No ((n, k, d < 127)) code can violate the quantum Hamming bound

TL;DR

This paper proves that all quantum error-correcting codes with parameters ((n,k,d < 127)) adhere to the quantum Hamming bound, resolving a long-standing open question in quantum coding theory.

Contribution

It combines analytical and numerical bounds to demonstrate that impure quantum codes cannot violate the quantum Hamming bound.

Findings

No ((n,k,d < 127)) code violates the bound

Impure codes are constrained by the quantum Hamming bound

Supports the universality of the quantum Hamming bound for certain code parameters

Abstract

It is well-known that pure quantum error correcting codes (QECCs) are constrained by a quantum version of the Hamming bound. Whether impure codes also obey such a bound, however, remains a long-standing question with practical implications for the efficacy of QECCs. We employ a combination of previously derived bounds on QECCs to demonstrate that a subset of all codes must obey the quantum Hamming bound. Specifically, we combine an analytical bound due to Rains with a numerical bound due to Li and Xing to show that no ((n,k,d < 127)) code can violate the quantum Hamming bound.