Complex Orthogonal Design (COD) codes are known to have the lowest detection complexity among Space-Time Block Codes (STBCs). However, the rate of square COD codes decreases exponentially with the number of transmit antennas. The Quasi-Orthogonal Design (QOD) codes emerged to provide a compromise between rate and complexity as they offer higher rates compared to COD codes at the expense of an increase of decoding complexity through partially relaxing the orthogonality conditions. The QOD codes were then generalized with the so called g-symbol and g-group decodable STBCs where the number of orthogonal groups of symbols is no longer restricted to two as in the QOD case. However, the adopted approach for the construction of such codes is based on sufficient but not necessary conditions which may limit the achievable rates for any number of orthogonal groups. In this paper, we limit ourselves to the case of Unitary Weight (UW)-g-group decodable STBCs for 2a transmit antennas where the weight matrices are required to be single thread matrices with non-zero entries ∈{± 1,± j} and address the problem of finding the highest achievable rate for any number of orthogonal groups. This special type of weight matrices guarantees full symbol-wise diversity and subsumes a wide range of existing codes in the literature. We show that in this case an exhaustive search can be applied to find the maximum achievable rates for UW-g-group decodable STBCs with g>;1. For this purpose, we extend our previously proposed approach for constructing UW-2-group decodable STBCs based on necessary and sufficient conditions to the case of UW-g-group decodable STBCs in a recursive manner.
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机译：众所周知，复杂的正交设计（COD）码在空时分组码（STBC）中具有最低的检测复杂度。但是，平方COD码的速率随着发射天线的数量呈指数下降。准正交设计（QOD）码的出现是为了在速率和复杂度之间做出折衷，因为与COD码相比，它们提供了更高的速率，但代价是通过部分放松正交性条件而增加了解码复杂度。然后使用所谓的g符号和g组可解码STBC来概括QOD代码，其中正交符号组的数量不再像QOD情况那样限制为两个。但是，采用的这种代码的构造方法是基于足够但不是必要的条件，该条件可能会限制任意数量的正交组的可达到速率。在本文中，我们将自己限制在2 a 发射天线的单位重量（UW）-g组可解码STBC的情况下，其中权重矩阵必须是具有非零项的单线程矩阵∈{±1，±j}并解决了为任意数量的正交组找到最高可实现速率的问题。这种特殊类型的权重矩阵可确保完全按符号进行分集，并包含文献中广泛的现有代码。我们表明，在这种情况下，可以应用穷举搜索来找到g>; 1的UW-g组可解码STBC的最大可实现速率。为此，我们以递归的方式将我们先前提出的基于必要和充分条件构建UW-2群可解码STBC的方法扩展到UW-g群可解码STBC的情况。
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