This paper is concerned with bounds for quantum error-correcting codes. Using the quantum MacWilliams (1972, 1977) identities, we generalize the linear programming approach from classical coding theory to the quantum case. Using this approach, we obtain Singleton-type, Hamming-type, and the first linear-programming-type bounds for quantum codes. Using the special structure of linear quantum codes, we derive an upper bound that is better than both Hamming and the first linear programming bounds on some subinterval of rates.
The quantum Hamming bound arises from a direct dimension estimate for exact quantum error-correcting codes. For nondegenerate codes, distinct correctable errors map the code space into orthogonal subspaces, so dimension counting directly gives the bound; degeneracy allows these subspaces to overlap and invalidates the argument. Although linear-programming, asymptotic, and structural approaches have advanced the problem from several directions, the general local-dimension case has remained open. Here we prove the quantum Hamming bound for every local dimension
q≥3. Building on the exact two-center Fourier reduction, we develop new overlap estimates for nonbinary Hamming spaces. For
q≥4, Jacobi estimates and monotonicity give a uniform gap; qutrits form the boundary case where the gap vanishes and are handled by a filtered correction. This completes the finite-length nonbinary case; together with the existing qubit result, it establishes the quantum Hamming bound for arbitrary local dimension.
Linear programming approaches have been applied to derive upper bounds on the size of classical and quantum codes. In this paper, we derive similar results for general quantum codes with entanglement assistance by considering a type of split weight enumerator. After deriving the MacWilliams identities for these enumerators, we are able to prove algebraic linear programming bounds, such as the Singleton bound, the Hamming bound, and the first linear programming bound. Our Singleton bound and Hamming bound are more general than the previous bounds for entanglement-assisted quantum stabilizer codes. In addition, we show that the first linear programming bound improves the Hamming bound when the relative distance is sufficiently large. On the other hand, we obtain additional constraints on the size of Pauli subgroups for quantum codes, which allow us to improve the linear programming bounds on the minimum distance of quantum codes of small length. In particular, we show that there is no [[27, 15, 5]] or [[28, 14, 6]] stabilizer code. We also discuss the existence of some entanglement-assisted quantum stabilizer codes with maximal entanglement. As a result, the upper and lower bounds on the minimum distance of maximal-entanglement quantum stabilizer codes with length up to 20 are significantly improved.
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.
In Part II we show that there exist quantum codes whose probability of undetected error falls exponentially with the length of the code and derive bounds on this exponent.The lower (existence) bound for stabilizer codes is proved by a counting argument for classical self-orthogonal quaternary codes. Upper bounds for any quantum codes are proved by linear programming. We present two general solutions of the LP problem. Together they give an upper bound on the exponent of undetected error. The upper and lower asymptotic bounds coincide for a certain interval of code rates close to 1.