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

Counting Steiner triple systems with classical parameters and prescribed rank

By a famous result of Doyen, Hubaut and Vandensavel \cite{DHV}, the 2-rank of a Steiner triple system on $2^n-1$ points is at least $2^n -1 -n$, and equality holds only for the classical point-line design in the projective geometry $PG(n-1,2)$. It follows from results of Assmus \cite{A} that, given any integer $t$ with $1 \leq t \leq n-1$, there is a code $C_{n,t}$ containing representatives of all isomorphism classes of STS$(2^n-1)$ with 2-rank at most $2^n -1 -n + t$. Using a mixture of coding theoretic, geometric, design theoretic and combinatorial arguments, we prove a general formula for the number of distinct STS$(2^n-1)$ with 2-rank at most $2^n -1 -n + t$ contained in this code. This generalizes the only previously known cases, $t=1$, proved by Tonchev \cite{T01} in 2001, $t=2$, proved by V. Zinoviev and D. Zinoviev \cite{ZZ12} in 2012, and $t=3$ (V. Zinoviev and D. Zinoviev \cite{ZZ13}, \cite{ZZ13a} (2013), D. Zinoviev \cite{Z16} (2016)), while also unifying and simplifying the proofs. This enumeration result allows us to prove lower and upper bounds for the number of isomorphism classes of STS$(2^n-1)$ with 2-rank exactly (or at most) $2^n -1 -n + t$. Finally, using our recent systematic study of the ternary block codes of Steiner triple systems \cite{JT}, we obtain analogous results for the ternary case, that is, for STS$(3^n)$ with 3-rank at most (or exactly) $3^n -1 -n + t$. We note that this work provides the first two infinite families of 2-designs for which one has non-trivial lower and upper bounds for the number of non-isomorphic examples with a prescribed $p$-rank in almost the entire range of possible ranks.Comment: 27 page

Self-Dual Codes

Self-dual codes are important because many of the best codes known are of this type and they have a rich mathematical theory. Topics covered in this survey include codes over F_2, F_3, F_4, F_q, Z_4, Z_m, shadow codes, weight enumerators, Gleason-Pierce theorem, invariant theory, Gleason theorems, bounds, mass formulae, enumeration, extremal codes, open problems. There is a comprehensive bibliography.Comment: 136 page

Breakup Conditions of Projectile Spectators from Dynamical Observables

Momenta and masses of heavy projectile fragments (Z >= 8), produced in collisions of 197Au with C, Al, Cu and Pb targets at E/A = 600 MeV, were determined with the ALADIN magnetic spectrometer at SIS. An analysis of kinematic correlations between the two and three heaviest projectile fragments in their rest frame was performed. The sensitivity of these correlations to the conditions at breakup was verified within the schematic SOS-model. The data were compared to calculations with statistical multifragmentation models and to classical three-body calculations. Classical trajectory calculations reproduce the dynamical observables. The deduced breakup parameters, however, differ considerably from those assumed in the statistical multifragmentation models which describe the charge correlations. If, on the other hand, the analysis of kinematic and charge correlations is performed for events with two and three heavy fragments produced by statistical multifragmentation codes, a good agreement with the data is found with the exception that the fluctuation widths of the intrinsic fragment energies are significantly underestimated. A new version of the multifragmentation code MCFRAG was therefore used to investigate the potential role of angular momentum at the breakup stage. If a mean angular momentum of 0.75$\hbar$/nucleon is added to the system, the energy fluctuations can be reproduced, but at the same time the charge partitions are modified and deviate from the data. PACS numbers: 25.70.Mn, 25.70.Pq, 25.75.Ld, 25.75.-qComment: 38 pages, RevTeX with 21 included figures; Also available from http://www-kp3.gsi.de/www/kp3/aladin_publications.htm

Syntactic Structures and Code Parameters

We assign binary and ternary error-correcting codes to the data of syntactic structures of world languages and we study the distribution of code points in the space of code parameters. We show that, while most codes populate the lower region approximating a superposition of Thomae functions, there is a substantial presence of codes above the Gilbert-Varshamov bound and even above the asymptotic bound and the Plotkin bound. We investigate the dynamics induced on the space of code parameters by spin glass models of language change, and show that, in the presence of entailment relations between syntactic parameters the dynamics can sometimes improve the code. For large sets of languages and syntactic data, one can gain information on the spin glass dynamics from the induced dynamics in the space of code parameters.Comment: 14 pages, LaTeX, 12 png figure