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New Explicit Constant-Degree Lossless Expanders

Louis Golowich

TL;DR

This work addresses the explicit construction of onesided lossless bipartite expanders with constant degree and a fixed ratio between the two vertex sets. The authors build a large unbalanced spectral expander and locally impose a fixed-size lossless gadget within each neighborhood, enabling a simpler analysis via counting arguments and the Expander Mixing Lemma. They prove that the resulting graphs are $(\mu,\epsilon)$-lossless with parameter $\mu= k^2\lambda_2^2$ and degree $D=O\big( (\log(1/\epsilon)+\log(1/\beta^{(2)}))/\epsilon^2 \big)$, offering an explicit alternative to the Capalbo–Reingold–Vadhan–Wigderson construction with improved degree at comparable regimes. The construction yields implications for locally testable codes and potentially quantum LDPC codes, and remains robust under near-Ramanujan conditions for the starting expander. Overall, the paper delivers a simpler, explicit route to lossless expanders with practical parameter ranges and broad applicability in coding theory and complexity.

Abstract

We present a new explicit construction of onesided bipartite lossless expanders of constant degree, with arbitrary constant ratio between the sizes of the two vertex sets. Our construction is simpler to state and analyze than the only prior construction of Capalbo, Reingold, Vadhan, and Wigderson (2002), and achieves improvements in some parameters. We construct our lossless expanders by imposing the structure of a constant-sized lossless expander "gadget" within the neighborhoods of a large bipartite spectral expander; similar constructions were previously used to obtain the weaker notion of unique-neighbor expansion. Our analysis simply consists of elementary counting arguments and an application of the expander mixing lemma.

New Explicit Constant-Degree Lossless Expanders

TL;DR

This work addresses the explicit construction of onesided lossless bipartite expanders with constant degree and a fixed ratio between the two vertex sets. The authors build a large unbalanced spectral expander and locally impose a fixed-size lossless gadget within each neighborhood, enabling a simpler analysis via counting arguments and the Expander Mixing Lemma. They prove that the resulting graphs are -lossless with parameter and degree , offering an explicit alternative to the Capalbo–Reingold–Vadhan–Wigderson construction with improved degree at comparable regimes. The construction yields implications for locally testable codes and potentially quantum LDPC codes, and remains robust under near-Ramanujan conditions for the starting expander. Overall, the paper delivers a simpler, explicit route to lossless expanders with practical parameter ranges and broad applicability in coding theory and complexity.

Abstract

We present a new explicit construction of onesided bipartite lossless expanders of constant degree, with arbitrary constant ratio between the sizes of the two vertex sets. Our construction is simpler to state and analyze than the only prior construction of Capalbo, Reingold, Vadhan, and Wigderson (2002), and achieves improvements in some parameters. We construct our lossless expanders by imposing the structure of a constant-sized lossless expander "gadget" within the neighborhoods of a large bipartite spectral expander; similar constructions were previously used to obtain the weaker notion of unique-neighbor expansion. Our analysis simply consists of elementary counting arguments and an application of the expander mixing lemma.
Paper Structure (7 sections, 5 theorems, 15 equations, 1 figure)

This paper contains 7 sections, 5 theorems, 15 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Construction
  4. General framework
  5. Choosing the parameters
  6. Proof of lossless expansion
  7. Acknowledgments

Key Result

Theorem 2

For every open interval $\beta=(\beta^{(1)},\beta^{(2)})\subseteq\mathbb{R}_+$ and every $\epsilon>0$, there exists a sufficiently large $D=D(\beta,\epsilon)\in\mathbb{N}$ and a sufficiently small $\mu=\mu(\beta,\epsilon)>0$ such that there is an infinite explicit family of $(\mu,\epsilon)$-lossless

Figures (1)

  • Figure 1: An illustration of how we construct our lossless expanders $G=G(X,G_0)$ as the union over $v\in R(X)$ of the gadgets $G_0^v\cong G_0$. A single such gadget is shown above, connecting the left vertices $L(G_0^v)=N_X(v)\cong L(G_0)$ to the right vertices $R(G_0^v)=\{v\}\times R(G_0)\cong R(G_0)$.

Theorems & Definitions (24)

  • Definition 1
  • Theorem 2
  • Remark 3
  • Definition 4
  • Definition 5
  • Lemma 6: Expander Mixing Lemma; see for instance Lemma 4.15 of vadhan_pseudorandomness_2012
  • Definition 7
  • Proposition 8
  • proof
  • Remark 9
  • ...and 14 more