Several Classes of Codes and Sequences Derived From a \BBZ4-Valued Quadratic Form.

Let m and k be positive integers with m/gcd(m,k) being odd, for a\in \BBR and b\in \BBL, the exponential sum \sumx\in \BBLi^{Tr(ax+2bx^{2^{k}+1})} is studied systematically in this paper, where i=\sqrt -1, \BBR =\BBG \BBR (4,m) is a Galois ring, \BBL is the Teichmüller set of \BBR and Tr(\cdot) is the trace function from the Galois ring \BBR to \BBZ4. Through the discussions on the solutions of certain equations and the newly developed theory of \BBZ4-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 \cal C^k over \BBZ4 and the correlation distribution of a quaternary sequence family \cal U^k, respectively. Furthermore, the Hamming weight distributions of the binary codes obtained from \cal C^k under the most significant bit (MSB) and Gray maps are also determined. For the MSB map sequences of \cal U^k, the nontrivial maximal correlation value is given and the correlation distribution is determined for the Gray map sequences of \cal U^k. It should be noted that the distribution of the exponential sum for the case \gcd (m,k)\ne 1 is obtained for the first time, and then the corresponding codes and sequences are novel. [ABSTRACT FROM AUTHOR]