective mass depends on the pion condensate. However, the strange quark condensate and the strange quark effective mass change slightly in the pion superfluid phase, which verifies the validity of the two-flavor models. The effective four-fermion interaction of the Kobayashi-Maskawa-'t Hooft term in the presence of the pion condensation is constructed. Due to the U$_{\rm A}(1)$ anomaly, the pion condensation generally induces scalar-pseudoscalar interaction. The Bethe-Salpeter equation for the mesonic excitations is established and the meson mass spectra are obtained at finite isospin chemical potential and temperature. Finally, the general expression for the topological susceptibility $\chi$ at finite isospin chemical potential $\mu_{\rm I}$ is derived. In contrast to the finite temperature effect which suppresses $\chi$, the isospin density effect leads to an enhancement of $\chi$.Comment: Version punlished in PR

Topological Susceptibility in Three-Flavor Quark Meson Model at Finite Temperature

We study $U_A(1)$ symmetry and its relation to chiral symmetry at finite temperature through the application of functional renormalization group to the $SU(3)$ quark meson model. Very different from the mass gap and mixing angel between $\eta$ and $\eta'$ mesons which are defined at mean field level and behavior like the chiral condensates, the topological susceptibility includes a fluctuations induced part which becomes dominant at high temperature. As a result, the $U_A(1)$ symmetry is still considerably broken in the chiral symmetry restoration phase.Comment: 9 pages, 5 figure

Sparse signal recovery by ℓq\ell_q minimization under restricted isometry property

In the context of compressed sensing, the nonconvex $\ell_q$ minimization with $0<q<1$ has been studied in recent years. In this paper, by generalizing the sharp bound for $\ell_1$ minimization of Cai and Zhang, we show that the condition $\delta_{(s^q+1)k}<\dfrac{1}{\sqrt{s^{q-2}+1}}$ in terms of \emph{restricted isometry constant (RIC)} can guarantee the exact recovery of $k$-sparse signals in noiseless case and the stable recovery of approximately $k$-sparse signals in noisy case by $\ell_q$ minimization. This result is more general than the sharp bound for $\ell_1$ minimization when the order of RIC is greater than $2k$ and illustrates the fact that a better approximation to $\ell_0$ minimization is provided by $\ell_q$ minimization than that provided by $\ell_1$ minimization