Lifting iso-dual algebraic geometry codes
For coding theorists, this provides a general construction technique for iso-dual AG codes over larger fields, enabling longer codes with self-duality properties.
This work presents a method to lift iso-dual algebraic geometry codes from a base function field to a finite separable extension, producing new iso-dual codes. The method is applied to construct long binary and ternary iso-dual codes from cyclotomic extensions and to lift codes over rational function fields to maximal function fields like Hermitian and Suzuki.
In this work we investigate the problem of producing iso-dual algebraic geometry (AG) codes over a finite field $\mathbb{F}_q$ with $q$ elements. Given a finite separable extension $\mathcal{M}/\mathcal{F}$ of function fields and an iso-dual AG-code $\mathcal{C}$ defined over $\mathcal{F}$, we provide a general method to lift the code $\mathcal{C}$ to another iso-dual AG-code $\tilde{\mathcal{C}}$ defined over $\mathcal{M}$ under some assumptions on the divisors $D$ and $G$ and on the parity of the involved different exponents. We apply this method to lift iso-dual AG-codes over the rational function field to elementary abelian $p$-extensions, like the maximal function fields defined by the Hermitian, Suzuki, and one covered by the $GGS$ function field. We also obtain long binary and ternary iso-dual AG-codes defined over cyclotomic extensions.