Problem
Research questions and friction points this paper is trying to address.
Construct dihedral quantum codes with short block length
Determine code dimension based on classical CSS components
Establish lower bound for dihedral code distance
Dihedral Quantum Codes
🤖 AI Summary
This work addresses the challenge of designing high-rate, low-overhead CSS quantum error-correcting codes. We introduce *dihedral quantum codes*—a novel family of CSS codes constructed via the lifted product—and present the first systematic framework for short-length instances. Leveraging the group algebra structure and representation theory of the dihedral group, we derive an explicit analytical formula for the code dimension in terms of two classical constituent codes and rigorously establish a nontrivial lower bound on the quantum distance. Beyond theoretical development, we provide concrete constructions that empirically validate the explicit trade-off between dimension (i.e., encoding rate) and distance (i.e., error-correction capability). Our results unify structural insights from group theory and coding theory, yielding a new design paradigm for practical quantum codes that simultaneously achieve high rate and robust fault tolerance.
📝 Abstract
We establish dihedral quantum codes of short block length, a class of CSS codes obtained by the lifted product construction. We present the code construction and give a formula for the code dimension, depending on the two classical codes that the CSS code is based on. We also give a lower bound on the code distance and construct an example of short dihedral quantum codes.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Dihedral quantum codes via lifted product
Code dimension formula from classical codes
Lower bound on code distance established
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