The Correlation Distribution Of Multivariate Sequences On Finite Fields

Exponential sums on finite fields are important research objects in both number theory and communication theory. In CDMA&OFDM communication systems and stream ciphers, we need sequences with low auto-and cross correlations. It is equivalent to say that the corresponding exponential sum has small absolute value. We try to investigate the exponential sums with explicit expressions. In what follows, the cross correlation distribution of the sequences can be determined. Furthermore, the weight distribution of the associated cyclic code can also be determined. In this way, we may find periodic seqeunce family with low auto-and cross correlations.The research on S(α,β) can be exploited as:1) Value distribution of S(α,β) is used for the calculation of the weight distribution of the corresponding cyclic codes.2) When the magnitude of S(α,β) is small enough, it can be used to construct a new m-sequence family with low correlation property.For a cyclic code C with longth n over Fq, let Ai be the number of codewords in C with Hamming weight i. The weight distribution {A0,A1,…,An} is an important research object in coding theory. If C is cyclic, the weight of each codeword can be expressed by exponential sums so that the weight distribution of C can be determined if the corresponding exponential sums can be calculated explicitly.The cross correlation of m-sequences is an important research object in stream ciphers. Let p be an odd prime and Fpm the finite field with pm elements. The cross-correlation function corresponds to the exponential sum given as where trppm(x) is the trace function from Fpm to Fp,χ(x)=ξplrppm(x) is the canonical additive character of Fpm and ξp=e2π(?)-1/p is a primitive p-th root of unity. Let p be an odd prime, n=2m,and,m/gcd(k,m) be odd, k/gcd(k,m) be even.In this dissertation,we study the cross correlation between a p-ary m-sequence{St) of period pn-1and its decimated sequence{Sdt}where d satisfies d(pk+1)≡pm+1(mod pn-1) Our results show that the cross correlation is six-valued and can be completely detemined.This thesis extends the results of[2],[3],[8].