Research proof index

Given a linear code of length ι and dimension dim over a field F, there exists a dim × ι matrix over F which generates the code. Theorem 2.2.7 [GRS25].

Let T be a finite subset of ι. If every word in a collection lies in the projected code C|[T], then so do all F-linear combinations of these.

Singleton bound for linear codes

A nonzero linear combination of linearly independent vectors is nonzero.

Let $1 < q \le 2$ and let $q'$ be its Hölder conjugate. If every phase-linearized nontangential operator has the required uniform $L^2$ bound, then for measurable $F,G$ and measurable $f$ with $\lVert f(x)\rVert \le \mathbf{1}_F(x)$, the linearized Carleson operator satisfies

$\int_G^+ \mathcal{C}^{\mathrm{lin}}_{Q,K}f(x)\,dx \le C(a,q)\,\mu(G)^{1/q'}\mu(F)^{1/q}.$

The Semiring morphism that maps m-variate polynomials onto univariate polynomials by evaluating them at (X^(2⁰), ... , X^(2ᵐ⁻¹)), i.e. sending aₑ X₀^σ(0) ⬝ ⋯ ⬝ Xₘ₋₁^σ(m-1) → aₑ (X^(2⁰))^σ(0) ⬝ ⋯ ⬝ (X^(2ᵐ⁻¹))^σ(m-1) for all σ : Fin m → ℕ -/ def powAlgHom : MvPolynomial (Fin m) F →ₐ[F] Polynomial F := aeval fun j => Polynomial.X ^ (2 ^ (j : ℕ)) lemma...