All the stabilizer codes of distance 3
Sixia Yu, Juergen Bierbrauer, Ying Dong, Qing Chen, and C.H. Oh

TL;DR
This paper characterizes all stabilizer codes of distance 3 for qubits, providing necessary and sufficient conditions for their existence, and offers explicit constructions for optimal codes at any length.
Contribution
It establishes exact existence criteria for stabilizer codes of distance 3 and provides explicit constructions for optimal codes at arbitrary lengths.
Findings
Necessary and sufficient conditions for stabilizer codes of distance 3.
Explicit construction methods for optimal codes.
Complete classification of such codes based on length.
Abstract
We give necessary and sufficient conditions for the existence of stabilizer codes of distance 3 for qubits: where if or for some integer and otherwise. Or equivalently, a code exists if and only if for even and for odd . Given an arbitrary length we present an explicit construction for an optimal quantum stabilizer code of distance 3 that saturates the above bound.
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Taxonomy
TopicsQuantum Computing Algorithms and Architecture · Coding theory and cryptography · graph theory and CDMA systems
