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

The twisted Grassmann graph is the block graph of a design

In this note, we show that the twisted Grassmann graph constructed by van Dam and Koolen is the block graph of the design constructed by Jungnickel and Tonchev. We also show that the full automorphism group of the design is isomorphic to the full automorphism group of the twisted Grassmann graph.Comment: 5 pages. A section on the automorphism group has been adde

High-rate self-synchronizing codes

Self-synchronization under the presence of additive noise can be achieved by allocating a certain number of bits of each codeword as markers for synchronization. Difference systems of sets are combinatorial designs which specify the positions of synchronization markers in codewords in such a way that the resulting error-tolerant self-synchronizing codes may be realized as cosets of linear codes. Ideally, difference systems of sets should sacrifice as few bits as possible for a given code length, alphabet size, and error-tolerance capability. However, it seems difficult to attain optimality with respect to known bounds when the noise level is relatively low. In fact, the majority of known optimal difference systems of sets are for exceptionally noisy channels, requiring a substantial amount of bits for synchronization. To address this problem, we present constructions for difference systems of sets that allow for higher information rates while sacrificing optimality to only a small extent. Our constructions utilize optimal difference systems of sets as ingredients and, when applied carefully, generate asymptotically optimal ones with higher information rates. We also give direct constructions for optimal difference systems of sets with high information rates and error-tolerance that generate binary and ternary self-synchronizing codes.Comment: 9 pages, no figure, 2 tables. Final accepted version for publication in the IEEE Transactions on Information Theory. Material presented in part at the International Symposium on Information Theory and its Applications, Honolulu, HI USA, October 201

The existence of optimal quaternary [28,20,6] and quantum [[28,12,6]] codes

The existence of a quantum $[[28,12,6]]$ code was one of the few cases for codes of length $n\le 30$ that was left open in the seminal paper by Calderbank, Rains, Shor, and Sloane \cite{CRSS}. The main result of this paper is the construction of a new optimal linear quaternary $[28,20,6]$ code which contains its hermitian dual code and yields an optimal linear quantum $[[28,12,6]]$ code

Quantum codes from caps

AbstractCaps in a finite projective geometry over GF(4) are used for the construction of some quantum error-correcting codes, including an optimal 〚27,13,5〛 code

Editor’s note

This Special Issue, entitled Algebraic Combinatorics and Applications, of the Journal of Algebra, Combinatorics, Discrete Structures, and Applications, contains selected papers submitted by conference participants at the "Algebraic Combinatorics and Applications: The First Annual Kliakhandler Conference", Houghton, Michigan, USA, August 26 - 30, 2015, as well as two additional papers submitted in response to our call for papers. The conference took place on the campus of Michigan Technological University, and was attended by 43 researchers and graduate and postdoctoral students from USA, Canada, Croatia, Japan, South Africa, and Turkey. Funding for the conference was provided by a generous gift of Igor Kliakhandler, and a grant from the National Science Foundation. The conference brought together researchers and students interested in combinatorics and its applications, to learn about the latest developments, and explore different visions for future work and collaborations. Over thirty talks were presented on various topics from combinatorial designs, graphs, finite geometry, and their applications to error-correcting codes, network coding, information security, quantum computing, DNA codes, mobile communications, and tournament scheduling. The current Special Issue contains papers on covering arrays and their applications, group divisible designs, automorphism groups of combinatorial designs, covering number of permutation groups, tournaments, large sets of geometric designs, partitions, quasi-symmetric functions, resolvable Steiner systems, and weak isometries of Hamming spaces