Cross-correlation Function Between An M-Sequence And Its Niho Decimation Sequences With Short Period

Let p be an arbitrary prime,mis a positive integer,letd = 2pm-1be the Niho decimation ovel Fp2m.Let{s(t)}denote a p-ary m-sequence of period p 2m-1.When pm(?)2(mod 3),we have gcd(d,p 2m-1)=1 and there is only one d-decimation sequence {s(dt)} of {s(t)} with the same period.In this case,the correlation distribution between{s(t}and{s(dt)}has been determined.However,when pm ? 2(mod 3),this situation is different and we havegcd(d,p 2m-1)=3.Then,there are three distinct d-decimation sequencests(dt+l)}of {s(t)}with short period p 2m-1/3,l = 0,1,2.The correlation distribution betweel{s(t)} and each d-decimation sequence{s(dt+l)} has not been determined yet.Determining the correlation distribution between {s(t)} and{s(dt+l)} for eachl ? {0,1,2} is our aim in this paper.More precisely,for each d-decimation sequence {s(dt+l)},we proved that the cross-correlation function between {s(t)} and {s(dt+l)} takes four values and the corresponding correlation distribution is completely determined.Our results extend the results of Niho and Helleseth about the Niho decimationd=2pm-1 in the case where gcd(d,p 2m-1)=1.