Since Shor proposed the first quantum error-correcting code [1], the theory of quantum error correction has been extensively developed. Today, quantum stabilizer codes [2, 3, 4, 5, 6] are the most widely-used class of quantum error-correcting codes. One reason for this is that the CSS and CRSS code constructions [7, 8, 2, 3] allow classical self-orthogonal codes to be easily transformed into quantum stabilizer codes. Bowen constructed the first entanglement-assisted quantum error-correcting (EAQEC) code from a three-qubit bit-flip code with the help of two pairs of maximally-entangled states [9]. Bowen’s code, which can correct an arbitrary one-qubit error, serves as an example that entanglement increases the error-correcting ability of quantum error-correcting codes. Brun, Devetak and Hsieh showed that if shared entanglement between the encoder and decoder is available, classical linear quaternary (and binary) codes that are not self-orthogonal can be transformed to EAQEC codes [10, 11]. An [ n, k, d ] classical linear quaternary code encodes k quaternary information digits into n quaternary digits and can correct up to ⌊ $^{d}$− 1 2 ⌋ quaternary digit errors, where d is called the minimum distance of the code. Brun, Devetak and Hsieh showed that an [ n, k, d ] classical linear quaternary code can be transformed to an [[ n, 2 k − n + c, d ; c ]] EAQEC code that encodes 2 k − n + c information qubits into n qubits with the help of c pairs of maximally- entangled states (ebits) for some c [10]. This EAQEC code can correct up to ⌊ $^{d}$− 1 2 ⌋ qubit errors and has the same minimum distance d as the classical code. If entanglement is used, it boosts the rate of the code. However, it has not been explored how entanglement can instead help increase the minimum distance. In addition, given parameters n, k, c , it is not clear how to construct an [[ n, k, d ; c ]] EAQEC code directly. We will answer these questions in the paper. An EAQEC code can be obtained from a standard QECC by changing one or more an- 2 1 Introduction