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In this paper, we consider the multiplicative orders of the MacWilliams matrix of order $N(M_{N})_{ij}$ and the Chebyshev matrix of order $N(D_{N})_{ij}$ according to modulo $m$ for $N\ge 1$. Consequently, we obtained the rules for the orders of the cyclic groups and semigroups generated by reducing the MacWilliams and Chebyshev matrices modulo $m$ and the deteminats of these matrices.

The Cyclic Groups and the Semigroups via MacWilliams and Chebyshev Matrices