Abstract
Extended 1-perfect codes in the Hamming scheme H(n,q) can be equivalently defined as codes that turn to 1-perfect codes after puncturing in any coordinate, as completely regular codes with certain intersection array, as uniformly packed codes with certain weight coefficients, as diameter perfect codes with respect to a certain anticode, as distance-4 codes with certain dual distances. We define extended 1-perfect bitrades in H(n,q) in five different manners, corresponding to the different definitions of extended 1-perfect codes, and prove the equivalence of these definitions of extended 1-perfect bitrades. For q=2m, we prove that such bitrades exist if and only if n=lq+2. For any q, we prove the nonexistence of extended 1-perfect bitrades if n is odd.
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