项目标书摘要：我们介绍了一种通过理想仿酉矩阵(desired para-unitary(PU))来构造大小为N的q-ary完全互补码(CCC)和互补序列集(CSS)的方法。介绍了种子仿酉矩阵(seed PU matrices)。并且给出了一种由种子仿酉矩阵的代数表达式得到所构造CCC和CSS的代数表达式的系统方法。这些代数表达式的一般形式仅仅依赖于从 Z_N到 Z_q的基本型和Butson型哈达玛矩阵(Butson type Hadamard(BH)matrices)的等价类中的代表集。特别是，从我们给出的一般形式中得到的格雷对恰好与标准格雷对一致。我们得出的互补集大小为3的三元互补序列有我们首次发现。我们得出的互补集大小为4的四元互补序列也几乎都是之前从未被发现的。进一步，我们给出了一般化的种子仿酉矩阵和迭代的构造方法，并相应给出了很多新的CSS和CCC的构造。从我们研究成果的角度看来，之前文献中已知的CSS和CCC的结果都是源自2阶Walsh矩阵。那么如果运用了更高阶的哈达玛矩阵进行构造，得到的低峰均比序列的数量将会指数级增长。最后研究所设计序列集的纠错能力和编解码算法，实现同时纠错和降低峰均比。此外，我们给出了一种构造长度为 2^m的4^q-QAM上的Golay互补序列。序列以q个QPSK上的标准Golay序列的加权和的形式给出。之前的Cases I-V可证明是我们构造的特殊情况。如果q是合数，会有相当多 generalized cases I-V 之外的Golay序列产生。
Application Abstract: A new method to construct$q$-ary complementary sequence sets(CSSs)and complete complementary codes(CCCs)of size$N$is proposed by using desired para-unitary(PU)matrices.The concept of seed PU matrices is introduced and a systematic approach on how to compute the explicit forms of the functions in constructed CSSs and CCCs from the seed PU matrices is given.A general form of these functions only depends on a basis of the functions from$\\Z_N$to$\\Z_q$and representatives in the equivalent class of Butson-type Hadamard(BH)matrices.Especially,the realization of Golay pairs from the our general form exactly coincides with the standard Golay pairs.The realization of ternary complementary sequences of size$3$is first reported here.For the realization of the quaternary complementary sequences of size 4,almost all the sequences derived here are never reported before.Generalized seed PU matrices and the recursive constructions of the desired PU matrices are also studied,and a large number of new constructions of CSSs and CCCs are given accordingly.From the perspective of this paper,all the known results of CSSs and CCCs with explicit GBF form in the literature(except non-standard Golay pairs)are constructed from the Walsh matrices of order 2.This suggests that the proposed method with the BH matrices of higher orders will yield a large number of new CSSs and CCCs with the exponentially increasing number of the sequences of low peak-to-mean envelope power ratio.What's more,we present two new constructions for$4^q$-QAM GCSs of length$2^{m}$,where the proposed sequences are also represented as the weighted sum of$q$quaternary standard GCSs.It is shown that the generalized cases I-V are special cases of these two constructions.In particular,if$q$is a composite number,a great number of new GCSs other than the sequences in the generalized cases I-V will arise.
