Quantum Hamming bound

Sphere-packing parameter bound for nondegenerate binary quantum codes.

Specification version 1Published 2010-12-091 evidence records

About this problem

The question, scope, and sources behind this Registry record.

Formal question

Bound parameters of a nondegenerate [[n,k,d]] binary quantum code by counting mutually orthogonal correctable Pauli errors.

Scope & constraints

code: nondegenerate [[n,k,d]]
alphabet: qubits
correctableErrors: t=floor((d-1)/2)
degenerateCodesExcluded: true

Primary sources

Quantum Computation and Quantum Information — Quantum Hamming bound <= Chapters 2, 6, 10 and 12

What we know

Current frontiers derived from accepted Claims.

KNOWN LOWER BOUNDKNOWN UPPER BOUND

Unknown

OPEN FRONTIER

sum_{j=0}^t 3^j binom(n,j) <= 2^(n-k), t=floor((d-1)/2)

Lower boundGap: UnknownUpper bound

Claims

Assertions tied to evidence, attribution, and review.

2026

CLM-QUANT-003313-1<= sum_{j=0}^t 3^j binom(n,j) <= 2^(n-k), t=floor((d-1)/2)UPPER BOUNDThe cited theorem establishes sum_{j=0}^t 3^j binom(n,j) <= 2^(n-k), t=floor((d-1)/2) under the recorded assumptions.Source: Quantum Computation and Quantum Information — Quantum Hamming bound <= ↗

PROVEN

Timeline

The frontier as it changed over time.

How this frontier is derived

Only accepted Claims matching the current specification contribute to the displayed bounds. Strict inequalities remain open; contradictory Claims require editorial review.

—No breakthrough events recorded yetEvents appear here when an accepted Claim tightens a bound or closes the frontier.