@article{Saif2026, 
author = {Sami H. Saif},
title = {Hermitian self-orthogonal infinitesimal evaluation codes over              F                      q        2              +  u            F                      q        2             and applications to quantum codes},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {6},
pages = {16952-16982},
keywords = {evaluation codes, Gray map, quantum codes, linear codes},
url = {https://www.sciopen.com/article/10.3934/math.2026694},
doi = {10.3934/math.2026694},
abstract = {In this paper, we introduced a new class of infinitesimal evaluation codes over the dual-number extension    R  =            F                      q        2              +  u            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    ′    (      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              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.}
}