arXiv subjects
Publications and source records attributed to Manuel Albrizzio.
In this paper, we attempt to find the weight spectrum of the Reed-Muller codes $RM(m-7,m)$. In the process, we tried to fully determine the weight spectrum of $RM(7,14)$. We found almost all of the weights, except for a select number of them. The remaining eight missing weights are necessary to give a positive answer to an open question on the weight spectrum of $RM(m-c,m)$ for $c=7$.
We consider the action on $\mathbb{F}_2^n$ by cyclic permutations ($\mathbb{Z}/n\mathbb{Z}$). Two elements $x, y\in \mathbb{F}_2^n$ are in the same orbit if they are cyclic shifts of each other. Cryptographic properties of rotation symmetric Boolean functions can be efficiently computed using the square matrix $_n\mathcal{A}$, the construction of which uses orbit representatives of the cyclic shifting action. In 2018, Ciungu and Iovanov proved that $_n\mathcal{A}^2=2^n\cdot I$, the identity matrix of dimension $g_n\times g_n$ where $g_n$ is the number of orbits. In this paper, we answer the open question of the precise number of positive and negative eigenvalues of $_n\mathcal{A}$.