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 $a_i+u{b}_i$. This evaluation produces a coupled value–derivative structure through the identity $f(a_i+u{b}_i)=f_0(a_i)+u(b_i{f}_0^{\prime }(a_i)+f_1(a_i)),$ 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.