Algebraic Expander Codes

TL;DR

This paper introduces Algebraic Expander Codes, a new family of Tanner codes with algebraic local constraints that maintain a positive rate across all regimes and achieve constant relative distance, using structured polynomial evaluations.

Contribution

The paper presents explicit algebraic Tanner codes with Reed--Solomon constraints that have bounded away from zero global rate for all local rates, including low-rate regimes, and are supported by spectral expansion analysis.

Findings

Codes have constant relative distance.

Global rate remains positive for all fixed local rates.

Codes are constructed via polynomial evaluations on structured orbits.

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.