基于扩展Clifford代数表示的空时块集
STBCs from Representation of Extended Clifford Algebras
摘要：
Karmakar等人给出了构造$λ$ -实符号最大似然（ML）可译码空时码的一组充分条件，满足这些充分条件的空时码被称为Clifford酉重量（CUW）码。在本文中，CUW码的最大速率（如测量在每个信道使用的复杂符号）$λ=2^a,a\in\mathbb{N}$得到使用表示理论的工具。实现这一最大速率的代码的两个代数结构也提供了。其中一个构造是利用有限群的线性表示得到的，而另一个构造是基于非交换环上右模代数的概念。据作者所知，这是第一个文件中，矩阵的非交换环被用来构造STBC。对Tirkkonen等人提出的ABBA结构和Karmakar等人提出的张量积结构给出了代数解释，并证明了Tirkkonen等人基于ABBA结构提出的4发射天线空时分组码，如果设计变量被置换，信号集被选择为适当的维数，则实际上是一个单复数符号ML可译码码。
英文摘要：
A set of sufficient conditions to construct $λ$-real symbol Maximum Likelihood (ML) decodable STBCs have recently been provided by Karmakar et al. STBCs satisfying these sufficient conditions were named as Clifford Unitary Weight (CUW) codes. In this paper, the maximal rate (as measured in complex symbols per channel use) of CUW codes for $λ=2^a,a\in\mathbb{N}$ is obtained using tools from representation theory. Two algebraic constructions of codes achieving this maximal rate are also provided. One of the constructions is obtained using linear representation of finite groups whereas the other construction is based on the concept of right module algebra over non-commutative rings. To the knowledge of the authors, this is the first paper in which matrices over non-commutative rings is used to construct STBCs. An algebraic explanation is provided for the 'ABBA' construction first proposed by Tirkkonen et al and the tensor product construction proposed by Karmakar et al. Furthermore, it is established that the 4 transmit antenna STBC originally proposed by Tirkkonen et al based on the ABBA construction is actually a single complex symbol ML decodable code if the design variables are permuted and signal sets of appropriate dimensions are chosen.
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