5,134 research outputs found
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
Some new results on the self-dual [120,60,24] code
The existence of an extremal self-dual binary linear code of length 120 is a
long-standing open problem. We continue the investigation of its automorphism
group, proving that automorphisms of order 30 and 57 cannot occur. Supposing
the involutions acting fixed point freely, we show that also automorphisms of
order 8 cannot occur and the automorphism group is of order at most 120, with
further restrictions. Finally, we present some necessary conditions for the
existence of the code, based on shadow and design theory.Comment: 23 pages, 6 tables, to appear in Finite Fields and Their Application
Symmetries of weight enumerators and applications to Reed-Muller codes
Gleason's 1970 theorem on weight enumerators of self-dual codes has played a
crucial role for research in coding theory during the last four decades. Plenty
of generalizations have been proved but, to our knowledge, they are all based
on the symmetries given by MacWilliams' identities. This paper is intended to
be a first step towards a more general investigation of symmetries of weight
enumerators. We list the possible groups of symmetries, dealing both with the
finite and infinite case, we develop a new algorithm to compute the group of
symmetries of a given weight enumerator and apply these methods to the family
of Reed-Muller codes, giving, in the binary case, an analogue of Gleason's
theorem for all parameters.Comment: 14 pages. Improved and extended version of arXiv:1511.00803. To
appear in Advances in Mathematics of Communication
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