  A Boolean function f of n variables is said to be q-locally correctable if,
given a black-box access to a function g which is "close" to an isomorphism
f_sigma(x)=f_sigma(x_1, ..., x_n) = f(x_sigma(1), ..., x_sigma(n)) of f, we can
compute f_sigma(x) for any x in {0,1}^n with good probability using q queries
to g. It is known that degree d polynomials are O(2^d)-locally correctable, and
that most k-juntas are O(k log k)-locally correctable, where the closeness
parameter, or more precisely the distance between g and f_sigma, is required to
be exponentially small (in d and k respectively).
  In this work we relax the requirement for the closeness parameter by allowing
the distance between the functions to be a constant. We first investigate the
family of juntas, and show that almost every k-junta is O(k log^2 k)-locally
correctable for any distance epsilon < 0.001. A similar result is shown for the
family of partially symmetric functions, that is functions which are
indifferent to any reordering of all but a constant number of their variables.
For both families, the algorithms provided here use non-adaptive queries and
are applicable to most but not all functions of each family (as it is shown to
be impossible to locally correct all of them).
  Our approach utilizes the measure of symmetric influence introduced in the
recent analysis of testing partial symmetry of functions.
