Coding Theory 05: Dual AG Codes and Residues

Coding theory / 05

Dual AG codes and residues

For AG evaluation codes, the orthogonal code is the corresponding residue code built from rational differentials.

Let \(\Omega(X)\) be the space of rational differentials on a smooth projective curve \(X\). For a divisor \(E\), set

\[\Omega(E)=\{\eta\in\Omega(X):\eta\ne0,\;(\eta)-E\ge0\}\cup\{0\}.\]

With \(D=P_1+\cdots+P_n\) and \(G\) disjoint from \(D\), the differential AG code is

\[C_\Omega(D,G)=\{(\operatorname{res}_{P_1}\eta,\ldots,\operatorname{res}_{P_n}\eta):\eta\in\Omega(G-D)\}.\]

Choose a canonical divisor \(K=(\omega)\). Multiplication by \(\omega\) identifies \(L(K+D-G)\) with \(\Omega(G-D)\). Thus the same code can be expressed through functions:

\[f\mapsto (\operatorname{res}_{P_i}(f\omega))_{i=1}^n.\]

Parameters

The parameter formula runs parallel to the evaluation-code formula. One has

\[k_\Omega=\ell(K+D-G)-\ell(K-G).\]

Under the standard degree hypotheses \(2g-2<\deg G<n\), this becomes

\[k_\Omega=n-(\deg G+1-g).\]

The designed distance satisfies

\[d_\Omega\ge \deg G-(2g-2).\]

Example 1: residues on \(\mathbb P^1\) recover parity checks

Let \(P_i\) be finite points on \(\mathbb P^1\). A differential with simple poles at the \(P_i\) and controlled pole behavior at infinity gives residue coordinates. The global residue theorem forces the sum of all residues, including infinity, to vanish, producing parity relations among coordinates.

Example 2: Goppa congruences as residue conditions

For support \(L_i\in\mathbb F_{q^m}\) and a Goppa polynomial \(g(x)\), the expression

\[\sum_i {c_i\over x-L_i}\equiv0\pmod{g(x)}\]

says that a rational differential has residues \(c_i\) at the support points and vanishing prescribed polar part modulo \(g\). Classical Goppa parity checks are genus-zero residue checks.

Duality

The crucial theorem is

\[C_\Omega(D,G)=C_L(D,G)^\perp.\]

The proof is a residue computation. For \(f\in L(G)\) and \(\eta\in\Omega(G-D)\), the differential \(f\eta\) has possible simple poles at the points \(P_i\) and no other residue contribution. The global residue theorem gives

\[\sum_i f(P_i)\operatorname{res}_{P_i}(\eta)=0.\]

Dimension equality then identifies the full orthogonal code.

Example 3: dual Reed-Solomon through differentials

The dual of a Reed-Solomon code is again a generalized Reed-Solomon code. In the AG language, the duality is explained by the canonical divisor of \(\mathbb P^1\) and residues of rational differentials with simple poles at the evaluation points.

Example 4: elliptic residue codes

On an elliptic curve, \(\deg K=0\). If \(G=mO\), \(D\) contains \(n\) points away from \(O\), and \(0<m<n\), then the residue code has designed distance at least \(m\) and dimension \(n-m\). The same genus correction that affected evaluation code dimension now appears in the dual direction.