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

TL;DR

This paper provides a new proof for the nonexistence of certain complex orthogonal designs with specific parameters, using the uniqueness of related designs and the impossibility of extending them with additional columns.

Contribution

The paper introduces an alternative proof for the nonexistence of specific CODs, based on the uniqueness and extension properties of related orthogonal designs.

Findings

Confirmed the nonexistence of COD with parameters $[inom{2m}{m-1}, 2m, inom{2m-1}{m-1}]$ for odd m.

Established the uniqueness of $[inom{2m}{m-1}, 2m-1, inom{2m-1}{m-1}]$ under equivalence.

Proved that adding an extra orthogonal column to the smaller design is impossible for odd m.

Abstract

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 $[\binom{2m}{m-1}, 2m, \binom{2m-1}{m-1}]$.