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

We study the coset weight distributions of the family of Z₄ -linear Goethals-like codes of length N=2^m+1, m⩾3 odd, constructed by Helleseth, Kumar, and Shanbhag (see Designs, Codes and Cryptography, vol.17, no.1-3, p.246-62, 1999). These codes have the same Lee weight distribution as the Z₄-linear Goethals code 𝒢₁, 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 (see ibid., vol.40, p.301-19, March 1994). The cosets of weight four is the most difficult case. In order to find 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 k. We prove that the distribution of solutions to such a system does not depend on k and, therefore, coincides with the case k=1 considered earlier by Helleseth and Zinoviev (see Designs, Codes and Cryptography, vol.17, no.1-3, p.246-62, 1999). For k=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