Near orders and codes

Core claim

The paper's central assertion is Theorem 5.7: for two well-agreeing near weights $\rho,\sigma$ on an $F$-algebra $R$, the code $C^m_\ell=(E^m_\ell)^\perp$, where $E^m_\ell$ is the image of $R^m_\ell=\{h:\rho(h)\le\ell,\ \sigma(h)\le m\}$ under an algebra morphism, satisfies $d(C^m_\ell)\ge d_{\mathrm{NORD}}(\ell,m)$. Here $d_{\mathrm{NORD}}(\ell,m)=\min\{\#N^m_r:r\ge\ell\}$ and $N^m_r$ collects pairs $(i,j)$ with $i+j=r+1$ and $\sigma(f_i)+\Sigma(j)\le m$. The proof runs through the syndrome matrix $S(y)=(h_i*h_j)\cdot y$: a word's weight is at least the rank of its syndrome matrix (Prop. 5.3), and for $y\in C^m_\ell\setminus C^m_{\ell+1}$ this rank is at least $\#N^m_\ell$ via a triangular minor (Cor. 5.5). For curve-induced near weights, $C^m_\ell$ is exactly the two-point Goppa code $C_\Omega(X,D,\ell Q_1+mQ_2)$, so the bound applies to those codes; Corollary 5.10 and Remark 5.11 show $d_{\mathrm{NORD}}$ is determined by the semigroup $H(\rho,\sigma)$ alone and computable in finite time. Example 5.14 exhibits a hyperelliptic case where $d_{\mathrm{NORD}}$ exceeds the Goppa bound $d_G(\ell,m)=\ell+m-2\gamma+2$.