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Abstract. Explicit equations are given for unramified coverings of the affine line in characteristic two whose Galois groups are the Mathieu groups of degrees 11 and 12 and the automorphism group of the Mathieu group of degree 12.

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A. K. Lenstra and E. R. Verheul in [2] proposed a very efficient way called XTR in which certain subgroup of the Galois field GF(p 6) can be represented by elements in GF(p 2). At the end of their paper [2], they briefly mentioned on a method of generalizing their idea to the field GF(p 6m). In this paper, we give a systematic design of this generalization and discuss about optimal choices for p and m with respect to performances. If we choose m large enough, we can reduce the size of p as small as the word size of common processors. In such a case, this extended XTR is well suited for the processors with optimized arithmetic on integers of word size.

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