Construction of Extremal Type II Codes over ℤ₄

In this paper, we give a pseudo-random method to construct extremal Type II codes over ℤ₄;. As an application, we give a number of new extremal Type II codes of lengths 24, 32 and 40, constructed from some extremal doubly-even self-dual binary codes. The extremal Type II codes of length 24 have the property that the supports of the codewords of Hamming weight 10 form 5-(24, 10, 36) designs. It is also shown that every extremal doubly-even self-dual binary code of length 32 can be considered as the residual code of an extremal Type II code over ℤ₄.