  Massless conformal scalar field in six dimensions corresponds to the minimal
unitary representation (minrep) of the conformal group SO(6,2). This minrep
admits a family of deformations labelled by the spin t of an SU(2)_T group,
which is the 6d analog of helicity in four dimensions. These deformations of
the minrep of SO(6,2) describe massless conformal fields that are symmetric
tensors in the spinorial representation of the 6d Lorentz group. The minrep and
its deformations were obtained by quantization of the nonlinear realization of
SO(6,2) as a quasiconformal group in arXiv:1005.3580. We give a novel
reformulation of the generators of SO(6,2) for these representations as
bilinears of deformed twistorial oscillators which transform nonlinearly under
the Lorentz group SO(5,1) and apply them to define higher spin algebras and
superalgebras in AdS_7. The higher spin (HS) algebra of Fradkin-Vasiliev type
in AdS_7 is simply the enveloping algebra of SO(6,2) quotiented by a two-sided
ideal (Joseph ideal) which annihilates the minrep. We show that the Joseph
ideal vanishes identically for the quasiconformal realization of the minrep and
its enveloping algebra leads directly to the HS algebra in AdS_7. Furthermore,
the enveloping algebras of the deformations of the minrep define a discrete
infinite family of HS algebras in AdS_7 for which certain 6d Lorentz covariant
deformations of the Joseph ideal vanish identically. These results extend to
superconformal algebras OSp(8*|2N) and we find a discrete infinite family of HS
superalgebras as enveloping algebras of the minimal unitary supermultiplet and
its deformations. Our results suggest the existence of a discrete family of
(supersymmetric) HS theories in AdS_7 which are dual to free (super)conformal
field theories (CFTs) or to interacting but integrable (supersymmetric) CFTs in
6d.
