Geodesic
Approximation of $k$-ary functions by functions from the given system
Fundamentalʹnaâ i prikladnaâ matematika, Tome 3 (1997) no. 3, pp. 653-674

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Problems of $k$-ary functions approximation by functions from the given system are investigated in this paper. In particular, generalization of Golomb theorem [1] is obtained in the case of ring $\mathbb{Z}/k$ or finite field $GF(q)$. The definition of $k$-ary functions equivalency with respect to the given functions system is introduced. Classes of equivalency with respect to the linear functions system over finite field or ring $\mathbb{Z}/4$ are described. Limit theorems on cardinality of random $k$-ary functions equivalency class are proved. Also in this paper we found functions which minimize maximum probability of coincidence with linear functions in one variable over finite ring with identity. 
@article{FPM_1997_3_3_a1,
     author = {A. S. Ambrosimov},
     title = {Approximation of $k$-ary functions by functions from the given system},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {653--674},
     publisher = {mathdoc},
     volume = {3},
     number = {3},
     year = {1997},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_1997_3_3_a1/}
}
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A. S. Ambrosimov. Approximation of $k$-ary functions by functions from the given system. Fundamentalʹnaâ i prikladnaâ matematika, Tome 3 (1997) no. 3, pp. 653-674. http://geodesic.mathdoc.fr/item/FPM_1997_3_3_a1/