Codes with the same coset weight distributions as the Z4-linear Goethals codes

We study the coset weight distributions of the family of Z(4)-linear Goethals-like codes of length N 2(m+1) m greater than or equal to 3 odd, constructed by Helleseth, Kumar, and Shanbhag. These codes have the same Lee weight distribution as the Z4-linear Goethals code G(1), and, therefore (taking into account the result of Hammons, Kumar, Sloane, Calderbank, and Sole), the binary images of all these codes by the Gray map have the same weight distribution as the binary Goethals code. We prove that all these codes have the same coset weight distributions as the Z4-linear Goethals code, constructed by Hammons, Kumar, Sloane, Calderbank, and Sole. The cosets of weight Four is the most difficult case. In order to End the number of codewords of weight four in a coset of weight four we have to solve a nonlinear system of equations over the Galois field GF(2(m)). Such a system (the degree of one of the equations) depends on le, We prove that the distribution of solutions to such a system does not depend on le and, therefore, coincides with the case kappa = 1 considered earlier by Helleseth and Zinoviev. For kappa = 1, we solved this system in the following sense: for all cases (of cosets of weight four) we have either an exact expression, or an expression in terms of the Kloosterman sums.