Uniform Three-Fold Product Expansion of Algebraic-Geometry Codes over a Fixed Field
Abstract
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.
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BibTeX
@misc{ding2026uniform,
title = {{Uniform Three-Fold Product Expansion of Algebraic-Geometry Codes over a Fixed Field}},
author = {Cassie Ding and Min-Hsiu Hsieh},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08172},
eprint = {2611.08172},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08172},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Uniform Three-Fold Product Expansion of Algebraic-Geometry Codes over a Fixed Field
AU - Cassie Ding
AU - Min-Hsiu Hsieh
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08172
UR - https://arxiv.org/abs/2611.08172
AB - A collection of classical codes C_1,…,C_m 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 C_i. 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 C_i of length N_i=728·27^ i−1 and pole order ⌊ N_i/5⌋ on the Garcia--Stichtenoth tower over 𝔽_729, the triple (C_i,C_i,C_i) has product expansion with a constant independent of i. The dual codes C_i^⊥ 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.
ER -