Hermitian Dickson Dualities for Codes over Near-Fields

A Dickson near-field is obtained from ${\mathbb{F}}_{p^2}$ by twisting multiplication so that distributivity holds only on the right. In this work, we develop a basic theory of right-linear codes of length n over $\mathrm{NF}\left(p^2\right)$. We show that every right-linear code is right-monomially equivalent to a code with a systematic generator matrix, obtained via one-sided row operations. Using Galois conjugation, we introduce the Hermitian Dickson inner product and define the associated dual code, giving an explicit parity-check description in the ${\mathbb{F}}_p$-systematic case. We also provide effective criteria for Hermitian Dickson LCD, self-orthogonal, and self-dual codes, and we classify Hermitian Dickson self-orthogonal codes in short lengths.