Analysis And Design Of The Sequences With Low Cross-Correlation

m-Sequence has been widely employed in many fields,its auto-correlation function takes exactly two values.However,there is still not a perfect solution to calculate its cross-correlation function.In addition,sequences with good correlation properties have important application in stream cipher and communication systems,m-sequences and their decimated sequences are widely used to design families of sequences with low-correlation.Therefore,the study of them has important theoretical significance and practical value.In this paper,we study the cross-correlation properties of m-sequences,and construct a new family of sequences with m-sequences:1)When the m-sequence{s（t）}and its d-decimated sequence{s（dt）} are expressed by the trace function Tr（·）,the values of cross-correlation between {s（t）} and{s（dt）} can be calculated through the values of the exponential sum on the finite field,moreover we can use the quadratic theory to get the cross-correlation,and then calculate the sum of the power of the cross-correlation,combining it with the cross-correlation values and the frequencies of the values,we can list the equations.Solving these equations,we can get the distribution of the cross correlation values.We proved that when the decimation factor takes d=[（pⁿ-1）（p^k+3）+2（^pk+1）[/4,the cross-correlation function Cd（t）between the p-ary m-sequence and its d-decimated sequence has three values,andCd（t）∈{-1,-1+p（n+e）/2,-1-p^（n-e）/2,-1-p^（n-e）/2},we also derive the distribution of these cross-correlation values.2){s（d₁t＋i）} and{s（d₂t）} are two decimated sequnces of m-sequence,whered₁=2 and d₂ = pm+1 We list the values of the cross-correlation function between {s（d₁t+i）} and{s（d₂t）}.When pm ≡1（mod4）,we derive the distribution of the values of the cross-correlation function between {s（d₁t +1）} and {s（d₂t）}.3)By shifting and adding the decimated sequences of m-sequence,we construct a class of large family of sequences with low correlation:The new sequences we constructed has large family size 28m and eleven-valued cross-correlations,and the maximum cross-correlation is 9· 2^m-1.