Several Classes of Codes and Sequences Derived From a $BBZ_{4}$ -Valued Quadratic Form

Let m and k be positive integers with m/gcd(m,k) being odd, for a ∈ R and b ∈ L, the exponential sum Σx∈LiTr(ax+2bx²k+1) is studied systematically in this paper, where i=√(-1), R = GR (4,m) is a Galois ring, L is the Teichmüller set of R and Tr(·) is the trace function from the Galois ring R to Z₄. Through the discussions on the solutions of certain equations and the newly developed theory of Z₄-valued quadratic forms, the distribution of the exponential sum is completely determined. As its applications, we can determine the Lee weight and Hamming weight distributions of a class of codes Ck over Z₄ and the correlation distribution of a quaternary sequence family Uk, respectively. Furthermore, the Hamming weight distributions of the binary codes obtained from Ck under the most significant bit (MSB) and Gray maps are also determined. For the MSB map sequences of Uk, the nontrivial maximal correlation value is given and the correlation distribution is determined for the Gray map sequences of Uk. It should be noted that the distribution of the exponential sum for the case gcd(m,k) ≠ 1 is obtained for the first time, and then the corresponding codes and sequences are novel.