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Counting The Generator Matrices of $\mathbb{Z}_{2}\mathbb{Z}_{8}$-Codes

In this paper, we count the number of matrices whose rows generate different $\mathbb{Z}_2\mathbb{Z}_8$ additive codes. This is a natural generalization of the well known Gaussian numbers that count the number of matrices whose rows generate vector spaces with particular dimension over finite fields. Due to this similarity we name this numbers as Mixed Generalized Gaussian Numbers (MGN). The MGN formula by specialization leads to the well known formula for the number of binary codes and the number of codes over $\mathbb{Z}_8,$ and for additive $\mathbb{Z}_2\mathbb{Z}_4$ codes. Also, we conclude by some properties and examples of the MGN numbers that provide a good source for new number sequences that are not listed in The On-Line Encyclopedia of Integer Sequences.