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On the Minimum Decoding Delay of Balanced Complex Orthogonal Design

Complex orthogonal design (COD) with parameter $[p, n, k]$ is a combinatorial design used in space-time block codes (STBCs). For STBC, $n$ is the number of antennas, $k/p$ is the rate, and $p$ is the decoding delay. A class of rate $1/2$ COD called balanced complex orthogonal design (BCOD) has been proposed by Adams et al., and they constructed BCODs with rate $k/p = 1/2$ and decoding delay $p = 2^m$ for $n=2m$. Furthermore, they prove that the constructions have optimal decoding delay when $m$ is congruent to $1$, $2$, or $3$ module $4$. They conjecture that for the case $m \equiv 0 \pmod 4$, $2^m$ is also a lower bound of $p$. In this paper, we prove this conjecture.