The Determinant of a Class of Irreducible Polynomials over Z[ζm] Related to Lattice-Based Diagonal Space-Time Block Codes

DOI：10.16205/j.cnki.cama.2021.0012

 作者 单位 杨仕椿 阿坝师范学院数学学院, 四川\ 汶川 623002. 廖群英 四川师范大学数学科学学院,成都 610066.

为实现信号在空间的分集, 关于格的空时分组码的设计近年来备受关注.通过研究与对角的格空时码相关的$\mathbb{Z}[\zeta_{m}]$上的一类二次不可约多项式的判别式$|\Delta|$,确定了$\mathbb{Z}[\zeta_{m}]$上的格空时编码的正规分集乘积的大小.进而, 利用Pell方程的解的性质, 构造性地证明了$m=5, 8, 10, 12$时,$|\Delta|$的值可以任意小. 最后,提出几个关于$\mathbb{Z}[\zeta_{m}]$上的二次不可约和三次不可约多项式的判别式大小的猜想.

To achieve the diversity of the signal in space, the design of the case of spacetime block codes has attracted much attention in recent years. By studying the discriminant of a class of quadratic irreducible polynomials over Z[ζm] related to lattice-based diagonal space-time block codes, the authors determine the size of the normalized diversity product for constructing the lattice space time code over Z[ζm]. Furthermore, based on the property for solutions of the Pell equation, it is proved that the absolute value of the discriminant can be arbitrarily small when m = 5, 8, 10, 12. And then for the quadratic or cubic irreducible polynomials over Z[ζm], some problems to be further studied are proposed.