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Storage codes on coset graphs with asymptotically unit rate

Alexander Barg, Moshe Schwartz, Lev Yohananov

TL;DR

This work presents a permutation-matrix framework to build large storage codes on triangle-free coset graphs derived from binary linear codes, achieving rates arbitrarily close to 1 over $\mathbb F_2$. By expressing the adjacency structure as sums of permutation matrices and exploiting block decompositions, the authors control the rank of the augmented adjacency matrix $\tilde{A}(G)$, driving $\text{rk}(\tilde{A}(G))/N$ to 0 in an explicit infinite family. They introduce a new family of codes based on Hamming parity-check matrices, with a hierarchical construction that yields asymptotic rates $R\ge 1-2^{-s}-2^{-r+1}$ for parameters $(s,r)$, and in the limit as $s,r\to\infty$ obtain storage codes on triangle-free graphs with rate approaching 1. The approach recovers and extends prior results (e.g., BargZemor2022) while clarifying the rank-control mechanism via permutation operations, with implications for hat-guessing games and linear index coding. Overall, the paper demonstrates a concrete path to near-optimal storage codes on sparse, triangle-free graphs and provides explicit constructions with provable rate guarantees.

Abstract

A storage code on a graph $G$ is a set of assignments of symbols to the vertices such that every vertex can recover its value by looking at its neighbors. We consider the question of constructing large-size storage codes on triangle-free graphs constructed as coset graphs of binary linear codes. Previously it was shown that there are infinite families of binary storage codes on coset graphs with rate converging to 3/4. Here we show that codes on such graphs can attain rate asymptotically approaching 1. Equivalently, this question can be phrased as a version of hat-guessing games on graphs (e.g., P.J. Cameron e.a., \emph{Electronic J. Comb.} 2016). In this language, we construct triangle-free graphs with success probability of the players approaching one as the number of vertices tends to infinity. Furthermore, finding linear index codes of rate approaching zero is also an equivalent problem. Another family of storage codes on triangle-free graphs of rate approaching 1 was constructed earlier by A. Golovnev and I. Haviv (36th Computational Complexity Conf., 2021) relying on a different family of graphs.

Storage codes on coset graphs with asymptotically unit rate

TL;DR

This work presents a permutation-matrix framework to build large storage codes on triangle-free coset graphs derived from binary linear codes, achieving rates arbitrarily close to 1 over . By expressing the adjacency structure as sums of permutation matrices and exploiting block decompositions, the authors control the rank of the augmented adjacency matrix , driving to 0 in an explicit infinite family. They introduce a new family of codes based on Hamming parity-check matrices, with a hierarchical construction that yields asymptotic rates for parameters , and in the limit as obtain storage codes on triangle-free graphs with rate approaching 1. The approach recovers and extends prior results (e.g., BargZemor2022) while clarifying the rank-control mechanism via permutation operations, with implications for hat-guessing games and linear index coding. Overall, the paper demonstrates a concrete path to near-optimal storage codes on sparse, triangle-free graphs and provides explicit constructions with provable rate guarantees.

Abstract

A storage code on a graph is a set of assignments of symbols to the vertices such that every vertex can recover its value by looking at its neighbors. We consider the question of constructing large-size storage codes on triangle-free graphs constructed as coset graphs of binary linear codes. Previously it was shown that there are infinite families of binary storage codes on coset graphs with rate converging to 3/4. Here we show that codes on such graphs can attain rate asymptotically approaching 1. Equivalently, this question can be phrased as a version of hat-guessing games on graphs (e.g., P.J. Cameron e.a., \emph{Electronic J. Comb.} 2016). In this language, we construct triangle-free graphs with success probability of the players approaching one as the number of vertices tends to infinity. Furthermore, finding linear index codes of rate approaching zero is also an equivalent problem. Another family of storage codes on triangle-free graphs of rate approaching 1 was constructed earlier by A. Golovnev and I. Haviv (36th Computational Complexity Conf., 2021) relying on a different family of graphs.
Paper Structure (8 sections, 20 theorems, 39 equations, 1 figure)

This paper contains 8 sections, 20 theorems, 39 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The construction of BargZemor2022 and our approach
  3. Properties of permutation matrices
  4. Partitioning the adjacency matrix
  5. Zero-codeword-only codes and repetition codes
  6. A new code family
  7. The case of ${s}=2$
  8. The case of ${s}=3$

Key Result

Theorem 1

(GolovnevHaviv2021) Let $r$ be a multiple of 6. Then the graph $K^<(r,\frac{r}{2},\frac{r}{6})$ contains no triangles, and its minimum rank over ${\mathbb F}_2$ is at most $2^{r h(1/3)}<2^{0.92r},$ where $h(x)=-x \log_2x -(1-x)\log_2(1-x)$ is the entropy function. Consequently, there exists a sequen

Figures (1)

  • Figure :

Theorems & Definitions (34)

  • Theorem 1
  • Theorem 2
  • Theorem 3: Gadouleau2011CM2011Cameron2016
  • Lemma 4
  • proof
  • Proposition 5
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • ...and 24 more