Improved Successive Cancellation Decoding of Polar Codes

As improved versions of successive cancellation (SC) decoding algorithm, successive cancellation list (SCL) decoding and successive cancellation stack (SCS) decoding are used to improve the finite-length performance of polar codes. Unified descriptions of SC, SCL and SCS decoding algorithms are given as path searching procedures on the code tree of polar codes. Combining the ideas of SCL and SCS, a new decoding algorithm named successive cancellation hybrid (SCH) is proposed, which can achieve a better trade-off between computational complexity and space complexity. Further, to reduce the complexity, a pruning technique is proposed to avoid unnecessary path searching operations. Performance and complexity analysis based on simulations show that, with proper configurations, all the three improved successive cancellation (ISC) decoding algorithms can have a performance very close to that of maximum-likelihood (ML) decoding with acceptable complexity. Moreover, with the help of the proposed pruning technique, the complexities of ISC decoders can be very close to that of SC decoder in the moderate and high signal-to-noise ratio (SNR) regime.Comment: This paper is modified and submitted to IEEE Transactions on Communication

An Extension of the Well-Posedness Concept for Fractional Differential Equations of Caputo's Type

It is well known that, under standard assumptions, initial value problems for fractional ordinary differential equations involving Caputo-type derivatives are well posed in the sense that a unique solution exists and that this solution continuously depends on the given function, the initial value and the order of the derivative. Here we extend this well-posedness concept to the extent that we also allow the location of the starting point of the differential operator to be changed, and we prove that the solution depends on this parameter in a continuous way too if the usual assumptions are satisfied. Similarly, the solution to the corresponding terminal value problems depends on the location of the starting point and of the terminal point in a continuous way too.Comment: 11 page

Maximal Area Triangles in a Convex Polygon

The widely known linear time algorithm for computing the maximum area triangle in a convex polygon was found incorrect recently by Keikha et. al.(arXiv:1705.11035). We present an alternative algorithm in this paper. Comparing to the only previously known correct solution, ours is much simpler and more efficient. More importantly, our new approach is powerful in solving related problems

Stationary Solutions of Neutral Stochastic Partial Differential Equations with Delays in the Highest-Order Derivatives

In this work, we shall consider the existence and uniqueness of stationary solutions to stochastic partial functional differential equations with additive noise in which a neutral type of delay is explicitly presented. We are especially concerned about those delays appearing in both spatial and temporal derivative terms in which the coefficient operator under spatial variables may take the same form as the infinitesimal generator of the equation. We establish the stationary property of the neutral system under investigation by focusing on distributed delays. In the end, an illustrative example is analysed to explain the theory in this work