Multi-Valued Cross-Correlation Functions between Two Maximal Linear Recursive Sequences

The cross-correlation function, A (y), between two maximal linear recursive sequences is defined by Ar(y)= i (.i) ^ + ) r XGGF(2n) for some r, GCD(r,2-l) = 1. Ar(y) is analyzed and evaluated for two types of decimation r. For the first type, r = 2 mod 2 -1. It is shown that Ar(y) is restricted to the form Ar{y) = 2 (j-l), 0< j< J, where j is the number of distinct solutions to the system of two equations over GF(2) and J is the degree of one of the two equations. For the second type, r = (2 +l)/(2 +1) and for this case Ar(y) is restricted to the form Ar(y) = 0, +2( )/, o < d < m-2, d odd where e = GCD(n,k) and d depends on the rank of a quadratic form over GF(2). The explicit evaluation of A (y) is equivalent to the explicit evaluation of weight distribution of the (2-l,2n) cyclic code whose dual code is generated by f1(x)f (x), the product of two primitive polynomials of degree n.