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By N.J.A. Sloane

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10) Therefore W is invariant under the group generated by T 1 and T 2 • This is a group G 1 (say), of order 16, isomorphic to the dihedral group of order 16. 11) and T1 , T 3 generate a group G 2 (say) of order 192. 4 Invariant Theory The problems stated at the end of the previous section are special cases of thc general problern of finding the invariant polynomials of a group of linear transformations. The classical Statement and solution of this problern are as follows (see [ 81, Ch. 1) The Problem Let G be a finite group of linear tr 11sformations on n (complex) variables x 1 ,x 2 , ...

J. (1963) A Theorem on the Distribution of Weights in a Systematic Code, BSTJ 42:79-94. J. (1956), A Note on the Mathieu Groups, CJM 9:15-18. [78 Perron, 0. (1952), Bemerkungen Ueber Die Verteilung Der Quadratischen Reste, Math. Zeit. 56: 122-130. S. (1968), On the Uniqueness of the Golay Codes, JCT 5:215-228. M. -L. Chen (1972), New Binary Codes, PGIT 18:503-510. 1 lntroduction In this chapter we prove two of the most remarkable results in all of coding theory. The first, due to Mrs. J. MacWilliams, says that the weigth cnumeration of the dual code ct1 is completely determined just by the weight cnumerator of CC.

78 Perron, 0. (1952), Bemerkungen Ueber Die Verteilung Der Quadratischen Reste, Math. Zeit. 56: 122-130. S. (1968), On the Uniqueness of the Golay Codes, JCT 5:215-228. M. -L. Chen (1972), New Binary Codes, PGIT 18:503-510. 1 lntroduction In this chapter we prove two of the most remarkable results in all of coding theory. The first, due to Mrs. J. MacWilliams, says that the weigth cnumeration of the dual code ct1 is completely determined just by the weight cnumerator of CC. M. Gleason, states that the weight enumerator of any sclf-dual code (in which the weigth of any codeward isamultiple of 4) is a polynomial in the weigth enumerators of the extended Hamming code and the extended Golay code.

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