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In this paper, we present a computer-supported method of searching for quantum caps. By means of this method and relevant knowledge of combinatorics, many quantum caps in <inline-formula> <tex-math notation="LaTeX">$PG(3,9)$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$PG(4,9)$ </tex-math></inline-formula> are constructively proven to exist. Then, according to the theorem that each quantum cap corresponds to a quantum error-correcting code with <inline-formula> <tex-math notation="LaTeX">$d=4$ </tex-math></inline-formula>, we obtain 278 quantum error-correcting codes. Most of these results break the GV bound, and a number of them are optimal quantum codes or have improved parameters.

New Quantum Codes Constructed by Quantum Caps in PG(3,9) and PG(4,9)