Extension-degree dependence of binary triorthogonal code trade-offs

Determine whether increasing the extension degree in the algebraic-geometry-code and alphabet-reduction constructions yields better rate–distance trade-offs, taking into account the resulting dependence of the parameters on the alphabet-reduction length and compatible-set cardinality.

Background

The paper constructs asymptotically good binary triorthogonal codes by starting with algebraic geometry codes over binary extension fields and applying additive alphabet-reduction maps. For a fixed extension field, the resulting rate–distance trade-off depends on the extension-field parameter as well as on the alphabet-reduction length s, the minimum inner-map weight d_in, and the compatible-set size t.

The paper compares explicit constructions over $\mathbb{F}_{64}$ and $\mathbb{F}_{256}$, finding that the $\mathbb{F}_{256}$ construction has better relative parameters over the range considered. It does not resolve whether using still larger extension degrees can improve the trade-offs, because increasing the degree may also worsen the auxiliary parameters s and t.