论文标题
三阶芦苇毛刺代码RM（3,7）的覆盖半径为20
The Covering Radius of the Third-Order Reed-Muller Code RM(3,7) is 20
论文作者
论文摘要
我们证明了三阶芦苇毛刺代码RM（3,7）的覆盖半径为20，以前已知在20至23之间（包含在内）。 RM的覆盖半径（3，7）是所有7个可添加布尔函数中最大的三阶非线性。众所周知，存在具有三阶非线性20的7个可变性布尔函数。我们证明，三阶非线性无法实现21。根据RM（6,6）/RM（3,6）商的分类，我们将所有7- Variable Boolean函数分类为66种。首先，我们证明62种类型（66种）不能具有三阶非线性21；其次，如果其三阶非线性为21，我们证明其余4种类型的功能可以转换为类型（6，10）函数。最后，我们将类型（6，10）的功能转换为特定形式，并证明该形式的功能无法实现三阶非线性21（在计算机的帮助下）。顺便说一句，我们证明了任何有限字段上的仿射转换组可以由两个元素生成。
We prove the covering radius of the third-order Reed-Muller code RM(3,7) is 20, which was previously known to be between 20 and 23 (inclusive). The covering radius of RM(3, 7) is the maximum third-order nonlinearity among all 7-variable Boolean functions. It was known that there exist 7-variable Boolean functions with third-order nonlinearity 20. We prove the third-order nonlinearity cannot achieve 21. According to the classification of the quotient space of RM(6,6)/RM(3,6), we classify all 7-variable Boolean functions into 66 types. Firstly, we prove 62 types (among 66) cannot have third-order nonlinearity 21; Secondly, we prove function of the remaining 4 types can be transformed into a type (6, 10) function, if its third-order nonlinearity is 21; Finally, we transform type (6, 10) functions into a specific form, and prove the functions in that form cannot achieve third-order nonlinearity 21 (with the assistance of computers). By the way, we prove that the affine transformation group over any finite field can be generated by two elements.
