Geodesic
On linear operators injective on arbitrary subsets
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 156 (2014) no. 3, pp. 132-141

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Linear operators that are injective on subsets of a linear space over $GF(p)$ are considered. Given any positive constant $\varepsilon$ and sufficiently large $n$, for any domain $D$ from $GF^n(p)$, there exists a linear operator injective on this domain whose rank is at most $(2+\varepsilon)\log_p|D|$ and whose complexity is $\mathcal O(n)$.
Keywords: perfect linear hashing, circuits of functional elements, circuit complexity. 
@article{UZKU_2014_156_3_a13,
     author = {A. V. Chashkin},
     title = {On linear operators injective on arbitrary subsets},
     journal = {U\v{c}\"enye zapiski Kazanskogo universiteta. Seri\^a Fiziko-matemati\v{c}eskie nauki},
     pages = {132--141},
     publisher = {mathdoc},
     volume = {156},
     number = {3},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/UZKU_2014_156_3_a13/}
}
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A. V. Chashkin. On linear operators injective on arbitrary subsets. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 156 (2014) no. 3, pp. 132-141. http://geodesic.mathdoc.fr/item/UZKU_2014_156_3_a13/