Two-Stage Constructions for the Rate-Compatible Shortened Polar Codes

In this paper,we propose the two-stage constructions for the rate-compatible shortened polar(RCSP)codes.For the Stage-I construction,the shortening pattern and the frozen bit are jointly designed to make the shortened bits be completely known by the decoder.Besides,a distance-greedy algorithm is presented to improve the minimum Hamming distance of the codes.To design the remaining Stage-II frozen bits,three different construction algorithms are further presented,called the Reed-Muller(RM)construction,the Gaussian Approximation(GA)construction,and the RM-GA construction.Then we give the row weight distribution numerical results of the generator matrix after the Stage-I and Stage-II constructions,which shows that the proposed constructions can efficiently increase the minimum Hamming distance.Simulation results show that the proposed RCSP codes have excellent frame error rate(FER)performances at different code lengths and code rates.More specifically,the RM-GA construction performs best and can achieve at most 0.8 dB gain compared to the Wang14 and the quasi-uniform puncturing(QUP)schemes.The RM construction is designed completely by the distance-constraint without channel evaluation thus has the simplest structure.Interestingly,it still has better FER performance than the existing shortening/puncturing schemes,especially at high signal noise ratio(SNR)region.

Asymptotic bound on binary self-orthogonal codes

We present two constructions for binary self-orthogonal codes. It turns out that our constructions yield a constructive bound on binary self-orthogonal codes. In particular, when the in-formation rate R = 1/2, by our constructive lower bound, the relative minimum distance δ≈ 0.0595 (for GV bound, δ≈ 0.110). Moreover, we have proved that the binary self-orthogonal codes asymptotically achieve the Gilbert-Varshamov bound.

On the Value Distributions of Walsh Spectrum of Two Classes of Plateaued Functions

Based on the properties of trace functions and quadratic forms, this paper presents value distributions of Walsh spectrum of the Plateaued functions of the form Tr（R（x）） with n=3r or 4r variables, where r 〉 1 is an odd integer. Our results can be used to determine the numbers of non-zero Walsh spectrum values and the nonlinearities of these functions, and estimate their resiliency orders. Especially, the value distributions can be used to deduce the tight lower bounds of the second order nonlinearity of two classes of Boolean functions. It is demonstrated that our bounds are better than the previously obtained bounds.

Analysis of affinely equivalent Boolean functions

By some basic transforms and invariant theory, we give two results： 1） an algorithm, which can be used to judge if two Boolean functions are affinely equivalent and to obtain the equivalence relationship if they are equivalent. This is useful in studying Boolean functions and in engineering. For example, we classify all 8-variable homogeneous bent functions of degree 3 into two classes; 2） Reed-Muller codes R（4,6）/R（1,6）, R（3,7）/R（1,7） are classified efficiently.