1,770 research outputs found
Generator Matrix Based Search for Extremal Self-Dual Binary Error-Correcting Codes
Self-dual doubly even linear binary error-correcting codes, often referred to
as Type II codes, are codes closely related to many combinatorial structures
such as 5-designs. Extremal codes are codes that have the largest possible
minimum distance for a given length and dimension.
The existence of an extremal (72,36,16) Type II code is still open. Previous
results show that the automorphism group of a putative code C with
the aforementioned properties has order 5 or dividing 24. In this work, we
present a method and the results of an exhaustive search showing that such
a code C cannot admit an automorphism group Z6.
In addition, we present so far unpublished construction of the extended
Golay code by P. Becker. We generalize the notion and provide example of
another Type II code that can be obtained in this fashion. Consequently, we
relate Becker's construction to the construction of binary Type II codes from
codes over GF(2^r) via the Gray map
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
New extremal binary self-dual codes of length 68 via short kharaghani array over f_2 + uf_2
In this work, new construction methods for self-dual codes are given. The
methods use the short Kharaghani array and a variation of it. These are
applicable to any commutative Frobenius ring. We apply the constructions over
the ring F_2 + uF_2 and self-dual Type I [64, 32, 12]_2-codes with various
weight enumerators obtained as Gray images. By the use of an extension theorem
for self-dual codes we were able to construct 27 new extremal binary self-dual
codes of length 68. The existence of the extremal binary self-dual codes with
these weight enumerators was previously unknown.Comment: 10 pages, 5 table
- ā¦