Algebraic Expander Codes

2026-03-25

AI Summary

This work proposes a new class of algebraic expander codes that overcome a fundamental limitation of classical Tanner codes: the inability to guarantee a positive global rate when the local code rate \( r \leq 1/2 \), which restricts their applicability in settings requiring algebraic local constraints such as Reed–Solomon codes. By evaluating structured polynomial subspaces over orbits of non-commutative affine (translation-scaling) subgroups, the construction seamlessly integrates sparse graphical structures with algebraic local codes. For the first time, it achieves constant relative distance and a global rate bounded away from zero for any fixed local rate \( r \in (0,1) \), thereby surpassing traditional constraint-counting lower bounds. Key technical ingredients include a novel notion of polynomial degree, dimension counting via polytope volumes, and an analytical framework combining spectral graph expansion, additive characters, and algebraic geometry.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Local Search
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

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.

Problem

• Expander codes
• Tanner codes
• Low-rate regime
• Algebraic constraints
• Global rate

Innovation

• Algebraic Expander Codes
• Tanner codes
• Reed–Solomon locality
• spectral expander
• polynomial degree