Several Classes of Codes and Sequences Derived From a Z(4)-Valued Quadratic Form
Let and m be k positive integers with m/gcd(m,k) being odd, for a is an element of R and b is an element of L, the exponential sum Sigma(x is an element of L) i(Tr(ax+2nx2k+1)) is studied systematically in this paper, where i = root-1, R = GR(4,m) is a Galois ring, L wis the Teichmuller set of R and Tr(.) is the trace function from the Galois ring to R to Z(4). Through the discussions on the solutions of certain equations and the newly developed theory of Z(4)-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 C-k over Z(4) and the correlation distribution of a quaternary sequence family U-k, respectively. Furthermore, the Hamming weight distributions of the binary codes obtained from C-k under the most significant bit (MSB) and Gray maps are also determined. For the MSB map sequences of U-k, the nontrivial maximal correlation value is given and the correlation distribution is determined for the Gray map sequences of U-k. It should be noted that the distribution of the exponential sum for the case gcd(m,k) not equal 1 is obtained for the first time, and then the corresponding codes and sequences are novel.
quadratic form galois ring lee weight hamming weight correlation distribution gray map most significant bit (msb) map