We establish a density variant of the Frankl–Rödl theorem on the sphere 𝕊^{n-1}, which concerns avoiding pairs of vectors with a specific distance, or equivalently, a prescribed inner product. In particular, we establish lower bounds on the probability that a randomly chosen pair of such vectors lies entirely within a measurable subset A ⊆ 𝕊^{n-1} of sufficiently large measure. Additionally, we prove a density version of spherical avoidance problems, which generalize from pairwise avoidance to broader configurations with prescribed pairwise inner products. Our framework encompasses a class of configurations we call inductive configurations, which include simplices with any prescribed inner product -1 < r < 1. As a consequence of our density statement, we show that all inductive configurations are sphere Ramsey.

We present an explicit construction of a sequence of rate$1/2$Wozencraft ensemble codes (over any fixed prime field$\mathbb {F}_{q}$) that achieve minimum distance$\Omega (\sqrt {k})$where k is the message length. The coefficients of the Wozencraft ensemble codes are constructed using Sidon Sets and the cyclic structure of$\mathbb {F}_{q^{k}}$where$k+1$is prime with q a primitive root modulo$k+1$. Assuming Artin’s conjecture, there are infinitely many such k for any prime q.

We present an explicit construction of a sequence of rate $1/2$ Wozencraft ensemble codes (over any fixed finite field $\mathbb{F}_q$) that achieve minimum distance $Ω(\sqrt{k})$ where $k$ is the message length. The coefficients of the Wozencraft ensemble codes are constructed using Sidon Sets and the cyclic structure of $\mathbb{F}_{q^{k}}$ where $k+1$ is prime with $q$ a primitive root modulo $k+1$. Assuming Artin's conjecture, there are infinitely many such $k$ for any prime power $q$.