Optimization of Code Constructions of Binary Sequences Class on a Basis of Set-Theoretical Representation

Abstract

In this paper there are solved two related problems: the development of the method and correspondent algorithm of code binary sequences synthesis with property of minimal level of side lobes of their aperiodic autocorrelation functions. The method is based on discrete representation of autocorrelation function in form of equations system considered on a set of integer numbers, set-theoretical representation of components of the sequence, their integer-number transformations, mutual properties and relations. There are obtained the expressions defining the dependences of absolute value of the sequence elements on a sum of levels of side lobes of aperiodic and periodic correlation functions; there are defined the necessary conditions of existence binary code sequences ; there are developed the mathematical models of binary code sequences components transformations.

The efficiency criterion of each proposed algorithm is the relation of amount of all possible variants of binary code sequences of definite size l to amount of defined by proposed algorithms; the expressions for these quantities are obtained. The efficiency of proposed method and algorithms is verified by simulation results and it increases with increase of size l.