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In this paper we show that Calderbank-Shor-Steane (CSS) codes with alphabet $q\\geq 5$ cannot beat the quantum Hamming bound. We prove a quantum version of the Griesmer bound for the CSS codes which allows us to strengthen the Rains' bound that an $[[n,k,d]]_2$ code cannot correct more than $\\floor{(n+1)/6}$ errors to $\\floor{(n-k+1)/6}$. 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