Woodin has demonstrated that, in the presence of large cardinals, there is a single model of ZFC which is maximal for Π_{2}-sentences over the structure ($H$($ω$_{2}), ∈, NS_{$ω$1}), in the sense that its ($H$($ω$_{2}), ∈, NS_{$ω$1}) satisfies every Π_{2}-sentence$σ$for which ($H$($ω$_{2}), ∈, NS_{$ω$1}) ⊨$σ$can be forced by set-forcing. In this paper we answer a question of Woodin by showing that there are two Π_{2}-sentences over the structure ($H$($ω$_{2}), ∈,$ω$_{1}) which can each be forced to hold along with the continuum hypothesis, but whose conjunction implies $$ {2^{{{\aleph_0}}}}={2^{{{\aleph_1}}}} $$ . In the process we establish that there are two preservation theorems for not introducing new real numbers by a countable support iterated forcing which cannot be subsumed into a single preservation theorem.

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Let $W$ be an extended affine Weyl group. We prove that the minimal length elements $w_{\co}$ of any conjugacy class $\co$ of $W$ satisfy some nice properties, generalizing results of Geck and Pfeiffer \cite{GP} on finite Weyl groups. We also study a special class of conjugacy classes, the straight conjugacy classes. These conjugacy classes are in a natural bijection with the Frobenius-twisted conjugacy classes of some $p$-adic group and satisfy additional interesting properties. Furthermore, we discuss some applications to the affine Hecke algebra $H$. We prove that $T_{w_\co}$, where $\co$ ranges over all the conjugacy classes of $W$, forms a basis of the cocenter $H/[H, H]$. We also introduce the class polynomials, which play a crucial role in the study of affine Deligne-Lusztig varieties \cite{H99}.

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Let $q=p^{e}$ be a prime power and $\ell$ be an integer with $0\leq \ell\leq e-1$. The $\ell$-Galois hull of classical linear codes is a generalization of the Euclidean hull and Hermitian hull. We provide a necessary and sufficient condition under which a codeword of a GRS code or an extended GRS code belongs to its $\ell$-Galois dual code, generalizing both the Euclidean case and Hermitian case in the literature. By using four different tools: 1) the norm mapping from $\mathbb{F}_{q}^{\ast}$ to $\mathbb{F}_{p^{\ell}}^{\ast}$; 2) the direct product of two cyclic subgroups; 3) the coset decomposition of a cyclic group; 4) an additive subgroup of $\mathbb{F}_{q}$ and its cosets, we construct eleven families of $q$-ary MDS codes with $\ell$-Galois hulls of arbitrary dimensions, and give the related eleven families of $[[n,k,d;c]]_{q}$ entanglement-assisted quantum error-correcting codes (EAQECCs) with relatively large minimum distance in the sense that $2d=n-k+2+c$. We show that developing the theory of $\ell$-Galois hulls of $q$-ary MDS codes in this paper enables us to obtain new $q$-ary EAQECCs with different kinds of length sets via different $\ell$, where $2\ell\mid e$.