Finite Commutative Rings with a MacWilliams Type Relation for the m-Spotty Hamming Weight Enumerators

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Let R be a finite commutative ring. We prove that a MacWilliams type relation between the m-spotty weight enumerators of a linear code over R and its dual hold, if and only if, R is a Frobenius (equivalently, Quasi-Frobenius) ring, if and only if, the number of maximal ideals and minimal ideals of R are the same, if and only if, for every linear code C over R, the dual of the dual C is C itself. Also as an intermediate step, we present a new and simpler proof for the commutative case of Wood's theorem which states that R has a generating character if and only if R is a Frobenius ring.