On Weight Distributions of Cosets of Goethals-Like Codes

We study weight distributions of shifts of codes from a well-known family: the 3-error-correcting binary nonlinear Goethals-like codes of length $n=2^m$, where $m\geq 6$ is even. These codes have covering radius $\rho=6$. We know the weight distribution of any shift of weight $i=1,2,3,5$, or 6. For a shift of weight 4, the weight distribution is uniquely defined by the number of leaders in this shift, i.e., the number of vectors of weight 4. We also consider the weight distribution of shifts of codes with minimum distance 7 obtained by deleting any one position of a Goethals-like code of length $n$.