The aim of this thesis is to describe basic concepts from the theory of linear codes, with an emphasis on the weight polynomials of the code and its dual code. The explicit formula between them is given by the identity of Jessie MacWilliams, which can be proven in several ways. The thesis presents some equivalent forms of this identity, as linear systems equalities for the coefficients, and the proof using those forms. For self-dual codes, with additional special properties, the MacWilliams identity is interpreted as the invariance of the weight polynomial under the action of a certain group of linear transformations. With this approach, proof of the non-existence of projective planes of order 8m + 6 is obtained. Furthermore, in the proof of the non-existence of a projective plane of order 10, the weight polynomial is of key importance as the basis for an extensive computer research that finally proved this claim. At the end of this thesis, it is briefly described how the non-existence of an 2–(22, 8, 4) block design was proven by an approach using coding theory, the research of which turned out to be even more demanding than for the projective plane of order 10.