论文标题
在小有限场上的超规模矩阵
Superregular matrices over small finite fields
论文作者
论文摘要
矩阵的微不足道的零小调是莱布尼兹公式中的所有项，等于零。如果其所有未微不足道零的未成年人均非零，则矩阵是超规则的。在编码理论的领域中，有限场上的超规模矩阵与具有最佳距离专有的代码相连。当超规则矩阵具有所有条目非零时，称为完整的超规则，这些矩阵用于构建最大距离可分开的块代码。在卷积代码的背景下，采用下三角托管超规则矩阵来构建具有最佳柱距离的卷积代码。尽管已知在小场上的完整超规模矩阵（例如考奇矩阵），但具有较低三角形toeplitz结构的这些矩阵的少数已知常规结构需要非常大的田地尺寸。在这项工作中，我们研究了小小的有限原始场上的下三角toeplitz超规则矩阵。在Hutchinson，Smarandache和Trumpf的工作之后，我们研究了这种矩阵所具有的最少数量的非平凡未成年人数量，并展示了这种超规模矩阵的具体构造。
A trivially zero minor of a matrix is a minor having all its terms in the Leibniz formula equal to zero. A matrix is superregular if all of its minors that are not trivially zero are nonzero. In the area of Coding Theory, superregular matrices over finite fields are connected with codes with optimum distance proprieties. When a superregular matrix has all its entries nonzero, it is called full superregular and these matrices are used to construct Maximum Distance Separable block codes. In the context of convolutional codes, lower triangular Toeplitz superregular matrices are employed to build convolutional codes with optimal column distance. Although full superregular matrices over small fields are known (e.g. Cauchy matrices), the few known general constructions of these matrices having a lower triangular Toeplitz structure require very large field sizes. In this work we investigate lower triangular Toeplitz superregular matrices over small finite prime fields. Following the work of Hutchinson, Smarandache and Trumpf, we study the minimum number of different nontrivial minors that such a matrix have, and exhibit concrete constructions of superregular matrices of this kind.
