Yet another approach to the extended ternary Golay code

A new proof of the uniqueness and of the existence of the extended ternary Golay code is presented. The proof connects the code to the projective plane of order 3 and is of an elementary nature. The available proofs of the uniqueness of the extended ternary Golay code [2,7] are much more complicated than the standard corresponding proof in the binary case [2]. The prevailing opinion seems to be that there are intrinsic reasons for this fact [1,2,3,5]. The purpose of this paper is to show that the extended ternary Golay code can be presented in a natural way that ooers a short and direct elementary proof of its uniqueness. Our departing point is the standard computation of the weight polynomial of an (11; 729; 5) ternary code C. While doing this, we shall also determine the polynomial in the case when C does not contain the zero vector. Considering properties of such codes seems to be the main novel feature of this paper. If u; v are n-vectors over F q , then u; v means u i v i. The weight of a vector u will be denoted by |u|. For a ternary 11-vector v; |v|= j, denote by a(i; j); 06i611 and 06j611, the number of vectors u with |u − v|62 and |u|=i. Then a(i; j)=0