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

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

Primitive Polynomials, Singer Cycles, and Word-Oriented Linear Feedback Shift Registers

Using the structure of Singer cycles in general linear groups, we prove that a conjecture of Zeng, Han and He (2007) holds in the affirmative in a special case, and outline a plausible approach to prove it in the general case. This conjecture is about the number of primitive $\sigma$-LFSRs of a given order over a finite field, and it generalizes a known formula for the number of primitive LFSRs, which, in turn, is the number of primitive polynomials of a given degree over a finite field. Moreover, this conjecture is intimately related to an open question of Niederreiter (1995) on the enumeration of splitting subspaces of a given dimension.Comment: Version 2 with some minor changes; to appear in Designs, Codes and Cryptography

Commutative association schemes

Association schemes were originally introduced by Bose and his co-workers in the design of statistical experiments. Since that point of inception, the concept has proved useful in the study of group actions, in algebraic graph theory, in algebraic coding theory, and in areas as far afield as knot theory and numerical integration. This branch of the theory, viewed in this collection of surveys as the "commutative case," has seen significant activity in the last few decades. The goal of the present survey is to discuss the most important new developments in several directions, including Gelfand pairs, cometric association schemes, Delsarte Theory, spin models and the semidefinite programming technique. The narrative follows a thread through this list of topics, this being the contrast between combinatorial symmetry and group-theoretic symmetry, culminating in Schrijver's SDP bound for binary codes (based on group actions) and its connection to the Terwilliger algebra (based on combinatorial symmetry). We propose this new role of the Terwilliger algebra in Delsarte Theory as a central topic for future work.Comment: 36 page

Steiner t-designs for large t

One of the most central and long-standing open questions in combinatorial design theory concerns the existence of Steiner t-designs for large values of t. Although in his classical 1987 paper, L. Teirlinck has shown that non-trivial t-designs exist for all values of t, no non-trivial Steiner t-design with t > 5 has been constructed until now. Understandingly, the case t = 6 has received considerable attention. There has been recent progress concerning the existence of highly symmetric Steiner 6-designs: It is shown in [M. Huber, J. Algebr. Comb. 26 (2007), pp. 453-476] that no non-trivial flag-transitive Steiner 6-design can exist. In this paper, we announce that essentially also no block-transitive Steiner 6-design can exist.Comment: 9 pages; to appear in: Mathematical Methods in Computer Science 2008, ed. by J.Calmet, W.Geiselmann, J.Mueller-Quade, Springer Lecture Notes in Computer Scienc