On coset weight distributions of the Z4-linear Goethalscodes

We study the coset weight distributions of two well-known families of codes: the three-error-correcting binary Z₄-linear Goethals codes of length N=2^m+1, m⩾3 odd, and the Z₄ -linear Goethals codes over Z₄ of length n=N/2=2^m . The hard case is the weight distributions of cosets of weight 4. To know the weight distribution of the coset of weight 4 we have to know the number of codewords of weight 4 in such a coset. Altogether, there are nine different types of cosets of weight 4. For six cases, we give the exact expressions for the number of codewords of weight 4, and for three other cases, we give such expressions in terms of Kloosterman sums