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Maximally Extendable Product Codes are Good Coboundary Expanders

Gleb Kalachev, Pavel Panteleev

TL;DR

The paper addresses whether product expansion extends to tensor products of an arbitrary number of codes, a property crucial for high-dimensional expander–based constructions in quantum and classical coding. It shows that random collections of $D$ linear codes over a sufficiently large field are $\rho$-product-expanding with high probability, and it develops a framework to transfer this expansion to dual codes via maximal extendability. The approach combines locally testable codes, extendable/inner-generated set theory, and a Schwartz–Zippel argument, plus the novel concept of maximally extendable product codes and a universal inheritance property. These results advance the construction of good quantum LTCs and LDPC codes in higher dimensions, potentially enabling four-code based quantum locally testable codes.

Abstract

We investigate the coboundary expansion property of tensor product codes, known as product expansion, which plays an important role in recent constructions of good quantum LDPC codes and classical locally testable codes. Prior research has shown that this property is equivalent to agreement testability and robust testability for products of two codes with linear distance. However, for products of more than two codes, product expansion is a strictly stronger property. In this paper, we prove that a collection of an arbitrary number of random codes over a sufficiently large field has good product expansion. We believe that, in the case of four codes, the same ideas can be used to construct good quantum locally testable codes, in a way similar to the current constructions that use only products of two codes.

Maximally Extendable Product Codes are Good Coboundary Expanders

TL;DR

The paper addresses whether product expansion extends to tensor products of an arbitrary number of codes, a property crucial for high-dimensional expander–based constructions in quantum and classical coding. It shows that random collections of linear codes over a sufficiently large field are -product-expanding with high probability, and it develops a framework to transfer this expansion to dual codes via maximal extendability. The approach combines locally testable codes, extendable/inner-generated set theory, and a Schwartz–Zippel argument, plus the novel concept of maximally extendable product codes and a universal inheritance property. These results advance the construction of good quantum LTCs and LDPC codes in higher dimensions, potentially enabling four-code based quantum locally testable codes.

Abstract

We investigate the coboundary expansion property of tensor product codes, known as product expansion, which plays an important role in recent constructions of good quantum LDPC codes and classical locally testable codes. Prior research has shown that this property is equivalent to agreement testability and robust testability for products of two codes with linear distance. However, for products of more than two codes, product expansion is a strictly stronger property. In this paper, we prove that a collection of an arbitrary number of random codes over a sufficiently large field has good product expansion. We believe that, in the case of four codes, the same ideas can be used to construct good quantum locally testable codes, in a way similar to the current constructions that use only products of two codes.
Paper Structure (10 sections, 22 theorems, 80 equations, 3 figures)

This paper contains 10 sections, 22 theorems, 80 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Definitions and Main Results
  3. Proof of the Main Result
  4. Locally Testable Codes
  5. Extendable and Inner-generated Sets
  6. Proof Outline of Theorem \ref{['th:rand-expanding']}
  7. Operations Preserving Product Expansion
  8. Products of LTCs are Expanding
  9. Maximally Extendable Product Codes
  10. LTCs of Arbitrary Length

Key Result

Theorem 2

For every tuple $(R_1,\dots,R_D)\in (0,1)^D$ there exists $\rho>0$ such that for each $n\in \mathbb{N}$ a tuple of codes $(\mathcal{C}_1,\dots,\mathcal{C}_D) \in {\sf Gr}_{2^t}(n,k_1)\times \cdots\times{\sf Gr}_{2^t}(n,k_D)$ picked uniformly at random, where $k_i \leqslant n R_i$, $i\in [D]$, is $\r

Figures (3)

  • Figure 1: (a) Example of a set $M\in [3]^2$ that is not inner-generated for the code $\mathcal{C}_1\boxplus \mathcal{C}_2$ where $\mathcal{C}_1=\mathcal{C}_2$ are the repetition $[3,1,3]$ codes. The points $(1,2)$ and $(2,1)$ shown as black dots do not belong to the lines contained in $M$ shown as the black lines. Therefore the codeword $c\in \mathcal{C}_1\boxplus \mathcal{C}_2$ shown in Subfigure (b) with non-zero elements on $M$ cannot be represented as a sum of codewords along the lines contained in $M$. Subfigure (c) illustrates the idea of Proposition \ref{['pr:1']}: the word $w$ on $M$ with one non-zero bit (the blue cell) cannot be extended to a codeword from the dual code $\mathcal{C}_1^\perp\otimes\mathcal{C}_2^\perp$ since it does not satisfy the check $c$ though satisfies all the checks along the lines in $M$, thus $M$ is not extendable in $\mathcal{C}_1^\perp\otimes\mathcal{C}_2^\perp$.
  • Figure 2: Steps for constructing $\varepsilon$-closure of a set $M\subseteq [n]^2$ shown for $\varepsilon=1/2$ and $n=6$. At each step we add to the set all lines intersecting with it in more than $\varepsilon n$ points.
  • Figure 3: $S$ is extendable in $\mathcal{C}=\ker H(\mathbf{a})$if‌ f$\ker H^S(\mathbf{a}) = \mathcal{C}|_S$if‌ f any information set $J$ of $\ker H^S(\mathbf{a})$ is contained in some information set $I$ of $\ker H(\mathbf{a})$. In this case, the gray submatrices of $H(\mathbf{a})$ and $H^S(\mathbf{a})$ both must be full-rank.

Theorems & Definitions (47)

  • Definition 1: product expansion
  • Theorem 2
  • Corollary 3
  • Theorem 4
  • Definition 5: inner-generated sets
  • Definition 6: extendable sets
  • Proposition 7
  • Definition 8: maximally extendable product code
  • Remark 9
  • Lemma 1: Lemma 6 in kalachevTwosidedRobustlyTestable2023
  • ...and 37 more