Well-Ordered Steiner Triple Systems and 1-Perfect Partitions of the n-cube

Binary 1-perfect codes which give rise to partitions of the n-cube are presented. The 1-perfect partitions are characterized as homomorphic images of simple algebraic structures on F and are constructed starting from a particular case of a structure defined in F. A special property (so-called well-ordering) of STS(n) is given in such a way that for this kind of STS it is possible to define the algebraic structure we need in F and to construct 1-perfect partitions of the n-cube. These 1-perfect partitions give us a kind of 1-perfect code for which it is easy to do the coding and decoding. Furthermore, there exists a syndrome which allows us to perform error correction. We present systematic codes of length n = 15 and we give examples of how to do the coding, decoding, and error correction.