  Transversal gates are logical gate operations on encoded quantum information
that are efficient in gate count and depth, and are designed to minimize error
propagation. Efficient encoding circuits for quantum codes that admit
transversal gates are thus crucial to reduce noise and realize useful quantum
computers. The class of punctured Quantum Reed-Muller codes admit transversal
gates. We construct resource efficient recursive encoders for the class of
quantum codes constructed from Reed-Muller and punctured Reed-Muller codes.
These encoders on $n$ qubits have circuit depth of $O(\log n)$ and lower gate
counts compared to previous works. The number of CNOT gates in the encoder
across bi-partitions of the qubits is found to be equal to the entanglement
entropy across these partitions, demonstrating that the encoder is optimal in
terms of CNOT gates across these partitions. Finally, connecting these ideas,
we explicitly show that entanglement can be extracted from QRM codewords.
