Tricolore designs in Type III codes

A split complete weight enumerator in six variables is used to study the colored designs held by codewords of xed composition in Type III codes containing the all one codeword In particular the ternary Golay code contains colored designs We conjecture that every weight class in a Type III code with the all one codeword holds colored designs Introduction Colored designs were introduced in to study the Harada designs held by the lifted Golay over Z They produce ordinary designs by bleaching and are easily understood in terms of split complete weight enumerators Colored t designs are known to exist when the per mutation part of the automorphism group is t transitive This is the case for instance of the doubled Golay code in which is invariant under the action of M In the present work we investigate the tricolore designs held by codewords of extremal Type III codes It is well known that the ternary Golay is invariant under the monomial action of the transitive group M The permutation part of that group however is not even transitive x We show that the codewords of given composition in the ternary Golay the symmetry code of length the extended quadratic residue code of length hold tricolore designs Similarly as observed in x the permutation part of the automorphism group of the symmetry code is not transitive We conjecture based on these results and computations in lengths that the code words of given composition in an extremal Type III code containing the all one codeword hold tricolore designs GECT Universit e de Toulon et du Var La Garde France Email bonnecaz univ tln fr CNRS I S ESSI BP Route des Colles Sophia Antipolis France Email ps essi fr Department of Mathematics City Campus Royal Melbourne Institute of Technology GPO Box V Melbourne VIC Australia Email rmapu gauss ma rmit edu au De nitions and Notations The composition of a vector c F is the triple n c n c n c where ni c counts the number of j n with cj i A ternary linear code C is a F subspace of F A Type III code is a ternary linear code self dual w r t the usual inner product It is said to be extremal if its minimum distance d bn c The complete weight enumerator of C denoted by cweC is then cweC x y z X