Constructions Of Perfect And Pseudorandom Punctured Binary Array Pairs

Punctured binary array pair(PBAP)is a kind of discrete signal with good periodic correlation,which is widely used in radar system,cryptography,frame synchronization,mismatch filtering and other engineering fields.Over the years,many people have studied its construction method.In this paper,we will use the knowledge of finite Abel group and finite multiple sets to give a new construction method for the perfect punctured binary array pair and the pseudorandom punctured binary array pair.Firstly,according to the definition and related knowledge of array pair in coding theory,the author gives its corresponding in math,that is and further gives the corresponding(f,fp),punctured binary array pair.According to the coding theory,in order to make the punctured binary array pair to be the punctured binary array pair or the pseudorandom punctured binary array pair,we need to calculate its energy and autocorrelation function,and make it meet some conditions,so we use the convolution method to calculate the autocorrelation function of the array pair,that is,formula(2-1)in proposition 3.2.Generally speaking,the calculation of convolution is relatively complex,so we use the Fourier transform of finite Abel group to find a simple calculation method,that is,the formula(2-2)in proposition 3.2,which also provides a more efficient tool for later proving the feasibility of our construction method.Next,we give the basic condition of making(f,fp)the perfect punctured binary array pair,that is,lemma 3.2.Using this condition,we have come to our first important conclusion,that is,the theorem 3.1.In theorem 3.1,a new construction method of the perfect punctured binary array pair is given.In addition,we generalize the theorem 3.1 by giving some special restrictions,and obtain several infinite families of the perfect punctured binary array pair,that is,corollary 3.1 and corollary 3.2.At the same time,we also give some concrete examples to verifly our results.Finally,on the basis of the previous discussion,we give the construction method of pseudorandom punctured binary array pair in theorem 3.2,and extend the theorem 3.2 with the same idea as the generalization of theorem 3.1.