135 research outputs found
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
On extremal self-dual ternary codes of length 48
All extremal ternary codes of length 48 that have some automorphism of prime
order are equivalent to one of the two known codes, the Pless code or
the extended quadratic residue code
The Automorphism Group of an Extremal [72,36,16] Code does not contain elements of order 6
The existence of an extremal code of length 72 is a long-standing open
problem. Let C be a putative extremal code of length 72 and suppose that C has
an automorphism g of order 6. We show that C, as an F_2-module, is the
direct sum of two modules, one easily determinable and the other one which has
a very restrictive structure. We use this fact to do an exhaustive search and
we do not find any code. This proves that the automorphism group of an extremal
code of length 72 does not contain elements of order 6.Comment: 15 pages, 0 figures. A revised version of the paper is published on
IEEE Transactions on Information Theor
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
Extremal doubly-even [80,40,16] codes with an automorphism of order 19
AbstractIn this paper, it is shown that there are 11 extremal doubly-even self-dual [80,40,16] codes with an automorphism of order 19, up to equivalence, and the codes are new. The automorphism groups of the codes are also obtained
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