Über die Identität von MacWilliams für die Gewichtsfunktion von Codes (Q1085138)

The identity of MacWilliams is established for linear codes with scalars out of the ring of integers modulo m. The weight enumerator, which is introduced in this paper, is a homogeneous polynomial in d independents, where d is the number of divisors of m. The connection between the weigth enumerators of a code and its dual (''Identity of MacWilliams'') is then described by an integral quadratic matrix \(T_ m\) of size d, involving the Möbius function. If m is a product of two factors a and b, which are relatively prime, then \(T_ m=T_ a\otimes T_ b\), and if m is a prime power, then \(T_ m\) can be determined by induction on the exponent. In order to motivate these investigations it is shown that a symmetric \((v,n+\lambda,\lambda)\) design with \(n=mm'\) and \(g.c.d.(m,m')=g.c.d.(m,\lambda)=1\) has a selfdual code of length \(v+1\) over the ring of integers modulo m.