2016/05/10 by Ashkan Nikseresht, Nikseresht, Ashkan
Computer Science · Engineering · Mathematics · #Coding theory and cryptography #graph theory and CDMA systems #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.1605.02870
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.