Code Parameters of Principal Ideals of Group Algebra of Group $(2,2,\dots,2)$ over Field of Characteristic 2

Assume that $G$ is the direct product of $m$ groups of order $2,\pi$ is a field of two elements; and $G$ is a group algebra of $G$ over $\pi$. Nonzero element $u\in\pi G$ is called nondecreasable if the code distance of the principal ideal $(u)$ is equal to the weight of element $u$. Necessary and sufficient nondecreasability conditions are derived for all elements of a second-order RM code and for those elements of a third-order RM code that are specified by a pair of quadratic forms over $\pi$. It is shown that those elements of an RM code of arbitrary order whose weight does not exceed twice the code distance of the smallest RM code containing the given element are nondecreasable.