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In this paper, we proposed a construction of a UD $k$-ary $T$-user codingscheme for MAAC. We first give a construction of $k$-ary $T^{f+g}$-user UD codefrom a $k$-ary $T^{f}$-user UD code and a $k^{\pm}$-ary $T^{g}$-user differenceset with its two component sets $\mathcal{D}^{+}$ and $\mathcal{D}^{-}$ {\em apriori}. Based on the $k^{\pm}$-ary $T^{g}$-user difference set constructedfrom a $(2k-1)$-ary UD code, we recursively construct a UD $k$-ary $T$-usercodes with code length of $2^m$ from initial multi-user codes of $k$-ary,$2(k-1)+1$-ary, \dots, $(2^m(k-1)+1)$-ary. Introducing multi-user codes withhiger-ary makes the total rate of generated code $\mathcal{A}$ higher than thatof conventional code.

A construction of UD $k$-ary multi-user codes from $(2^m(k-1)+1)$-ary codes for MAAC