On unbalanced Boolean functions with best correlation immunity

It is known that the order of correlation immunity of a nonconstant unbalanced Boolean function in $n$ variables cannot exceed $2n/3-1$; moreover, it is $2n/3-1$ if and only if the function corresponds to an equitable $2$-partition of the $n$-cube with an eigenvalue $-n/3$ of the quotient matrix. The known series of such functions have proportion $1:3$, $3:5$, or $7:9$ of the number of ones and zeros. We prove that if a nonconstant unbalanced Boolean function attains the correlation-immunity bound and has ratio $C:B$ of the number of ones and zeros, then $CB$ is divisible by $3$. In particular, this proves the nonexistence of equitable partitions for an infinite series of putative quotient matrices. We also establish that there are exactly $2$ equivalence classes of the equitable partitions of the $12$-cube with quotient matrix $[[3,9],[7,5]]$ and $16$ classes, with $[[0,12],[4,8]]$. These parameters correspond to the Boolean functions in $12$ variables with correlation immunity $7$ and proportion $7:9$ and $1:3$, respectively (the case $3:5$ remains unsolved). This also implies the characterization of the orthogonal arrays OA$(1024,12,2,7)$ and OA$(512,11,2,6)$.Comment: v3: final; title changed; revised; OA(512,11,2,6) discusse

Evidence of the virtual Anderson transition in a narrow impurity band of p-GaAs/AlGaAs quantum wells: ϵ4\epsilon_4 conductivity and electric breakdown at low temperatures

In highly doped uncompensated p-type layers within the central part of GaAs/AlGaAs quantum wells at low temperatures we observed an activated behavior of the conductivity with low activation energies (1-3) meV which can not be ascribed to standard mechanisms. We attribute this behavior to the delocalization of hole states near the maximum of the narrow impurity band in the sense of the Anderson transition. Low temperature conduction $\epsilon_4$ is supported by an activation of minority carriers - electrons (resulting from a weak compensation by back-ground defects) - from the Fermi level to the band of delocalized states mentioned above. The corresponding behavior can be specified as virtual Anderson transition. Low temperature transport ($<4$ K) exhibits also strong nonlinearity of a breakdown type characterized in particular by S-shaped I-V curve. The nonlinearity is observed in unexpectedly low fields ($<10$ V/cm). Such a behavior can be explained by a simple model implying an impact ionization of the localized states of the minority carriers mentioned above to the band of Anderson-delocalized states.Comment: 9 pages, 6 fi