The Perfect Binary One-Error-Correcting Codes of Length 15: Part II--Properties

A complete classification of the perfect binary one-error-correcting codes of length 15 as well as their extensions of length 16 was recently carried out in [P. R. J. \"Osterg{\aa}rd and O. Pottonen, "The perfect binary one-error-correcting codes of length 15: Part I--Classification," IEEE Trans. Inform. Theory vol. 55, pp. 4657--4660, 2009]. In the current accompanying work, the classified codes are studied in great detail, and their main properties are tabulated. The results include the fact that 33 of the 80 Steiner triple systems of order 15 occur in such codes. Further understanding is gained on full-rank codes via switching, as it turns out that all but two full-rank codes can be obtained through a series of such transformations from the Hamming code. Other topics studied include (non)systematic codes, embedded one-error-correcting codes, and defining sets of codes. A classification of certain mixed perfect codes is also obtained.Comment: v2: fixed two errors (extension of nonsystematic codes, table of coordinates fixed by symmetries of codes), added and extended many other result

Operation of weaving partial Steiner triple systems

We introduce an operation of a kind of product which associates with a partial Steiner triple system another partial Steiner triple system, the starting one being a quotient of the result. We discuss relations of our product to some other product-like constructions and prove some preservation/non-preservation theorems. In particular, we show series of anti-Pasch Steiner triple systems which are obtained that way

Coding Theory and Algebraic Combinatorics

This chapter introduces and elaborates on the fruitful interplay of coding theory and algebraic combinatorics, with most of the focus on the interaction of codes with combinatorial designs, finite geometries, simple groups, sphere packings, kissing numbers, lattices, and association schemes. In particular, special interest is devoted to the relationship between codes and combinatorial designs. We describe and recapitulate important results in the development of the state of the art. In addition, we give illustrative examples and constructions, and highlight recent advances. Finally, we provide a collection of significant open problems and challenges concerning future research.Comment: 33 pages; handbook chapter, to appear in: "Selected Topics in Information and Coding Theory", ed. by I. Woungang et al., World Scientific, Singapore, 201

Schreier extensions of Steiner loops and extensions of Bol loops arising from Bol reflections

This dissertation explores two constructions of loop extensions: Schreier extensions of Steiner loops and a new extension formula for right Bol loops arising from Bol reflections.Steiner loops are a key tool in studying Steiner triple systems. We investigate extensions of Steiner loops, focusing in particular on the case of Schreier extensions, which provides a powerful method for constructing Steiner triple systems containing Veblen points. We determine the number of the Steiner triple systems of sizes 19, 27 and 31 with Veblen points, presenting some examples.Furthermore, we study a new extension formula for right Bol loops. We prove the necessary and sufficient conditions for the extension to be right Bol as well. We describe the most important invariants: right multiplication group, nuclei, center. We show that the core is an involutory quandle which is the disjoint union of two isomorphic involutory quandles. Lastly, we derive further results on the structure group of the core of the extension

The classification of flag-transitive Steiner 3-designs

We solve the long-standing open problem of classifying all 3-(v,k,1) designs with a flag-transitive group of automorphisms (cf. A. Delandtsheer, Geom. Dedicata 41 (1992), p. 147; and in: "Handbook of Incidence Geometry", ed. by F. Buekenhout, Elsevier Science, Amsterdam, 1995, p. 273; but presumably dating back to 1965). Our result relies on the classification of the finite 2-transitive permutation groups.Comment: 27 pages; to appear in the journal "Advances in Geometry