Barron Han

Barron Han, Victoria Kostina, Babak Hassibi

We study online linear-quadratic regulation (LQR) with unknown dynamics under communication rate constraints. Classical networked control quantizes the plant state at every time step, requiring $O(T)$ total bits while injecting persistent quantization noise that limits control performance. We consider a setting where the plant observes its state locally and can estimate system dynamics via ordinary least squares, while a remote controller possesses knowledge of the control cost. Rather than quantizing the raw state, the plant transmits learned dynamics estimates over a rate-limited uplink, and the controller returns the optimal control policy so that the plant can compute actions locally using its superior state knowledge. We first prove a fundamental information-theoretic lower bound: any scheme achieving $O(T^α)$ regret for $α\in [1/2,1)$ compared to the optimal infinite horizon LQR controller that knows the true system dynamics must transmit at least $Ω(\log T)$ bits. We then design the \textbf{Quantized Certainty Equivalent (QCE-LQR)} algorithm, which matches this bound. The resulting regret bound contains inflation factors $Q_{\mathrm{slow}}(\varrho)$ and $Q_{\mathrm{fast}}(\varrho)$ that vanish as the codebook resolution increases, smoothly recovering the unquantized baseline regret. Numerical experiments on four benchmark systems -- from a scalar unstable plant to a 24-parameter Boeing 747 lateral model -- confirm that a variant of QCE-LQR achieves regret comparable to an unquantized certainty equivalent controller over a horizon of $T=10{,}000$ steps.

Barron Han, Hikmet Yildiz, Babak Hassibi

We study linear codes that maximize minimum distance subject to arbitrary support constraints on the parity-check matrix. Such constraints arise naturally in the design of LDPC codes, locally repairable codes, and hardware-constrained systems where each parity check must involve only a limited number of code symbols. They are also essential in quantum error correction, where sparse stabilizers reduce measurement noise and respect the connectivity constraints of physical qubit architectures. We derive the optimal minimum distance possible given support constraints on the parity-check matrix and show it is achievable over sufficiently large fields. When this maximum distance coincides with the Singleton bound for unconstrained parity check matrices, the dual GM-MDS construction yields generalized Reed--Solomon codes obeying the mask. In the generator-matrix setting, the GM-MDS theorem guarantees that the optimal distance can always be achieved by a subcode of a generalized Reed--Solomon code while satisfying arbitrary support constraints. We show that this is not true for the parity-check setting. We exhibit a set of support constraints, derived from the vertex-edge incidence of $K_{6,6}$, for which the optimal minimum distance cannot be realized by any subcode of a generalized Reed--Solomon code over any field. We also analyze structured constraint families -- regular, balanced, and cyclic masks -- through numerical optimization, providing design guidance for practical code constructions.