Optimal Locality and Parameter Tradeoffs for Subsystem Codes (2503.22651v1)

Abstract: We study the tradeoffs between the locality and parameters of subsystem codes. We prove lower bounds on both the number and lengths of interactions in any $D$-dimensional embedding of a subsystem code. Specifically, we show that any embedding of a subsystem code with parameters $[[n,k,d]]$ into $\mathbb{R}^D$ must have at least $M^*$ interactions of length at least $\ell^*$, where [ M^* = \Omega(\max(k,d)), \quad\text{and}\quad \ell^* = \Omega\bigg(\max\bigg(\frac{d}{n^\frac{D-1}{D}}, \bigg(\frac{kd^\frac{1}{D-1}}{n}\bigg)^\frac{D-1}{D}\bigg)\bigg). ] We also give tradeoffs between the locality and parameters of commuting projector codes in $D$-dimensions, generalizing a result of Dai and Li. We provide explicit constructions of embedded codes that show our bounds are optimal in both the interaction count and interaction length.