**The non-existence of projective plane of order 6**

In this short talk, we will define projective planes of finite order as well as some other crucial concepts in the field of finite geometries and demonstrate several different proofs of the non-existence of a projective plane of order 6.
We will define symmetric designs and talk about various statements about the conditions of their existence. The most important among such results is the Bruck-Ryser-Chowla theorem, which yields the non-existence of a projective plane of order 6 as an immediate consequence. Further, we will define latin squares and the concept of their orthogonality. We will show that the existence of a complete M OLS(n) is equivalent to the existence of a projective plane of order n. After that, we gave the proof that a pair of orthogonal latin squares of order 6 cannot exist, which means thet there is no projective plane of order 6. Finally, we will mention coding theory and binary linear codes, selfdual codes, minimal weight and the weight polynomial of a code. By investigation of the properties of the codes generated by incidence matrices of finite projective planes we obtain, by yet another method, the result that a projective plane of order 6 does not exist.