Research On The Constructions And Applications Of Zero-difference Balanced Functions

As a new type of combinatorial function,the zero-difference balanced function not only closely related to the difference set,the perfect nonlinear function and the permutation polynomial,but also is widely used in the fields of cryptography,coding theory and communication.Therefore,it is of great significance to construct zero-difference balanced functions with new parameters.In this thesis,we constructed several classes of zero-difference balanced func-tions with new parameters based on the Zeng-Cai-Tang generalized cyclotomic class-es,based on which optimal constant composition codes,optimal and perfect differ-ence systems of sets and optimal frequency-hopping sequences can be constructed.In detail,following topics were addressed including the construction of a new class of zero-difference balanced functions constructed from two classes of known zero-difference balanced functions,the construction of a new class of near zero-difference balanced functions was constructed by improving the construction of Cai,and the construction of two classes of generalized zero-difference balanced functions were constructed with new parameters by using Zeng-Cai-Tang generalized cyclotomic classes modulo 2pm.Our main research works axe summarized as follows.1.Using the Zeng-Cai-Tang generalized cyclotomic classes,we present a new con-struction of zero-difference balanced functions with the parameters((2e-1)n,(2e-1)n-1/e+1,e-1).Different from known methods,the selection and partitionof the support sets are different,and the new construction is also satisfied in the noncyclic case.At the same time,we analyse the applications of the new construction,and give optimal constant composition codes,optimal and perfect difference systems of sets and optimal frequency-hopping sequences.2.By improving the construction of Cai,we give a new class of near zero-difference balanced functions with new parameters(en,en+n-1/e+1,{e-2,e-1}).More-over,we construct optimal constant composition codes and optimal frequency-hopping sequences with new parameters from the construction.3.Based on Zeng-Cai-Tang generalized cyclotomic classes modulo 2pm,we con-struct two classes generalized zero-difference balanced functions,which the pa-rameters are(2pm,2pm-2/e+2,{2e-2,0})and(2pm,pm-1/e+1,{4e-2,2,0})and are new.