论文标题
总和BCH代码和循环旋转循环代码
Sum-Rank BCH Codes and Cyclic-Skew-Cyclic Codes
论文作者
论文摘要
在这项工作中，引入了循环旋转循环代码和总和BCH代码。环状循环代码的特征是合适的非交通有限环的左理想，该理想是在多项式（反之亦然）上使用偏斜多项式构建的。找到了这种左派理想的单发器，它们用于构造相应代码的生成器矩阵。介绍了定义集的概念，使用对poynomials顶部的偏斜多项式的根对。对于循环旋转循环代码，给出了最小总和级距离上的下限（称为总和BCH结合），其定义集包含某些连续对。然后将具有规定最小总和距离的总和BCH代码定义为最大的循环旋转环境代码，其定义集包含此类连续对。描述了总和BCH代码的定义集，并获得了其尺寸的下限。因此，提供了表格，表明总和BCH代码击败了以前已知的代码，以二进制$ 2 \ times 2 $矩阵（即其代码字列表$ 2 \ times 2 $二进制矩阵的列表，用于与代码长度相对应的较宽列表长度的代码）。最后，获得了总和BCH代码的解码器，最多可获得一半的规定距离。
In this work, cyclic-skew-cyclic codes and sum-rank BCH codes are introduced. Cyclic-skew-cyclic codes are characterized as left ideals of a suitable non-commutative finite ring, constructed using skew polynomials on top of polynomials (or vice versa). Single generators of such left ideals are found, and they are used to construct generator matrices of the corresponding codes. The notion of defining set is introduced, using pairs of roots of skew polynomials on top of poynomials. A lower bound (called sum-rank BCH bound) on the minimum sum-rank distance is given for cyclic-skew-cyclic codes whose defining set contains certain consecutive pairs. Sum-rank BCH codes, with prescribed minimum sum-rank distance, are then defined as the largest cyclic-skew-cyclic codes whose defining set contains such consecutive pairs. The defining set of a sum-rank BCH code is described, and a lower bound on its dimension is obtained. Thanks to it, tables are provided showing that sum-rank BCH codes beat previously known codes for the sum-rank metric for binary $ 2 \times 2 $ matrices (i.e., codes whose codewords are lists of $ 2 \times 2 $ binary matrices, for a wide range of list lengths that correspond to the code length). Finally, a decoder for sum-rank BCH codes up to half their prescribed distance is obtained.
