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Complex orthogonal designs (CODs) are used to construct space-time blockcodes. COD $\mathcal{O}_z$ with parameter $[p, n, k]$ is a $p\times n$ matrix,where nonzero entries are filled by $\pm z_i$ or $\pm z^*_i$, $i = 1, 2,...,k$, such that $\mathcal{O}^H_z \mathcal{O}_z =(|z_1|^2+|z_2|^2+...+|z_k|^2)I_{n \times n}$. Adams et al. in "The final caseof the decoding delay problem for maximum rate complex orthogonal designs,"IEEE Trans. Inf. Theory, vol. 56, no. 1, pp. 103-122, Jan. 2010, first provedthe nonexistence of $[\binom{2m}{m-1}, 2m, \binom{2m-1}{m-1}]$, $m$ odd, COD.Combining with the previous result that decoding delay should be an integermultiple of $\binom{2m}{m-1}$, they solved the final case $n \equiv 2 \pmod 4$of the decoding delay problem for maximum rate complex orthogonal designs.  In this paper, we give another proof of the nonexistence of COD withparameter $[\binom{2m}{m-1}, 2m, \binom{2m-1}{m-1}]$, $m$ odd. Our new proof isbased on the uniqueness of $[\binom{2m}{m-1}, 2m-1, \binom{2m-1}{m-1}]$ underequivalence operation, where an explicit-form representation is proposed tohelp the proof. Then, by proving it's impossible to add an extra orthogonalcolumn on COD $[\binom{2m}{m-1}, 2m-1, \binom{2m-1}{m-1}]$ when $m$ is odd, wecomplete the proof of the nonexistence of COD $[\binom{2m}{m-1}, 2m,\binom{2m-1}{m-1}]$.

On the nonexistence of $[\binom{2m}{m-1}, 2m, \binom{2m-1}{m-1}]$, $m$ odd, complex orthogonal design