This paper presents four classes of binary quantum codes with minimum distance 3 and 4, namely Class-I, Class-II, Class-III and Class-IV. The classes Class-I and Class-II are constructed based on self-dual orientable embeddings of the complete graphs K4r+1\documentclass[12pt]{minimal}
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\begin{document}$$K_{4r+1}$$\end{document} and K4s\documentclass[12pt]{minimal}
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\begin{document}$$K_{4s}$$\end{document} and by current graphs and rotation schemes. The parameters of two classes of quantum codes are [[2r(4r+1),2r(4r-3),3]]\documentclass[12pt]{minimal}
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\begin{document}$$[[2r(4r+1),2r(4r-3),3]]$$\end{document} and [[2s(4s-1),2(s-1)(4s-1),3]]\documentclass[12pt]{minimal}
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\begin{document}$$[[2s(4s-1),2(s-1)(4s-1),3]]$$\end{document}, respectively, where r≥1\documentclass[12pt]{minimal}
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\begin{document}$$r\ge 1$$\end{document} and s≥2\documentclass[12pt]{minimal}
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\begin{document}$$s\ge 2$$\end{document}. For these quantum codes, the code rate approaches 1 as r and s tend to infinity. The Class-III with minimum distance 4 is constructed by using self-dual embeddings of complete bipartite graphs. The parameters of this class are rs,(r-2)(s-2)2,4\documentclass[12pt]{minimal}
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\begin{document}$$\left[ \left[ rs,\frac{(r-2)(s-2)}{2},4\right] \right] $$\end{document}, where r and s are both divisible by 4. The proposed Class-IV is the minimum distance 3 and code length n=(2r+1)s2\documentclass[12pt]{minimal}
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\begin{document}$$n=(2r+1)s^{2}$$\end{document}. This class is constructed based on self-dual embeddings of complete tripartite graph Krs,s,s\documentclass[12pt]{minimal}
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\begin{document}$$K_{rs,s,s}$$\end{document}, and its parameters are [[(2r+1)s2,(rs-2)(s-1),3]]\documentclass[12pt]{minimal}
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\begin{document}$$[[(2r+1)s^{2},(rs-2)(s-1),3]]$$\end{document}, where r≥2\documentclass[12pt]{minimal}
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\begin{document}$$r\ge 2$$\end{document} and s≥2\documentclass[12pt]{minimal}
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\begin{document}$$s\ge 2$$\end{document}.
