Abstract:
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In addition, we present so far unpublished construction of the extended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.
