It is known that it is possible to encode a logical qubit over many physical qubits such that it is immune to the effects of collective decoherence, and it is possible to perform universal quantum computation using these 'decoherence-free' qubits. However, current proposed methods of performing gates on these encoded qubits could be difficult to implement, or could take too much time to perform. Here we investigate whether exploiting ring-exchange interactions, which may be naturally present,

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... n simplify the implementation of these gates in any way. Using a ring exchange interaction, we have found a way to create a controlled-Z gate on the 4-qubit decoherence-free subspace and the 3-qubit decoherence-free subsystem using a sequence with 5 pulses. This could be useful in situations where simplicity is important or where ring exchange interactions are prominent. We also investigate how timing errors and magnetic field fluctuations affect the fidelity of this gate.