This note is an erratum for the paper On the Inequalities of Projected Volumes and the Constructible Region [ SIAM J. Discrete Math., 33 (2019), pp. 694--711]. There is a mistake in the proof of Theorem 4 in Section 3. However, Theorem 4 is itself correct (a proof can be found in [ Inequalities on Projected Volumes, arxiv.org/abs/1909.12858, 2019] by Leader, Randelović, and Räty), and all the other results of the paper stand.

We study the following geometry problem: given a $2^n-1$ dimensional vector $\pi=\{\pi_S\}_{S\subseteq [n], S\ne \emptyset}$, is there an object $T\subseteq\mathbb{R}^n$ such that $\log(\mathrm{vol}(T_S))= \pi_S\,\,\forall S\subseteq [n]$, where $T_S$ is the projection of $T$ onto the subspace spanned by the axes in $S$ and $\mathrm{vol}(T_S)$ is its $|S|$-dimensional volume? If $\pi$ does correspond to an object in $\mathbb{R}^n$, we say that $\pi$ is constructible. We use $\Psi_n$ to denote the constructible region, i.e., the set of all constructible vectors in $\mathbb{R}^{2^n-1}$. In 1995, Bollobás and Thomason showed that $\Psi_n$ is contained in a polyhedral cone and defined a class of so-called uniform-cover inequalities. We propose a new set of inequalities, called nonuniform-cover inequalities, which generalizes the uniform-cover inequalities. We show that any linear inequality that all points in $\Psi_n$ satisfy must be a nonuniform-cover inequality. Based on this result and an example by Bollobás and Thomason, we show that the constructible region $\Psi_n$ is nonconvex for $n\geq 4$ and thus cannot be fully characterized by linear inequalities. We further show that some subclasses of the nonuniform-cover inequalities are not satisfied by all constructible vectors via various combinatorial constructions, which refutes a previous conjecture about $\Psi_n$. Finally, we conclude with an interesting conjecture regarding the convex hull of $\Psi_n$.

Cyclic codes with two zeros and their dual codes have been a subject of study for many years. However, their weight distributions are known only for a few cases. In this paper, the weight distributions of the duals of the cyclic codes with two zeros are settled for a few more cases.

Cyclic codes with two zeros and their dual codes have been a subject of study for many years. However, their weight distributions are known only for a few cases. In this paper, the weight distributions of the duals of the cyclic codes with two zeros are settled for a few cases. The weight distributions of punctured versions of these codes are also determined for several special cases.