Expander (Tanner) codes combine sparse graphs with local constraints, enabling linear-time decoding and asymptotically good distance--rate tradeoffs. A standard constraint-counting argument yields the global-rate lower bound $R\ge 2r-1$ for a Tanner code with local rate $r$, which gives no positive-rate guarantee in the low-rate regime $r\le 1/2$. This regime is nonetheless important in applications that require algebraic local constraints (e.g., Reed--Solomon locality and the Schur-product/multiplication property). We introduce \emph{Algebraic Expander Codes}, an explicit algebraic family of Tanner-type codes whose local constraints are Reed--Solomon and whose global rate remains bounded away from $0$ for every fixed $r\in(0,1)$ (in particular, for $r\le 1/2$), while achieving constant relative distance. Our codes are defined by evaluating a structured subspace of polynomials on an orbit of a non-commutative subgroup of $\mathrm{AGL}(1,\mathbb{F})$ generated by translations and scalings. The resulting sparse coset geometry forms a strong spectral expander, proved via additive character-sum estimates, while the rate analysis uses a new notion of polynomial degree and a polytope-volume/dimension-counting argument.

翻译：扩展（Tanner）码将稀疏图与局部约束相结合，支持线性时间译码和渐近良好的距离-速率权衡。对于局部率为$r$的Tanner码，标准约束计数论证给出全局率下界$R\ge 2r-1$，该下界在低率区域$r\le 1/2$时无法保证正率。然而，该区域在需要代数局部约束（例如Reed–Solomon局部性及Schur积/乘法性质）的应用中至关重要。我们提出\emph{代数扩展码}——一类显式构造的代数Tanner型码，其局部约束采用Reed–Solomon码，且对于任意固定的$r\in(0,1)$（特别是$r\le 1/2$），全局率始终远离$0$，同时实现常数相对距离。该码通过在一个由平移和缩放生成的$\mathrm{AGL}(1,\mathbb{F})$非交换子群轨道上评估结构化多项式子空间来定义。由此产生的稀疏陪集几何构成强谱扩展结构（通过加法特征和估计证明），而率分析采用新的多项式次数概念及多面体体积/维度计数论证。