论文标题
全部重量枚举者和二进制自动偶数代码的新平衡原理的衍生物
Derivatives of Complete Weight Enumerators and New Balance Principle of Binary Self-Dual Codes
论文作者
论文摘要
令H为二级订单的标准HADAMARD矩阵，令K = 2^{ - 1/2} h。众所周知，二进制二进制$ n $的二进制自d $ $ n $的全部重量枚举$ \ w $是一个特征向量，对应于kronecker power $ k^{[n]}的特征值1。 $ w _ {<t>} $位于$ \ 1 $的特征空间中，矩阵$ k^{[n-t]}。我们计算了长度为24的长度24的订单$ n-5 $的导数，长度为48的延长二次残留守则以及推定的[72,24,12]代码，并表明它们在$ \ 1 $ $ \ 1 $的矩阵$ k^{5]}的特征范围内，我们使用范围的范围。重量在选定的坐标位置中具有1个。例如，我们使用平衡方程式消除了一些枚举二进制自偶代码的枚举者的候选者。
Let H be the standard Hadamard matrix of order two and let K=2^{-1/2}H. It is known that the complete weight enumerator $\ W$ of a binary self-dual code of length $n$ is an eigenvector corresponding to an eigenvalue 1 of the Kronecker power $K^{[n]}.$ For every integer $t$ in the interval [0,n] we define the derivative of order $t$, $W_{<t>},$ of $W$ in such a way that $W_{<t>}$ is in the eigenspace of $\ 1$ of the matrix $K^{[n-t]}.$ For large values of $t,$ $W_{<t>}$ contains less information about the code but has smaller length while $W_{<0>}=W$ completely determines the code. We compute the derivative of order $n-5$ for the extended Golay code of length 24, the extended quadratic residue code of length 48, and the putative [72,24,12] code and show that they are in the eigenspace of $\ 1$ of the matrix $% K^{[5]}.$ We use the derivatives to prove a new balance equation which involves the number of code vectors of given weight having 1 in a selected coordinate position. As an example, we use the balance equation to eliminate some candidates for weight enumerators of binary self-dual codes of length eight.
