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

Hypertoric varieties, WW-Hilbert schemes, and Coulomb branches

We study transverse equivariant Hilbert schemes of affine hypertoric varieties equipped with a symplectic action of a Weyl group. In particular, we show that the Coulomb branches of Braverman, Finkelberg, and Nakajima can be obtained either as such Hilbert schemes or Hamiltonian reductions thereof. Furthermore, we propose that the Coulomb branches for representations of non-cotangent type are also obtained in this way. We also investigate the putative complete hyperk\"ahler metrics on these objects. We describe their twistor spaces and, in the case when the symplectic quotient construction of the hypertoric variety is $W$-equivariant (which includes Coulomb branches of cotangent type), we show that the hyperk\"ahler metric can be described as the natural $L^2$-metric on a moduli space of solutions to modified Nahm's equations on an interval with poles at both ends and a discontinuity in the middle, with the latter described by a new object: a hyperspherical variety canonically associated to a hypertoric variety.Comment: Main changes in v2 are: 1) a discussion of $W$-invariant hypertoric varieties which do not arise via a $W$-equivariant symplectic quotient construction; 2) a construction of Coulomb branches of non-cotangent type; 3) moduli spaces of solutions to Nahm's equations on a star-shaped graph have been replaced by moduli spaces of solutions to modified Nahm's equations on an interva