Designs and Perfect Codes
A Steiner triple system of order n (briefly STS(n)) is a family of 3-element blocks (subsets or triples) of the set N={1,2,..., n} such that each not ordered pair of elements of N appears in exactly one block. 
 
Two STS-s of order n are called isomorphic if there exists a permutation on the set N which transforms them into one another. It is well known that STS(n) exists if and only if n≡1 or 3 (mod 6).
There exist Steiner triple systems of order 15 that do not occur in a perfect binary one‐error‐correcting code
The codewords at distance three from a particular codeword of a perfect binary one‐error‐correcting code (of length 2m−1) form a Steiner triple system. It is a longstanding open problem whether every Steiner triple system of order 2m−1 occurs in a perfect code. It turns out that this is not the case; relying on a classification of the Steiner quadruple systems of order 16 it is shown that the unique anti‐Pasch Steiner triple system of order 15 provides a counterexample. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 465–468, 2007
On Hilbert's thirteenth problem and related questions
Hilbert's thirteenth problem involves the study of solutions of algebraic equations. The object is to obtain a complexity estimate for an algebraic function. As of now, the problem remains open. There are only a few partial algebraic results in this connection, but at the same time the problem has stimulated a series of studies in the theory of functions with their subsequent applications. The most brilliant result in this cycle is Kolmogorov's theorem on superpositions of continuous functions.