Self-dual doubly even linear binary error-correcting codes, often referred toas Type II codes, are codes closely related to many combinatorial structuressuch as 5-designs. Extremal codes are codes that have the largest possibleminimum distance for a given length and dimension.The existence of an extremal (72,36,16) Type II code is still open. Previousresults show that the automorphism group of a putative code C withthe aforementioned properties has order 5 or dividing 24. In this work, wepresent a method and the results of an exhaustive search showing that sucha code C cannot admit an automorphism group Z6.In addition, we present so far unpublished construction of the extendedGolay code by P. Becker. We generalize the notion and provide example ofanother Type II code that can be obtained in this fashion. Consequently, werelate Becker's construction to the construction of binary Type II codes fromcodes over GF(2^r) via the Gray map.
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