Subsystem codes are a generalization of noiseless subsystems, decoherence-free subspaces and stabilizer codes. We generalize the quantum Singleton bound to q -linear subsystem codes. It follows that no subsystem code over a prime field can beat the quantum Singleton bound. On the other hand, we show the remarkable fact that there exist impure subsystem codes beating the quantum Hamming bound. A number of open problems concern the comparison in the performance of stabilizer and subsystem codes. One of the open problems suggested by Poulin's work asks whether a subsystem code can use fewer syndrome measurements than an optimal q -linear maximum distance separable stabilizer code while encoding the same number of qudits and having the same distance. We prove that linear subsystem codes cannot offer such an improvement under complete decoding.
We prove a conjecture of Yu, Lai, and Oh: their strengthened quantum Hamming bounds, a family that interpolates between Hamming-type bounds and the Singleton bound, hold for all quantum codes of dimension
K>1, including impure and nonadditive codes. The result covers every integer local dimension
p≥2, not necessarily a prime power, every integer
t≥1, both distances
2t+1 and
2t+2, and every interpolation parameter
e∈{0,…,t}. The case
e=0 implies the ordinary quantum Hamming bound for general codes, recently proved by Zhang and Chen. Yu, Lai, and Oh proved their bounds for pure codes and for general codes of distance three or four. More generally, for every integer
p≥2 and every quantum code of length
n, dimension
K>1 and distance
d≥3, we prove
Kf(0)≤pnEf(X), where
X∼Bin(n,1−p−2), for every real polynomial
f of degree less than
d that is nonnegative on
{0,…,n} and positive at zero. For pure codes this inequality is implicit in the argument of Yu, Lai, and Oh. The proof uses the linear-programming method with quantum weight enumerators.
The quantum Hamming bound is the standard finite-length sphere-packing bound for exact correction of arbitrary qubit errors. Whether degeneracy can evade this bound has remained unresolved in full generality for nearly three decades: distinct correctable errors may act identically on the code space, so the usual disjoint-sphere argument breaks down. We prove that every exact binary quantum subspace code with
K>1 obeys the bound, without assuming either nondegeneracy or additivity. Our proof turns the Li--Xing linear-programming polynomial into an exact intersection count for quaternary Hamming balls. Monotonicity in block length and in ball-center separation then reduces the problem to a local node--edge charging inequality at the shortest admissible length. Thus degeneracy can merge correctable error sectors, but cannot enlarge the finite-length binary Hamming bound.