Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Algebraic Expander Codes

Swastik Kopparty, Itzhak Tamo

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.

Algebraic Expander Codes

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 for a Tanner code with local rate , which gives no positive-rate guarantee in the low-rate regime . 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 for every fixed (in particular, for ), 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 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.

Paper Structure

This paper contains 51 sections, 19 theorems, 100 equations, 1 table.

Table of Contents

  1. Introduction
  2. Main theorem (informal).
  3. What is new.
  4. Connection to Locally Recoverable Codes and Graph Density
  5. Overview of Results
  6. Main Results.
  7. Limitations.
  8. Techniques
  9. Establishing the Rate via $u$-Base Degree.
  10. Spectral Expansion via Character Sums.
  11. Graph Sparsity via Mixed-Type Subgroups.
  12. Two explicit instantiations.
  13. Related Work
  14. Expander/Tanner codes and expander-based coding.
  15. Algebraic locality via group actions.
  16. ...and 36 more sections

Key Result

Lemma 2.2

Let $u(X)\in\mathbb{F}[X]$ be a nonconstant polynomial. Then for all $f,g\in\mathbb{F}[X]$,

Theorems & Definitions (45)

  • Remark 1.1
  • Definition 2.1: Base-$u$ Degree
  • Lemma 2.2: Subadditivity of the $u$-base degree
  • proof
  • Remark 2.3
  • Proposition 2.4
  • Remark 2.5
  • Remark 2.6
  • Theorem 3.1
  • proof
  • ...and 35 more