The concept of asymmetric entanglement-assisted quantum error-correcting code (asymmetric EAQECC) is introduced in this article. Codes of this type take advantage of the asymmetry in quantum errors since phase-shift errors are more probable than qudit-flip errors. Moreover, they use pre-shared entanglement between encoder and decoder to simplify the theory of quantum error correction and increase the communication capacity. Thus, asymmetric EAQECCs can be constructed from any pair of classical linear codes over an arbitrary field. Their parameters are described and a Gilbert-Varshamov bound is presented. Explicit parameters of asymmetric EAQECCs from BCH codes are computed and examples exceeding the introduced Gilbert-Varshamov bound are shown.
The Gilbert--Varshamov (GV) bound is a central benchmark in coding theory, establishing existential guarantees for error-correcting codes and serving as a baseline for both Hamming and quantum fault-tolerant information processing. Despite decades of effort, improving the GV bound is notoriously difficult, and known improvements often rely on technically heavy arguments and do not extend naturally to the quantum setting due to additional self-orthogonality constraints. In this work we develop a concise probabilistic method that yields an improvement over the classical GV bound for q-ary linear codes. For relative distance δ=d/n<1−1/q, we show that an [n,k,d]q linear code exists whenever q−1qk−1<Volq(n,d−1)cδnqn, for positive constant cδ depending only on δ, where Volq(n,d−1) denotes the volume of a q-ary Hamming ball. We further adapt this approach to the quantum setting by analyzing symplectic self-orthogonal structures. For δ<1−1/q2, we obtain an improved quantum GV bound: there exists a q-ary quantum code [[n,n−k,d]] provided that q−1q2n−k−1<∑i=0d−1(in)(q2−1)icδn⋅q2n. In particular, our result improves the standard quantum GV bound by an Ω(n) multiplicative factor.