  Dimension witness provides a device-independent certification of the minimal
dimension required to reproduce the observed data without imposing assumptions
on the functioning of the devices used to generate the experimental statistics.
In this paper, we provide a family of Bell expressions where Alice and Bob
perform $2^{n-1}$ and $n$ number of dichotomic measurements respectively which
serve as the device-independent dimension witnesses of Hilbert space of $2^{m}$
dimensions with $m=1,2..2^{\lfloor n/2\rfloor}$. The family of Bell expressions
considered here determines the success probability of a communication game
known as $n$-bit parity-oblivious random access code. The parity obliviousness
constraint is equivalent to preparation non-contextuality assumption in an
ontological model of an operational theory. For any given $n\geq 3$, if such a
constraint is imposed on the encoding scheme of the random-access code, then
the local bound of the Bell expression reduces to the preparation
non-contextual bound. We provide explicit examples for $n=4,5$ case to
demonstrate that the relevant Bell expressions certify the qubit and two-qubit
system, and for $n=6$ case, the relevant Bell expression certifies the qubit,
two-qubit and three-qubit systems. We further demonstrate the sharing of
quantum preparation contextuality by multiple Bobs sequentially to examine
whether number of Bobs sharing the preparation contextuality is dependent on
the dimension of the system. We provide explicit example of $n=5$ and $6$ to
demonstrate that number of Bobs sequentially sharing the contextuality remains
same for any of the $2^{m}$ dimensional systems.
