Perfect binary codes: classification and properties

An r-perfect binary code is a subset of ℤ2n such that for any word, there is a unique codeword at Hamming distance at most r. Such a code is r-error-correcting. Two codes are equivalent if one can be obtained from the other by permuting the coordinates and adding a constant vector. The main result of this thesis is a computer-aided classification, up to equivalence, of the 1-perfect binary codes of length 15. In an extended 1-perfect code, the neighborhood of a codeword corresponds to a Steiner quadruple system. To utilize this connection, we start with a computational classification of Steiner quadruple systems of order 16. This classification is also used to establish the nonexistence of Steiner quintuple systems S(4, 5, 17). The classification of the codes is used for computational examination of their properties. These properties include occurrences of Steiner triple and quadruple systems, automorphisms, ranks, structure of i-components and connections to orthogonal arrays and mixed perfect codes. It is also proved that extended 1-perfect binary codes are equivalent if and only if their minimum distance graphs are isomorphic

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

Efficient Two-Stage Group Testing Algorithms for Genetic Screening

Efficient two-stage group testing algorithms that are particularly suited for rapid and less-expensive DNA library screening and other large scale biological group testing efforts are investigated in this paper. The main focus is on novel combinatorial constructions in order to minimize the number of individual tests at the second stage of a two-stage disjunctive testing procedure. Building on recent work by Levenshtein (2003) and Tonchev (2008), several new infinite classes of such combinatorial designs are presented.Comment: 14 pages; to appear in "Algorithmica". Part of this work has been presented at the ICALP 2011 Group Testing Workshop; arXiv:1106.368

Constructing Optimal Authentication Codes with Perfect Multi-fold Secrecy

We establish a construction of optimal authentication codes achieving perfect multi-fold secrecy by means of combinatorial designs. This continues the author's work (ISIT 2009) and answers an open question posed therein. As an application, we present the first infinite class of optimal codes that provide two-fold security against spoofing attacks and at the same time perfect two- fold secrecy.Comment: 4 pages (double-column); to appear in Proc. 2010 International Zurich Seminar on Communications (IZS 2010, Zurich

Computational complexity of reconstruction and isomorphism testing for designs and line graphs

Graphs with high symmetry or regularity are the main source for experimentally hard instances of the notoriously difficult graph isomorphism problem. In this paper, we study the computational complexity of isomorphism testing for line graphs of $t$-$(v,k,\lambda)$ designs. For this class of highly regular graphs, we obtain a worst-case running time of $O(v^{\log v + O(1)})$ for bounded parameters $t,k,\lambda$. In a first step, our approach makes use of the Babai--Luks algorithm to compute canonical forms of $t$-designs. In a second step, we show that $t$-designs can be reconstructed from their line graphs in polynomial-time. The first is algebraic in nature, the second purely combinatorial. For both, profound structural knowledge in design theory is required. Our results extend earlier complexity results about isomorphism testing of graphs generated from Steiner triple systems and block designs.Comment: 12 pages; to appear in: "Journal of Combinatorial Theory, Series A

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