Uniform Three-Fold Product Expansion of Algebraic-Geometry Codes over a Fixed Field

A collection of classical codes C₁,,Cₘ has product expansion if every low-weight word in the sum of their extensions to the product space can be written as a sum of words supported on few lines, each line being a codeword of the corresponding Cᵢ. For $m=2$ product expansion is equivalent to robust testability of the tensor code, while for m≥ 3 it is strictly stronger; it is the hypothesis on the local codes behind recent constructions of good quantum LDPC codes, good quantum locally testable codes, and quantum codes with transversal non-Clifford gates. No explicit family over a fixed field was previously known to achieve constant rate, linear distance for both the code and its dual, and 3-fold product expansion at growing length. We prove that for the one-point evaluation codes Cᵢ of length Nᵢ=728·27\,i-1 and pole order Nᵢ/5 on the Garcia--Stichtenoth tower over F₇₂₉, the triple (Cᵢ,Cᵢ,Cᵢ) has product expansion with a constant independent of i. The dual codes Cᵢ^⊥ then form an explicit fixed-field family of rate tending to $109/130$ and relative distance at least $8/65$. The proof is geometric: we view the words of the sum code as arrays on the product of three copies of the i-th curve of the tower, where the parity checks are sections of a line bundle. An exact generation theorem describes the sum-code words supported on the grid footprint of a suitable divisor. A support-localisation argument then shows that every low-weight array lies on such a divisor, after deleting a small number of coordinates. We also give a sufficient criterion for three-fold product expansion in terms of degree and genus ratios, yielding explicit families over every non-prime finite field of order at least $512$. The principal conclusions have been formally verified in Lean 4.