Hermitian self-orthogonal infinitesimal evaluation codes over $ \mathbb{F}_{q^2}+u\mathbb{F}_{q^2} $ and applications to quantum codes

In this paper, we introduced a new class of infinitesimal evaluation codes over the dual-number extension $ R = \mathbb{F}_{q^2}+u \mathbb{F}_{q^2}, u^2 = 0, $ obtained by evaluating polynomials at perturbed points $ a_i+ub_i $. This evaluation produces a coupled value–derivative structure through the identity $ f(a_i+ub_i) = f_0(a_i)+u\bigl(b_i f_0'(a_i)+f_1(a_i)\bigr), $ which enriches classical evaluation codes with first-order infinitesimal corrections. We established the Hermitian duality theory for these codes and showed that Hermitian orthogonality over $ R $ decomposes into a residue-layer condition over $ \mathbb{F}_{q^2} $ together with a correction equation involving the infinitesimal parameters. This yields explicit criteria for Hermitian self-orthogonality. Using these criteria, we constructed several families of Hermitian self-orthogonal infinitesimal evaluation codes, including multiplier perturbation, locator perturbation, and subgroup–coset constructions. Via the Gray map and the Hermitian construction, these codes produce new families of $ q $-ary quantum stabilizer codes.