关于$[\binom{2m}{m-1}, 2m, \binom{2m-1}{m-1}]$、$m$奇、复正交设计的不存在性
On the nonexistence of $[\binom{2m}{m-1}, 2m, \binom{2m-1}{m-1}]$, $m$ odd, complex orthogonal design
摘要：
利用复正交设计（CODs）构造空时分组码。具有参数$[p, n, k]$的COD $\mathcal{O}_z$是$p\times n$矩阵，其中非零条目由$\pm z_i$或$\pm z^*_i$、$i = 1, 2,..., k$填充，使得$\mathcal{O}^H_z \mathcal{O}_z = (|z_1|^2+|z_2|^2+...+|z_k|^2)I_{n \times n}$。Adams等人在“The final case of the decoding delay problem for maximum rate complex orthogonal designs”，IEEE Trans. Inf. Theory，vol.56，no.1，pp. 103-122，2010年1月，首次证明了不存在$[\binom{2m}{m-1}, 2m, \binom{2m-1}{m-1}]$，$m$奇数，COD。结合前面的解码延迟应为$\binom{2m}{m-1}$的整数倍的结果，解决了最大速率复正交设计的解码延迟问题的最终情形$n \equiv 2 \pmod 4$。本文给出了参数为$[\binom{2m}{m-1}, 2m, \binom{2m-1}{m-1}]$，$m$奇的COD不存在的另一个证明.我们的新证明是基于等价运算下$[\binom{2m}{m-1}, 2m-1, \binom{2m-1}{m-1}]$的唯一性，其中提出了一个显式表示，以帮助证明。然后，通过证明当$m$为奇数时，不可能在COD $[\binom{2m}{m-1}, 2m-1, \binom{2m-1}{m-1}]$上增加额外的正交列，完成了COD $[\binom{2m}{m-1}, 2m, \binom{2m-1}{m-1}]$不存在的证明.
英文摘要：
Complex orthogonal designs (CODs) are used to construct space-time block codes. 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 case of the decoding delay problem for maximum rate complex orthogonal designs," IEEE Trans. Inf. Theory, vol. 56, no. 1, pp. 103-122, Jan. 2010, first proved the 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 integer multiple 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 with parameter $[\binom{2m}{m-1}, 2m, \binom{2m-1}{m-1}]$, $m$ odd. Our new proof is based on the uniqueness of $[\binom{2m}{m-1}, 2m-1, \binom{2m-1}{m-1}]$ under equivalence operation, where an explicit-form representation is proposed to help the proof. Then, by proving it's impossible to add an extra orthogonal column on COD $[\binom{2m}{m-1}, 2m-1, \binom{2m-1}{m-1}]$ when $m$ is odd, we complete the proof of the nonexistence of COD $[\binom{2m}{m-1}, 2m, \binom{2m-1}{m-1}]$.
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